On the Random Wave Conjecture for Dihedral Maaß Forms

We prove two results on arithmetic quantum chaos for dihedral Maaß forms, both of which are manifestations of Berry’s random wave conjecture: Planck scale mass equidistribution and an asymptotic formula for the fourth moment. For level 1 forms, these results were previously known for Eisenstein series and conditionally on the generalised Lindelöf hypothesis for Hecke–Maaß eigenforms. A key aspect of the proofs is bounds for certain mixed moments of L-functions that imply hybrid subconvexity.


Introduction
The random wave conjecture of Berry [Ber77] is the heuristic that the eigenfunctions of a classically ergodic system ought to evince Gaussian random behaviour, as though they were random waves, in the large eigenvalue limit. In this article, we study and resolve two manifestations of this conjecture for a particular subsequence of Laplacian eigenfunctions, dihedral Maaß forms, on the surface Γ 0 (q)\H.

1.1
The rate of equidistribution for quantum unique ergodicity. Given a positive integer q and a Dirichlet character χ modulo q, denote by L 2 (Γ 0 (q)\H, χ) the space of measurable functions f : H → C satisfying and f, f q < ∞, where ·, · q denotes the inner product f, g q := with dμ(z) = y −2 dx dy on any fundamental domain of Γ 0 (q)\H.
The first author is supported by the European Research Council Grant Agreement 670239. The second author is supported by the Simons Foundation, award number 630985.
Mathematics Subject Classification: 11F12 (primary); 58J51, 81Q50 (secondary) Quantum unique ergodicity in configuration space for L 2 (Γ 0 (q)\H, χ) is the statement that for any subsequence of Laplacian eigenfunctions g ∈ L 2 (Γ 0 (q)\H, χ) normalised such that g, g q = 1 with eigenvalue λ g = 1/4 + t 2 g tending to infinity, for every f ∈ C b (Γ 0 (q)\H), or equivalently for every indicator function f = 1 B of a continuity set B ⊂ Γ 0 (q)\H. This is known to be true (and in a stronger form, in the sense of quantum unique ergodicity on phase space), provided each eigenfunction g is a Hecke-Maaß eigenform, via the work of Lindenstrauss [Lin06] and Soundararajan [Sou10]. One may ask whether the rate of equidistribution for quantum unique ergodicity can be quantified in some way; Lindenstrauss' proof is via ergodic methods and does not address this aspect. One method of quantification is to give explicit rates of decay as λ g tends to infinity for the terms for a fixed Hecke-Maaß eigenform f or incomplete Eisenstein series E a (z, ψ); optimal decay rates for these integrals, namely O q,f,ε (t Another quantification of the rate of equidistribution, closely related to the spherical cap discrepancy discussed in [LS95], is small scale mass equidistribution. Let B R (w) denote the hyperbolic ball of radius R centred at w ∈ Γ 0 (q)\H with volume 4π sinh 2 (R/2). Two small scale refinements of quantum unique ergodicity were studied in [You16] and [Hum18] respectively, namely the investigation of the rates of decay in R, with regards to the growth of the spectral parameter t g ∈ [0, ∞)∪i(0, 1/2), for which either the asymptotic formula 1 vol(B R ) BR(w) |g(z)| 2 dμ(z) = 1 vol(Γ 0 (q)\H) + o q,w (1) (1.2) or the bound vol w ∈ Γ 0 (q)\H : holds as t g tends to infinity along any subsequence of g ∈ B * 0 (q, χ), the set of L 2normalised newforms g of weight zero, level q, nebentypus χ, and Laplacian eigenvalue λ g = 1/4 + t 2 g . X g;R (w) := 1 vol(B R ) BR(w) |g(z)| 2 dμ(z), which has expectation 1/ vol(Γ 0 (q)\H). The asymptotic formula (1.2) is equivalent to the pointwise convergence of X g;R to 1, while (1.3) is simply the convergence in probability of X g;R to 1, a consequence of the bound Var(X g;R ) = o(1). One could ask for further refinements of these problems, such as asymptotic formulae for this variance and a central limit theorem, as studied in [WY19] for toral Laplace eigenfunctions, though we do not pursue these problems.
For q = 1, Young [You16, Proposition 1.5] has shown that (1.2) holds when R t −δ g with 0 < δ < 1/3 under the assumption of the generalised Lindelöf hypothesis, and that an analogous result with 0 < δ < 1/9 is true unconditionally for the Eisenstein series g(z) = E(z, 1/2 + it g ) [You16, Theorem 1.4]. One expects that this is true for 0 < δ < 1, but the method of proof of [You16, Proposition 1.5] is hindered by an inability to detect cancellation involving a spectral sum of terms not necessarily all of the same sign; see [You16,p. 965].
This hindrance does not arise for (1.3), and so we are lead to the following conjecture on Planck scale mass equidistribution, which roughly states that quantum unique ergodicity holds for almost every shrinking ball whose radius is larger than the Planck scale λ −1/2 g . Conjecture 1.5. Suppose that R t −δ g with 0 < δ < 1. Then (1.3) holds as t g tends to infinity along any subsequence of newforms g ∈ B * 0 (q, χ).
Via Chebyshev's inequality, the left-hand side of (1.3) is bounded by c −2 Var(g; R), where Var(g; R) := This reduces the problem to bounding this variance. For q = 1, the first author showed that if R t −δ g with 0 < δ < 1, then Var(g; R) = o(1) under the assumption of the generalised Lindelöf hypothesis [Hum18, Proposition 5.1]; an analogous result is also proved unconditionally for g(z) equal to an Eisenstein series E(z, 1/2 + it g ) [Hum18, Proposition 5.5]. The barrier R t −1 g is the Planck scale, at which equidistribution need not hold [Hum18, Theorem 1.14]; as discussed in [HR92,Section 5.1], the topography of Maaß forms below this scale is "essentially sinusoidal" and so Maaß forms should not be expected to exhibit random behaviour, such as mass equidistribution, at such minuscule scales.

The fourth moment of a Maaß form.
Another manifestation of Berry's conjecture is the Gaussian moments conjecture (see [Hum18,Conjecture 1.1]), which states that the (suitably normalised) n-th moment of a real-valued Maaß newform g restricted to a fixed compact subset K of Γ 0 (q)\H should converge to the n-th moment of a real-valued Gaussian random variable with mean 0 and variance 1 as t g tends to infinity. A similar conjecture may also be posed for complex-valued Maaß newforms, as well as for holomorphic newforms in the large weight limit; cf. [BKY13, Conjectures 1.2 and 1.3]. A closely related conjecture, namely essentially sharp upper bounds for L p -norms of automorphic forms, has been posed by Sarnak [Sar03,Conjecture 4]. For n = 2, the Gaussian moments conjecture is simply quantum unique ergodicity, and for small values of n, this is also conjectured to be true for noncompact K (but not for large n; cf. [Hum18, Section 1.1.2]).
The fourth moment is of particular interest, for, as first observed by Sarnak [Sar03,p. 461], it can be expressed as a spectral sum of L-functions. The conjecture takes the following form for K = Γ 0 (q)\H. Conjecture 1.6. As t g tends to infinity along a subsequence of real-valued newforms g ∈ B * 0 (q, χ), This has been proven for q = 1 conditionally under the generalised Lindelöf hypothesis by Buttcane and the second author [BuK17b, Theorem 1.1], but an unconditional proof currently seems well out of reach (cf. [Hum18,Remark 3.3] and Remark 1.24). Djanković and the second author have formulated [DK18a] and subsequently proven [DK18b, Theorem 1.1] a regularised version of this conjecture for Eisenstein series, improving upon earlier work of Spinu [Spi03, Theorem 1.1 (A)] that proves the upper bound O ε (t ε g ) in this setting. Numerical investigations of this conjecture for the family of dihedral Maaß newforms have also been undertaken by Hejhal and Strömbergsson [HS01], and the upper bound O q,ε (t ε g ) for dihedral forms has been proven by Luo [Luo14,Theorem] (cf. Remark 1.23). Furthermore, bounds for the fourth moment in the level aspect have also been investigated by many authors [Blo13,BuK15,Liu15,LMY13].
1.3 Results. This paper gives the first unconditional resolutions of Conjectures 1.5 and 1.6 for a family of cusp forms. We prove these two conjectures in the particular case when q = D ≡ 1 (mod 4) is a fixed positive squarefree fundamental discriminant, χ = χ D is the primitive quadratic character modulo D, and t g tends to infinity along any subsequence of dihedral Maaß newforms g = g ψ ∈ B * 0 (D, χ D ). as the spectral parameter t g tends to infinity along any subsequence of dihedral Maaß newforms g ψ ∈ B * 0 (D, χ D ). Consequently, vol w ∈ Γ 0 (D)\H : 1 vol(B R ) BR(w) |g ψ (z)| 2 dμ(z) − 1 vol(Γ 0 (D)\H) > c tends to zero as t g tends to infinity for any fixed c > 0.
Theorem 1.9. Let D ≡ 1 (mod 4) be a positive squarefree fundamental discriminant and let χ D be the primitive quadratic character modulo D. Then there exists an absolute constant δ > 0 such that as t g tends to infinity along any subsequence of dihedral Maaß newforms g ψ ∈ B * 0 (D, χ D ). Dihedral newforms form a particularly thin subsequence of Maaß forms; the number of dihedral Maaß newforms with spectral parameter less than T is asymptotic to c 1,D T , whereas the number of Maaß newforms with spectral parameter less than T is asymptotic to c 2,D T 2 , where c 1,D , c 2,D > 0 are constants dependent only on D. We explain in Section 1.8 the properties of dihedral Maaß newforms, not shared by nondihedral forms, that are crucial to our proofs of Theorems 1.7 and 1.9.
Remark 1.11. Previous work [Blo13, BuK15, BuK17a, Liu15, LMY13, Luo14] on the fourth moment has been subject to the restriction that D be a prime. We weaken this restriction to D being squarefree. The additional complexity that arises is determining explicit expressions for the inner product of |g| 2 with oldforms. Removing the squarefree restriction on D, while likely presently feasible, would undoubtedly involve significant extra work.
Remark 1.12. An examination of the proofs of Theorems 1.7 and 1.9 shows that the dependence on D in the error terms in (1.8) and (1.10) is polynomial.
Notation. Throughout this article, we make use of the ε-convention: ε denotes an arbitrarily small positive constant whose value may change from occurrence to occurrence. Results are stated involving level D when only valid for positive squarefree D ≡ 1 (mod 4) and are stated involving level q otherwise. The primitive quadratic character modulo D will always be denoted by χ D . Since we regard D as being fixed, all implicit constants in Vinogradov and big O notation may depend on D unless otherwise specified. We write N 0 := N ∪ {0} for the nonnegative integers. A dihedral Maaß newform will be written as g ψ ∈ B * 0 (D, χ D ); this is associated to a Hecke Größencharakter ψ of Q( √ D) as described in Appendix A. 1.4 Elements of the Proofs. The proofs of Theorems 1.7 and 1.9, which we give in Section 2, follow by combining three key tools; the approach that we follow is that first pioneered by Sarnak [Sar03,p. 461] and Spinu [Spi03]. First, we spectrally expand the variance and the fourth moment, obtaining the following explicit formulae.
Proposition 1.13. Let q be squarefree and let χ be a primitive Dirichlet character modulo q. Then for a newform g ∈ B * 0 (q, χ), the variance Var(g; R) is equal to where B * 0 (Γ 0 (q 1 )) f is an orthonormal basis of the space of newforms of weight zero, level q 1 , and principal nebentypus, normalised such that f, f q = 1, E ∞ (z, s) denotes the Eisenstein series associated to the cusp at infinity of Γ 0 (q)\H, and Similarly, the fourth moment Γ0(q)\H |g(z)| 4 dμ(z) is equal to The arithmetic functions ω, ν, ϕ are defined by ω(n) := # {p | n}, ν(n) := n p|n (1 + p −1 ), and ϕ(n) := n p|n (1 − p −1 ). We have written L p (s, π) for the p-component of the Euler product of an L-function L(s, π), while where Λ(s, π) := q(π) s/2 L ∞ (s, π)L(s, π) denotes the completed L-function with conductor q(π) and archimedean component L ∞ (s, π). Next, we obtain explicit expressions in terms of L-functions for the inner products | |g| 2 , f q | 2 and | |g| 2 , E ∞ (·, 1/2 + it) | 2 ; this is the Watson-Ichino formula. Proposition 1.16. Let q = q 1 q 2 be squarefree and let χ be a primitive Dirichlet character modulo q. Then for g ∈ B * 0 (q, χ) and for f ∈ B * 0 (Γ 0 (q 1 )) of parity f ∈ {1, −1} normalised such that g, g q = f, f q = 1, . (1.17) Similarly, (1.18) Now we specialise to g = g ψ ∈ B * 0 (D, χ D ). Observe that ad g ψ is equal to the (noncuspidal) isobaric sum χ D g ψ 2 , where g ψ 2 ∈ B * 0 (D, χ D ) is the dihedral Maaß newform associated to the Hecke Größencharakter ψ 2 of Q( √ D), and so which can readily be seen by comparing Euler factors. Then the identity (1.17) holds with 1 + f replaced by 2 as both sides vanish when f is odd: the right-hand side vanishes due to the fact that for Lemma A.2 shows that the root number in both cases is −1, while the left-hand side vanishes since one can make the change of variables z → −z in the integral over Γ 0 (D)\H, which leaves |g ψ (z)| 2 unchanged but replaces f (z) with −f (z). We have thereby reduced both problems to subconvex moment bounds. To this end, for a function h : R ∪ i(−1/2, 1/2) → C, we define the mixed moments (1.20) We prove the following bounds for these terms for various choices of function h.
Remark 1.22. For the purposes of proving Theorem 1.7, the exact identities in Propositions 1.13 and 1.16 as well as the asymptotic formula in Proposition 1.21 (2) are superfluous, for we could make do with upper bounds in each case in order to prove the desired upper bound for Var(g ψ ; R). These identities, however, are necessary to prove the desired asymptotic formula for the fourth moment of g ψ in Theorem 1.9.
Remark 1.23. The large sieve yields with relative ease the bounds O ε ((T t g ) 1+ε ) and O ε (t ε g ) for Proposition 1.21 (1) and (2) respectively; dropping all but one term then only yields the convexity bound for the associated L-functions. These weaker bounds imply that the variance Var(g ψ ; R) and the fourth moment of g ψ are both O ε (t ε g ), with the latter being a result of Luo [Luo14,Theorem] and the former falling just short of proving small scale mass equidistribution.
1.5 A sketch of the proofs and the structure of the paper. We briefly sketch the main ideas behind the proofs of Propositions 1.13, 1.16, and 1.21.
The proof of Proposition 1.13, given in Section 3, uses the spectral decomposition of L 2 (Γ 0 (q)\H) and Parseval's identity to spectrally expand the variance and the fourth moment. We then require an orthonormal basis in terms of newforms and translates of oldforms together with an explicit description of the action of Atkin-Lehner operators on these Maaß forms in order to obtain (1.14) and (1.15). Proposition 1.16 is an explicit form of the Watson-Ichino formula, which relates the integral of three GL 2 -automorphic forms to a special value of a triple product L-function; we present this material in Section 4. To ensure that the identities (1.17) and (1.18) are correct not merely up to multiplication by an unspecified constant requires a careful translation of the adèlic identity [Ich08, Theorem 1.1] into the classical language of automorphic forms. Moreover, this identity involves local constants at ramified primes, and the precise set-up of our problem involves determining such local constants, which is undertaken in Section 5. This problem of the determination of local constants in the Watson-Ichino formula is of independent interest; see, for example, [Col18,Col19,Hu16,Hu17,Wat08].
The proof of Proposition 1.21 takes up the bulk of this paper, for it is rather involved and requires several different strategies to deal with various ranges. The many (predominantly) standard automorphic tools used in the course of the proof, such as the approximate functional equation, the Kuznetsov formula, and the large sieve, are relegated to Appendix A; we recommend that on first reading, the reader familiarise themself with these tools via a quick perusal of Appendix A before continuing on to the proof of Proposition 1.21 that begins in Section 6. Proposition 1.21 (1), proven in Section 9, requires three different treatments for three different parts of the short initial range. We may use hybrid subconvex bounds for L(1/2, f ⊗ g ψ 2 ) and |L(1/2 + it, g ψ 2 )| 2 due to Michel and Venkatesh [MV10] to treat the range T ≤ t β g for an absolute constant β > 0. For t β g < T ≤ t 1/2 g , we use subconvex bounds for L(1/2, f ⊗ χ D ) and |L(1/2 + it, χ D )| 2 due to Young [You17] together with bounds proven in Section 6 for the first moment of L(1/2, f ⊗ g ψ 2 ) and of |L(1/2 + it, g ψ 2 )| 2 . This approach relies crucially on the nonnegativity of L(1/2, f ⊗ g ψ 2 ) (see, for example, the discussion on this point in [HT14, Section 1.1]). Bounds for the remaining range t 1/2 g < T ≤ t 1−α g for Proposition 1.21 (1) are shown in Sections 7 and 8 to follow from the previous bounds for the range t α g T t 1/2 g . This is spectral reciprocity: via the triad of Kuznetsov, Voronoȋ, and Kloosterman summation formulae (the latter being the Kuznetsov formula in the formulation that expresses sums of Kloosterman sums in terms of Fourier coefficients of automorphic forms), bounds of the form with h(t) = 1 E∪−E (t) for E = [T, 2T ] are essentially implied by the same bounds with E = [t g /T, 2t g /T ] together with analogous bounds for moments involving holomorphic cusp forms of even weight k ∈ [t g /T, 2t g /T ]. The proof of Proposition 1.21 (2) for the bulk range, appearing in Section 10, mimics that of the analogous result for Eisenstein series given in [DK18b]. As such, we give a laconic sketch of the proof, highlighting mainly the slight differences compared to the Eisenstein case.
Proposition 1.21 (3) is proven in Section 13 and relies upon the Cauchy-Schwarz inequality; the resulting short second moment of Rankin-Selberg L-functions is bounded via the large sieve, while a bound is also required for a short mixed moment of four L-functions. This latter bound is again a consequence of spectral reciprocity, akin to [Jut01,Theorem], and is detailed in Sections 11 and 12.

Further heuristics.
We give some very rough back-of-the-envelope type calculations to go along with the sketch above. Proposition 1.21 requires the evaluation of a mean value of L-functions looking essentially like tf <2tg where we pretend that D equals 1, since it is anyway fixed. The goal is to extract the main term with an error term bounded by a negative power of t g . The expression remains unchanged if the summand is multiplied by the parity f = ±1 of f , because L(1/2, f) = 0 when f = −1. Summing over t f using the opposite-sign case of the Kuznetsov formula gives, in the dyadic range t f ∼ T , an off-diagonal of the shape 1 t 1/2 where d(n) is the divisor function. Note that for the sake of argument, we use approximate functional equations, although our proof works with Dirichlet series in regions of absolute convergence and continues meromorphically at the last possible moment. Consider the case t α g ≤ T ≤ 2t g − t 1−α g , which includes the short initial and bulk ranges, so that m ∼ t 2 g and c ∼ t g . Applying the Voronoȋ summation formula to both n and m returns a sum like  Note that c ∼ (T /2t g ) √ mn, so applying the Kloosterman summation formula gives This can be recast as essentially The phenomenon of the same mean value of L-functions reappearing but with the range of summation now reciprocated to t f < 2t g /T is spectral reciprocity, as alluded to above.
When T ∼ t g , the bulk range, we immediately get a satisfactory estimate by inserting subconvexity bounds. When T < t 1−α g , the short initial range, we are not done right away, but we at least reduce to the case T < t 1/2 g . In this range, we must use a new approach. The idea is to bound, using nonnegativity of central values, L(1/2, f) 2 by subconvexity bounds and then to estimate the first moment . This is not an easy task because the sum over t f is very short. We expand the first moment using approximate functional equations, apply the Kuznetsov formula, use the Voronoȋ summation formula, and then estimate; this turns out to be sufficient. Finally, it remains to consider the short transition range Here the strategy is to apply the Cauchy-Schwarz inequality and consider tf L(1/2, f) 4 and tf L(1/2, f ⊗ g ψ 2 ) 2 , the latter of which can be estimated sharply using the spectral large sieve, while the former can be bounded once again via spectral reciprocity.

Related results for the fourth moment and spectral reciprocity.
Bounds of the form O ε (t ε g ) for the fourth moment of the truncation of an Eisenstein series E(z, 1/2 + it g ) or for a dihedral Maaß form g = g ψ have been proven by Spinu [Spi03] and Luo [Luo14] respectively; the proofs use the Cauchy-Schwarz inequality and the large sieve to bound moments of L-functions and rely on the factorisation of the L-functions appearing in the Watson-Ichino formula. In applying the large sieve to the bulk range, this approach loses the ability to obtain an asymptotic formula.
Sarnak and Watson [Sar03, Theorem 3(a)] noticed that via the GL 3 Voronoȋ summation formula coupled with the convexity bound for L(1/2, f ⊗ sym 2 g), one could prove the bound O ε (t ε g ) for the bulk range of the spectral expansion of the fourth moment of a Maaß cusp form (cf. [Hum18, Remark 3.3]). This approach was expanded upon by Buttcane and the second author [BuK17b], where an asymptotic for this bulk range was proven under the assumption of the generalised Lindelöf hypothesis. Asymptotics for a moment closely related to that appearing in Proposition 1.21 (2) are proven in [BuK17a]; the method is extremely similar to that used in GAFA ON THE RANDOM WAVE CONJECTURE FOR DIHEDRAL MAASS FORMS 45 [BuK17b]. Finally, asymptotics for the bulk range appearing in the spectral expansion of the regularised fourth moment of an Eisenstein series are proven in [DK18b] (and Proposition 1.21 (2) is proven via minor modifications of this proof). These results all follow via the triad of Kuznetsov, Voronoȋ, and Kloosterman summation formulae, and are cases of spectral reciprocity: the moment of L-functions in the bulk range is shown to be equal to a main term together with a moment of L-functions that is essentially extremely short, namely involving forms f for which t f t ε g . This nonetheless leaves the issue of dealing with the short initial and transition ranges. Assuming the generalised Lindelöf hypothesis, it is readily seen that these are negligible. Spectral reciprocity in the short initial range is insufficient to prove this, since it merely replaces the problem of bounding the contribution from the range [T, 2T ] with that of the range [T /t g , 2T /t g ]. Our key observation is that spectral reciprocity reduces the problem to the range T < t 1/2 g , at which point we may employ a different strategy, namely subconvex bounds for L(1/2, f)L(1/2, f ⊗ χ D ) together with a bound for the first moment of L(1/2, f ⊗ g ψ 2 ). This approach, albeit in a somewhat disguised form, is behind the success of the unconditional proofs of the negligibility of the short initial and transition ranges for the regularised fourth moment of an Eisenstein series. These follow from the work of Jutila [Jut01] and Jutila and Motohashi [JM05]; see [Hum18, Lemmata 3.7 and 3.8].

Connections to subconvexity.
Quantifying the rate of equidistribution for quantum unique ergodicity in terms of bounds for (1.1) is, via the Watson-Ichino formula, equivalent to determining subconvex bounds for L(1/2, f ⊗ ad g) in the t gaspect. Such bounds are yet to be proven except in a select few cases, namely when g is dihedral or an Eisenstein series, where L(1/2, f ⊗ ad g) factorises as Indeed, quantum unique ergodicity was already known for Eisenstein series [LS95] before the work of Lindenstrauss [Lin06] and Soundararajan [Sou10], and for dihedral Maaß forms [Blo05] with quantitative bounds for (1.1) shortly thereafter (see also [Sar01,LY02,LLY06a,LLY06b]). The proofs of Theorems 1.7 and 1.9, as well as their Eisenstein series counterparts [DK18b,Hum18], rely crucially on these factorisations, and the chief hindrance behind the lack of an unconditional proof of these theorems for an arbitrary Maaß cusp form is the lack of such a factorisation. In proving Theorem 1.7, on the other hand, we require bounds for the moments given in Proposition 1.21, most notably in the range E = [T, 2T ] with T < t 1−α g . Dropping all but one term in this range implies the hybrid subconvex bounds |t|t 1−δ g for these products of L-functions with analytic conductors (t f t g ) 4 and (|t|t g ) 4 respectively. Such bounds for product L-functions were previously known, and at various points in the proof of Proposition 1.21 we make use of known subconvex bounds for individual L-functions in this product; what is noteworthy is that individual subconvex bounds are insufficient for proving Theorems 1.7 and 1.9, but rather bounds for moments that imply subconvexity are required.
Remark 1.24. This demonstrates the difficulty of proving Theorems 1.7 and 1.9 unconditionally for arbitrary Hecke-Maaß eigenforms g: as mentioned in [BuK17b,p. 1493], we would require a subconvex bound of the form L(1/2, f ⊗ ad g) ( It follows that (2.5) We recall the bound L(1, g ψ 2 ) 1/ log t g , as well as [Hum18, Lemma 4.2], which states that as R tends to zero, if Rt tends to infinity, where J ν (z) denotes the Bessel function of the first kind. Moreover, h R (t) 1 if R 1 and t ∈ i(0, 1/2). We bound M Maaß (h) + M Eis (h) by breaking this up into intervals for which we can apply Proposition 1.21 and using the bounds (2.5) and (2.6): for the short initial and tail ranges, we use dyadic intervals, while for the short transition range, we divide into intervals of the form has polynomial decay in t when t is in the bulk range; the proof of Theorem 1.7 is thereby complete. Theorem 1.9 is proven much in the same way, as the fourth moment is equal to the sum of 1/ vol(Γ 0 (D)\H), (2.1), and (2.2) with h R (t) replaced by 1. We find that the short initial, short transition, tail, and exceptional ranges all contribute at most O(t −δ g ), while the bulk range contributes 2/ vol(Γ 0 (D)\H) + O(t −δ g ).

48
P. HUMPHRIES AND R. KHAN GAFA Remark 2.7. The method of proof also gives Var(g ψ ; R) ∼ 2/ vol(Γ 0 (D)\H) if R t −δ g with δ > 1, while a modification of Proposition 1.21 (2) implies that there exists an absolute constant α > 0 such that for t −1−α where p F q denotes the generalised hypergeometric function. This corrects an erroneous asymptotic formula in [Hum18, Remark 5.4].

An orthonormal basis of Maaß cusp forms for squarefree levels.
The proof of Proposition 1.13, which we give in Section 3.4, invokes the spectral decomposition of L 2 (Γ 0 (q)\H), which involves a spectral sum indexed by an orthonormal basis B 0 (Γ 0 (q)) of the space of Maaß cusp forms of weight zero, level q, and principal nebentypus. This space has the Atkin-Lehner decomposition where (ι f )(z) := f ( z), but this decomposition is not orthogonal for q > 1. Nevertheless, an orthonormal basis can be formed using linear combinations of elements of this decomposition.
Lemma 3.1 ([ILS00, Proposition 2.6]). An orthonormal basis of the space of Maaß cusp forms of weight zero, squarefree level q, and principal nebentypus is given by Proof. In [ILS00, Proposition 2.6], this is proved with Using the fact that λ f (p) 2 = λ f (p 2 ) + 1 and for p q 1 , this simplifies to the desired identity.

GAFA ON THE RANDOM WAVE CONJECTURE FOR DIHEDRAL MAASS FORMS 49
We record here the following identities, which follow readily from the multiplicativity of the summands involved.

An orthonormal basis of Eisenstein series for squarefree levels.
A similar orthonormal basis exists for Eisenstein series. Instead of the usual orthonormal basis we may form an orthonormal basis out of Eisenstein series newforms and oldforms: a basis of the space of Eisenstein series of weight zero, level q, and principal nebentypus is given by Here where E(z, s) is the usual Eisenstein series on Γ\H, defined for (s) > 1 by with Γ := SL 2 (Z) and Γ ∞ := {γ ∈ Γ : γ∞ = ∞} the stabiliser of the cusp at infinity. For t ∈ R \ {0}, this has the Fourier expansion with W α,β the Whittaker function, The Eisenstein series E(z, 1/2 + it) is normalised such that its formal inner product with itself on Γ\H is 1 (in the sense of [Iwa02, Proposition 7.1]), and so the formal inner product of E 1 (z, 1/2 + it) with itself on Γ 0 (q)\H is 1.
This basis is not orthogonal for q > 1, but Young [You19] has shown that there exists an orthonormal basis derived from this basis just as for Maaß cusp forms, as in Lemma 3.1.

Lemma 3.3 ([You19, Section 8.4]
). An orthonormal basis of the space of Eisenstein series of weight 0, level q, and principal nebentypus is given by As with Lemma 3.2, we have the following identities.
Lemma 3.4. For squarefree q and | q, we have that

Inner products with oldforms and Eisenstein series.
To deal with inner products involving oldforms and Eisenstein series, we use Atkin-Lehner operators. For squarefree q, write q = vw, and denote by the Atkin-Lehner operator on Γ 0 (q) associated to w, where a, b, c, d ∈ Z and det W w = adw − bcv = 1. We denote by B * hol (q, χ) the set of holomorphic newforms f of level q, nebentypus χ, and arbitrary even weight k f ∈ 2N; again, we write B * hol (Γ 0 (q)) when χ is the principal character. . Let q = vw be squarefree and let χ be a Dirichlet character of conductor q χ dividing q, so that we may write In particular, |η g (w)| = 1. Moreover, the same result holds for g ∈ B * hol (q, χ), so that

GAFA ON THE RANDOM WAVE CONJECTURE FOR DIHEDRAL MAASS FORMS 51
We call η g (w) the Atkin-Lehner pseudo-eigenvalue; note that it is independent of a, b, c, d ∈ Z when either χ is the principal character or a ≡ 1 (mod v) and b ≡ 1 (mod w), or equivalently d ≡ w (mod v) and c ≡ v (mod w).
Lemma 3.6. Let q = q 1 q 2 be squarefree, let χ be a Dirichlet character modulo q, and let g ∈ B * 0 (q, χ) and f ∈ B * 0 (Γ 0 (q 1 )). Then for vw = q 2 , so that Proof. Since the Atkin-Lehner operators normalise Γ 0 (q), and so as f is invariant under the action of Γ 0 (q 1 ), We now prove an analogous result for Eisenstein series. In this case, we may use Eisenstein series indexed by cusps (though later we will find it advantageous to work with Eisenstein newforms and oldforms). As q is squarefree, a cusp a of Γ 0 (q)\H has a representative of the form 1/v for some divisor v of q, and every cusp has a unique representative of this form; when a ∼ ∞, for example, we have that v = q. We define the Eisenstein series which converges absolutely for (s) > 1 and z ∈ H, where is the stabiliser of the cusp a, and the scaling matrix σ a ∈ SL 2 (R) is such that the Atkin-Lehner operator on Γ 0 (q) associated to w, where dw − bv = 1. 52 P. HUMPHRIES AND R. KHAN GAFA Lemma 3.7. Let g ∈ B * 0 (q, χ) with q squarefree, and let a ∼ 1/v be a cusp of Γ 0 (q)\H. Then Proof. By unfolding, using Lemma 3.5, and folding, we find that Finally, we claim that twisting g leaves these inner products unchanged. Alas, we do not know a simple proof of this fact; as such, the proof is a consequence of calculations in Sections 4 and 5.
Lemma 3.8. For q = q 1 q 2 squarefree and g ∈ B * 0 (q, χ) with χ primitive, we have that Furthermore, for f ∈ B * 0 (Γ 0 (q 1 )) and w | q 2 , Proof. The former is a consequence of Corollary 4.9, while the latter follows upon combining Lemma 3.6 with Corollary 4.19.

Proof of Proposition 1.13.
Proof of Proposition 1.13. An application of Parseval's identity, using the spectral decomposition of L 2 (Γ 0 (q)\H) [IK04, Theorem 15.5], together with the fact that for any Laplacian eigenfunction f [Hum18, Lemma 4.3], yields Similarly, Lemmata 3.7 and 3.8 imply that for any t ∈ R. This gives the desired spectral expansion for Var(g; R), while the spectral expansion for the fourth moment of g follows similarly, noting that the constant term 1/ vol(Γ 0 (q)\H) in the spectral expansion gives rise to the term 1/ vol(Γ 0 (q)\H) in (1.15).

The Watson-Ichino formula for Eisenstein series.
We require explicit expressions in terms of L-functions for | |g| 2 , f q | 2 and | |g| 2 , E ∞ (·, 1/2+it) q | 2 . This is the contents of the Watson-Ichino formula. In the latter case, this result is simply the Rankin-Selberg method, which far predates the work of Watson and Ichino; it can be proven by purely classical means via unfolding the Eisenstein series, as we shall now detail. Recall that a Maaß newform g ∈ B * 0 (q, χ) has the Fourier expansion about the cusp at infinity of the form where the Fourier coefficients ρ g (n) satisfy ρ g (n) = g ρ g (−n), with the parity g of g equal to 1 if g is even and −1 if g is odd. The Hecke eigenvalues λ g (n) of g satisfy for all m, n ≥ 1, (4.1) λ g (n) = χ(n)λ g (n) for all n ≥ 1 with (n, q) = 1, (4.2) Lemma 4.4. Let g ∈ B * 0 (q 1 , χ) with q 1 q 2 = q and q 1 ≡ 0 (mod q χ ), where q χ is the conductor of χ. We have that Proof. Unfolding the integral and using Parseval's identity and (4.3) yields Lemma 4.6. Let q be squarefree, and let g ∈ B * 0 (q 1 , χ) with q 1 q 2 = q and q 1 ≡ 0 (mod q χ ). We have that for (s) > 1 and that Proof. We recall that Using (4.1) and (4.2) together with the fact that we obtain (4.7). Next, we take the residue of (4.5) at s = 1, noting that E ∞ (z, s) has residue 1 vol(Γ 0 (q)\H) = 3 πν(q) at s = 1 independently of z ∈ Γ 0 (q)\H. This yields the desired identity (4.8). Corollary 4.9. Let q be squarefree, and let g ∈ B * 0 (q 1 , χ) with q 1 q 2 = q and q 1 ≡ 0 (mod q χ ), where g is normalised such that g, g q = 1. We have that Note that Corollary 4.9 remains valid when g is replaced by g ⊗ χ v for v | q χ , since the level is unchanged and ad(g ⊗ χ v ) = ad g.

The Adèlic
Watson-Ichino formula for Maaß newforms. Now we consider the inner product | |g| 2 , f q | 2 . The Watson-Ichino formula is an adèlic statement: the integral over Γ 0 (q)\H is replaced by an integral over Z(A Q ) GL 2 (Q)\ GL 2 (A Q ), and g and f are replaced by functions on GL 2 (Q)\ GL 2 (A Q ) that are square integrable modulo the centre Z(A Q ) and are elements of cuspidal automorphic representations of GL 2 (A Q ). In Section 4.3, we translate this adèlic statement into a statement in the classical language of automorphic forms.
Let F be a number field, and let For each place v of F with corresponding local field F v , we also let is normalised as follows: is the Haar probability measure on the compact group SO(2).
• A similar definition can also be given for F v ∼ = C, though we do not need this, since we will eventually take F = Q.
Here d F denotes the discriminant of F , and we recall that the conductor of the Dedekind zeta function is |d F |, so that the completed Dedekind zeta function is The quantity I v (ϕ v ⊗ ϕ v ) is often called the local constant. When ϕ 1 , ϕ 2 , ϕ 3 are pure tensors consisting of local newforms in the sense of Casselman (or in some cases translates of local newforms; see [Hu17] and [Col19, Section 2.1]), then these local constants depend only (but sensitively!) on the representations π 1,v , π 2,v , π 3,v . The local constants have been explicitly determined for many different combinations of representations π 1,v , π 2,v , π 3,v of GL 2 (F v ) (cf. [Col19, Sections 2.2 and 2.3]). We require several particular combinations of representations for our applications. For Now let F v be a nonarchimedean local field with uniformiser v and cardinality q v of the residue field. In Section 5, we prove the following.
This also holds if either or both ϕ 3,v and ϕ 3,v are translates of local newforms by Remark 4.18. The latter local constant has also been determined by Collins  and similarly let f ∈ B 0 (q, χ ) be a Hecke-Maaß eigenform such that f and f are both associated to the same newform. We assume additionally that χ 1 χ 2 χ 3 = χ 0(q) , the principal character modulo q. Letting ϕ 1 , ϕ 2 , ϕ 3 and ϕ 1 , ϕ 2 , ϕ 3 denote the adèlic lifts of the Hecke-Maaß eigenforms f 1 , f 2 , f 3 and f 1 , f 2 , f 3 , we have that This adèlic-to-classical interpretation of the Watson-Ichino formula uses the fact that Λ(2) = π/6 and vol(Γ 0 (q)\H) = πν(q)/3, as well as the identity ; the factor 2 is present for this is the Tamagawa number of Z(A Q ) GL 2 (Q)\ GL 2 (A Q ).
Corollary 4.19. For squarefree q = q 1 q 2 , g ∈ B * 0 (q, χ) with χ primitive, f ∈ B * 0 (q 1 ) normalised such that g, g q = f, f q = 1, and w 1 , w 2 | q 2 , we have that Proof. We have the isobaric decomposition g ⊗ g = 1 ad g, so that g ⊗ g ⊗ f = f f ⊗ ad g, while f = f implies that ad f = sym 2 f , and ad g = ad g. Consequently, the conductor q(g ⊗ g ⊗ f ) also factorises as q(f )q(f ⊗ ad g). The conductors of f , f ⊗ ad g, ad g, and sym 2 f are q 1 , q 4 q 1 , q 2 , and q 2 1 respectively (cf. Lemma A.2). We denote by π g , π g , π f the cuspidal automorphic representations of GL 2 (A Q ) associated to g, g, f respectively; note that π g = π g . The Watson-Ichino formula gives It remains to determine the local constants I p (ϕ p ⊗ ϕ p ). We observe the following: • When p | q 1 , the local component π g,p of g is a unitarisable ramified principal series representation ω 1,p ω 1,p , where the unitary characters ω 1,p , ω 1,p of Q × p have conductor exponents c(ω 1,p ) = 1 and c(ω 1,p ) = 0. The local component π f,p of f is a special representation ω 3,p St, where ω 3,p is either the trivial character or the unramified quadratic character of Q × p . Finally, ϕ 1,p , ϕ 2,p , ϕ 3,p , ϕ 1,p , ϕ 2,p , ϕ 3,p are all local newforms. • When p | q 2 but p [w 1 , w 2 ], the local component π g,p of g is of the same form as for p | q 1 . The local component π f,p of f is a unitarisable unramified principal series representation ω 3,p ω −1 3,p , where c(ω 3,p ) = 0 and p −1/2 < |ω 3,p (p)| < p 1/2 . Once again, all local forms are newforms.
• When p | (w 1 , w 2 ), the setting is as above except both ϕ 3,p and ϕ 3,p are translates of local newforms by π 3,p p −1 0 0 1 and π 3,p p −1 0 0 1 respectively. • When p | w 1 but p w 2 , the setting is as above except only ϕ 3,p is the translate of the local newform. • Finally, when p | w 2 but p w 1 , the setting is as above except instead only ϕ 3,p is the translate of the local newform.
For the former case, we apply Proposition 4.16 with F v = Q p and q v = p, while Proposition 4.17 is applied to the remaining cases. This gives the result.

Proof of Proposition
and finally the approximate functional equation for L(1/2, F ⊗ g) given in [Liu15, Proof of Lemma 3.2] ought to involve a sum over n ≤ q 3/2+ε , not q 1+ε (which is to say that the conductor of F ⊗ g is q 3 , not q 2 ; see Lemma A.2). The first of these two errata is readily rectified; the second, however, means that the exponent in [Liu15, Theorem 1.1] is subsequently weakened to −2/3 − δ/3 + ε rather than −11/12 − δ/3 + ε.

Local Constants in the Watson-Ichino Formula
This section is devoted to the proofs of Propositions 4.16 and 4.17. Since every calculation is purely local, we drop the subscripts v. Let F be a nonarchimedean local field with ring of integers O F , uniformiser , and maximal ideal We set K := GL 2 (O F ) and define the congruence subgroup for any nonnegative integer m. We normalise the additive Haar measure da on F to give O F volume 1, while the multiplicative Haar measure

Reduction to formulae for Whittaker functions.
For π equal to a principal series representation ω ω or a special representation ωSt, and given a vector ϕ π in the induced model of π, we let For generic irreducible unitarisable representations π 1 , π 2 , π 3 with π 1 a principal series representation, and for ϕ 1 in the induced model of π 1 , W 2 ∈ W(π 2 , ψ), and W 3 ∈ W(π 3 , ψ −1 ), we define the local Rankin-Selberg integral RS (ϕ 1 , W 2 , W 3 ) to be . The importance of this quantity is the following identity of Michel and Venkatesh.

5.2.1
The case π 3 = ω 3 St. In this section, we deal with the first case, so that π 3 = ω 3 St.
Lemma 5.12 (Cf. [Hu17, Lemma 2.13]). We have that Proof. Let Combining (5.1) and (5.7) yields W π1 a 0 0 1 Upon making the change of variables x → x − a and using (5.11), the identity for W π1 is derived. The identity for W π2 follows by taking complex conjugates. Finally, combining (5.1) and (5.8) shows that The result then follows via (5.5) after the change of variables x → x − a.

Proofs of Propositions 4.16 and 4.17.
To prove Propositions 4.16 and 4.17, we use Lemma 5.2 to reduce the problem to evaluating local Rankin-Selberg integrals. We then use the identities in Section 5.2 for values of ϕ π and W π together with the following lemma.
Proof of Proposition 4.16. Lemmata 5.9, 5.12, and 5.18 imply that The integral is readily seen to be equal to qω −1 3 ( )L(1, ω 3 ) via the change of variables a → −1 a; Lemma 5.2 then gives the identity and Lemma 5.9 implies that

P. HUMPHRIES AND R. KHAN GAFA
We conclude that On the other hand, we have the isobaric decomposition Moreover, while ad π 3 is the special representation of GL 3 (F ) associated to the trivial character, so that L(s, ad π 3 ) = ζ F (s + 1).

P. HUMPHRIES AND R. KHAN GAFA
Since W π is right K 1 (p)-invariant, Lemma 5.18 together with the Iwasawa decomposition imply that I(ϕ ⊗ ϕ)/ ϕ, ϕ is equal to where b = ( a x 0 1 ) with a ∈ F × , x ∈ F , and db = |a| −1 d × a dx. One can then use Lemmata 5.9 and 5.12 and the fact that where π 1 , π 2 , π 3 are as in Proposition 4.16. Inserting these identities into (5.21) and evaluating the resulting integrals thereby reproves Proposition 4.16; similar calculations yield Proposition 4.17.

The First Moment in the Short Initial Range
The main results of this section are bounds for the first moments which will be required in the course of the proof of Proposition 1.21 (1).
Were we to replace g ψ 2 with an Eisenstein series E(z, 1/2+2it g ), so that L(1/2, f⊗ g ψ 2 ) would be replaced by |L(1/2 + 2it g , f)| 2 , then we would immediately obtain the desired bound via the large sieve, Theorem A.32. Thus this result is of similar strength to the large sieve; in particular, dropping all but one term returns the convexity bounds for L(1/2, f ⊗ g ψ 2 ) and |L(1/2 + it, g ψ 2 )| 2 for T t 1/2 g . However, we cannot proceed via the large sieve as in the Eisenstein case because we do not know how to bound L(1/2, f ⊗ g ψ 2 ) by the square of a Dirichlet polynomial of length t 2 g , and if we were to instead first apply the Cauchy-Schwarz inequality and then use the large sieve, we would only obtain the bound O ε (T 2+ε +t 2+ε g ), which is insufficient for our requirements.
Our approach to prove Proposition 6.1 is to first use the approximate functional equation to write the L-functions involved as Dirichlet polynomials and then apply the Kuznetsov and Petersson formulae in order to express M Maaß (h) + M Eis (h) and M hol (h hol ) in terms of a delta term, which is trivially bounded, and sums of Kloosterman sums. We then open up the Kloosterman sums and apply the Voronoȋ summation formula. The proof is completed via employing a stationary phase-type argument to the ensuing expression.
Remark 6.2. This strategy is used elsewhere to obtain results that are similar to Proposition 6.1. Holowinsky and Templier use this approach in order to prove [HT14, Theorem 5], which gives a hybrid level aspect bound for a first moment of Rankin-Selberg L-functions involving holomorphic forms of fixed weight; the moment involves a sum over holomorphic newforms f of level N , while g ψ is of level M , and the bound for this moment is a hybrid bound in terms of N and M (with unspecified polynomial dependence on the weights of f and g ψ ). The first author and Radziwi l l have recently proven a hybrid bound [HR19, Proposition 2.28] akin to Proposition 6.1 where g ψ is replaced by the Eisenstein newform E χ,1 (z) := E ∞ (z, 1/2, χ D ) of level D and nebentypus χ D ; the bound for this moment is a hybrid bound in terms of T and D, and the method is also valid for cuspidal dihedral forms g ψ (with unspecified polynomial dependence on the weight or spectral parameter of g ψ ).
In applying the approximate functional equation in order to prove Proposition 6.1, we immediately run into difficulties because the length of the approximate functional equation depends on the level, and the Kuznetsov and Petersson formulae involve cusp forms of all levels dividing D. Since we are evaluating a first moment rather than a second moment, we cannot merely use positivity and oversum the Dirichlet polynomial coming from the approximate functional equation.
One possible approach to overcome this obstacle would be to use the Kuznetsov and Petersson formulae for newforms; see [HT14, Lemma 5] and [You19, Section 10.2]. Instead, we work around this issue by using the Kuznetsov and Petersson formulae associated to the pair of cusps (a, b) with a ∼ ∞ and b ∼ 1. As shall be seen, this introduces the root number of f ⊗ g ψ 2 in such a way to give approximate functional equations of the correct length for each level dividing D.
We will give the proof of Proposition 6.1 (1), then describe the minor modifications needed for the proof of Proposition 6.1 (2). Via the positivity of L(1/2, f ⊗g ψ 2 ), it suffices to prove the result with h replaced by We remind the reader that from here onwards, we will make use of many standard automorphic tools that are detailed in Appendix A.
Lemma 6.4. The first moment
Proof. We take m = 1 and h = V 1 2 (n 2 /D 3/2 , ·)h T in the Kuznetsov formula, Theorem A.10, using the explicit expressions in Lemma A.8, which we then multiply by χ D ( )/2 √ n and sum over n, ∈ N and over both the same sign and opposite sign Kuznetsov formulae. After making the change of variables n → w 2 n, using the fact that λ g ψ 2 (w 2 n) = λ g ψ 2 (n) for all w 2 | D via Lemma A.1, and simplifying the resulting sum over v 2 w 2 = using the multiplicativity of the summands, the spectral sum ends up as We do the same with the Kuznetsov formula associated to the (∞, 1) pair of cusps, Theorem A.16, using the explicit expressions in Lemma A.9, obtaining We add these two expressions together and use the approximate functional equation, Lemma A.5, with X = √ d 2 /w 2 . Recalling Lemma 3.2, this yields M Maaß (h T ). Similarly, the sum of the Eisenstein terms is M Eis (h T ). Upon noting that the delta term only arises when we take n = 1 in the same sign Kuznetsov formula with the (∞, ∞) pair of cusps, the desired identity follows.
Lemma 6.6. Both of the terms Proof. The strategy is to apply the Voronoȋ summation formula, Lemma A.30, to the sum over n, and then to bound carefully the resulting dual sum using a stationary phase-type argument (although this will be masked by integration by parts). We only cover the proof for the first term, since the second term follows by the exact same argument save for a slightly different formulation of the Voronoȋ summation formula, which gives rise to Ramanujan sums in place of Gauss sums. Dividing the n-sum and the r-integral in the definition of K + , (A.13), into dyadic intervals, we consider the sum for any N < t 2+ε g , where W and h are smooth functions compactly supported on (1, 2). Here the function h T has been absorbed into h. By Stirling's formula (2.4), we have that for j, k ∈ N 0 , where we follow the ε-convention. To understand the transform K + , we refer to [BuK17a, Lemma 3.7]. By [BuK17a, (3.61)], we must bound by O ε (t 1+ε g ). We make the substitutions r → rT and u → u/T . Repeated integration by parts with respect to r, recalling (6.7) and using (d/dr) k (tanh πrT ) k e −T for k ≥ 1, shows that we may restrict to |u| < T ε , up to a negligible error. After making this restriction, using tanh(πrT ) = 1 + O(e −T ), and taking the Taylor expansion of cosh(πu/T ), we need to show

rT rh(r)e(−ur) dr du
is O ε (t 1+ε g ). Now we integrate by parts multiple times with respect to u, differentiating the exponential e( 2 √ n c ( 1 2! ( πu T ) 2 + 1 4! ( πu T ) 4 + · · · )) and integrating the exponential e (−ur). This shows that we may restrict the summation over c to c < √ N/T 2−ε , because the contribution of the terms not satisfying this condition will be negligible. In particular, we may assume that N > T 4−ε , for otherwise the c-sum is empty. Also, the contribution of the endpoints u = ±T ε after integration by parts is negligible by repeated integration by parts with respect to r (the same argument which allowed us to truncate the u-integral in the first place). Thus we have shown that it suffices to prove that g ) for any smooth function Ω satisfying Ω (j) j 1 for j ∈ N 0 and any r ∈ (1, 2).
We now open up the Kloosterman sum and apply the Voronoȋ summation formula, Lemma A.30. Via Mellin inversion, (6.8) is equal to for any σ ≥ 0, where J ± 2tg is as in (A.14) with Mellin transform J ± 2tg given by (A.24) and (A.26). Repeated integration by parts in the x integral, integrating x −s and differentiating the rest and recalling (6.7), shows that up to negligible error, we may restrict the s-integral to (6.10) Moving the line of integration in (6.9) far to the right and using the bounds in Corollary A.27 for the Mellin transform of J ± 2tg , we may crudely restrict to n < t 2+ε g . Upon fixing σ = 0 in (6.9), so that the s-integral is on the line s = it and x −s = e(− t log x 2π ), and making the substitution x → x 2 , it suffices to prove that 74 P. HUMPHRIES AND R. KHAN GAFA is O ε (t 1+ε g ), where we have used Lemma A.31 to reexpress the sum over d as a sum over a | (c/D, n ∓ 1), and We write Ξ = Ξ 1 + Ξ 2 , where Ξ 1 is the same expression as Ξ but with the t-integral further restricted to and Ξ 2 is the same expression as Ξ but with the t-integral further restricted to (6.11) Thus Ξ 1 keeps close to the stationary point of the x-integral in the definition of I(t), while Ξ 2 keeps away.
We first bound Ξ 1 . Using the bound J ± 2tg (2(1 + it)) ε t 1+ε g in the range (6.10) from Corollary A.27 and the trivial bound I(t) 1, we get upon making the change of variables n → an ± 1 and recalling that N < t 2+ε g . We now turn to bounding Ξ 2 . The difference here is that we will not trivially bound the integral I(t). Keeping in mind the restriction (6.11), we write

ON THE RANDOM WAVE CONJECTURE FOR DIHEDRAL MAASS FORMS 75
We integrate by parts k-times with respect to x, differentiating the product of terms on the first line above and integrating the product of terms on the second line. This leads to the bound where the first term in the upper bound comes from the derivatives of Ω( √ Nx cT 2 ), while the second term comes from the derivatives of ( 2 √ N c − t πx ) −1 . By (6.10) and (6.11), the second term in this upper bound is negligible. The first term is negligible unless But the contribution to Ξ 2 of t in this range is which is trivially bounded, using the fact that J ± 2tg (2(1 + it)) ε t 1+ε g , by which is more than sufficient.
Lemma 6.12. Both of the terms Proof. The strategy is the same: to apply the Voronoȋ summation formula to the sum over n, and then to bound trivially. This time, however, there will be no stationary phase analysis, so the proof is more straightforward. Again, we will only detail the proof of the bound for the first term. Dividing as before the n-sum and the r-integral in the definition of K − into dyadic intervals, we consider the sum for any N < t 2+ε g , where W and h are smooth functions compactly supported on (1, 2), with the function h T having been absorbed into h. To understand the transform K − , we refer to [BuK17a, Lemma 3.8]. By [BuK17a,(3.68)] and the fact that by O ε (t 1+ε g ). We make the substitutions r → T r and u → u/T . Repeated integration by parts with respect to r shows that we may restrict to |u| < T ε , up to a negligible error. After making this restriction and taking the Taylor expansion of sinh(πu/T ), we need to prove that

rT rh(r)e(−ur) dr du
is O ε (t 1+ε g ). We integrate by parts multiple times with respect to u, differentiating the exponential e(− 2 √ n c ( πu T + 1 3! ( πu T ) 3 +· · · )) and integrating the exponential e(−ur). This shows that we may restrict the summation over c to c < √ N/T 1−ε , because the contribution of the terms not satisfying this condition will be negligible. In particular, we may assume that N > T 2−ε , for otherwise the c-sum is empty. Thus we have shown that it suffices to prove that is O ε (t 1+ε g ) for any smooth function Ω satisfying Ω (j) j 1 for j ∈ N 0 and any r ∈ (1, 2).
We now open up the Kloosterman sum and apply the Voronoȋ summation formula, Lemma A.30. Via Mellin inversion, (6.13) is equal to for any σ ≥ 0. We again use Lemma A.31 to write the Gauss sum over d as a sum over a | (c/D, n ± 1). Repeated integration by parts in the x-integral shows that the s-integral may be restricted to Moving the line of integration in (6.14) far to the right and using the bounds in Corollary A.27 for J ± 2tg , we may once again restrict to n < t 2+ε g . Upon fixing σ = 0 in (6.14) and bounding the resulting integral trivially by √ N cT t 1+ε g , since J ± 2tg (2(1 + it)) ε t 1+ε g , we arrive at the bound a ε t 1+ε g upon making the change of variables n → an ∓ 1 and recalling that N < t 2+ε g . Proof of Proposition 6.1 (1). It is clear that the first term in (6.5) is O ε (T 2+ε ). Lemmata 6.6 and 6.12 then bound the second and third terms by O ε (t 1+ε g ).
Proof of Proposition 6.1 (2). A similar identity to (6.5) for M hol (h hol ) may be obtained by using the Petersson formula, Theorems A.17 and A.19, instead of the Kuznetsov formula, namely (6.15) Here K hol is as in (A.18) and The first term in (6.15) is bounded by O ε (T 2+ε ). For the latter two terms, we use the methods of [Iwa97, Section 5.5] to understand K hol in place of [BuK17a, Lemmata 3.7 and 3.8] to understand K ± : this gives terms of the form n D 3/2 , r + 1 h hol (r + 1)re(−ur) dr du as well as the counterparts involving sums over c ∈ N with (c, D) = 1. The former term is then treated via the same methods as Lemma 6.6, while the latter is treated as in Lemma 6.12.

Spectral Reciprocity for the Short Initial Range
The main result of this section is an identity for We will take h to be an admissible function in the sense of [BlK19b, Lemma 8b)], namely h(t) is even and holomorphic in the horizontal strip | (t)| < 500, in which it satisfies h(t) (1 + |t|) −502 and has zeroes at ±(n + 1/2)i for nonnegative integers n < 500, while h hol (k) ≡ 0. We will later make the choice for some fixed large integer N ≥ 500 and T > 0; suffice it to say, one may read the rest of this section with this test function in mind.
Proposition 7.1. For an admissible function h, we have the identity Here L ± and L hol are as in (A.21), N and K − as in (A.13), and J ± r as in (A.14). The proof of Proposition 7.1, which we give at the end of this section, is via the triad of Kuznetsov, Voronoȋ, and Kloosterman summation formulae. Following the work of Blomer, Li, and Miller [BLM19] and Blomer and the second author [BlK19a,BlK19b], we avoid using approximate functional equations but instead use Dirichlet series in regions of absolute convergence to obtain an identity akin to (7.2), and then extend this identity holomorphically to give the desired identity.
Remark 7.6. This approach obviates the need for complicated stationary phase estimates and any utilisation of the spectral decomposition of shifted convolution which is used in [DK18b,Hum18] in the proofs of Theorems 1.7 and 1.9 for Eisenstein series. Indeed, the method of proof of spectral reciprocity in Proposition 7.1 could be used to give a simpler proof (and slightly stronger version) of [JM05, Theorem 2].
Remark 7.7. Structurally, Proposition 7.1 is proven in a similar way to [BuK17a, Theorem 1.1], where an asymptotic with a power savings is given for a moment of L-functions that closely resembles M − (h); see in particular the sketch of proof in [BuK17a, Section 2], which highlights the process of Kuznetsov, Voronoȋ, and Kloosterman summation formulae. The chief difference is the usage of Dirichlet series in regions of absolute convergence coupled with analytic continuation in place of approximate functional equations.
We define We additionally set Lemma 7.8. For admissible h and 5/4 < (s 1 ), (s 2 ) < 3/2, we have that and T + s1,s2,tg h := L + H + s1,s2,tg , L hol H + s1,s2,tg , The proof of this is similar to the proofs of analogous results in [BLM19, BlK19a,BlK19b]; as such, we will be terse at times in justifying various technical steps, especially governing the absolute convergence required for the valid shifting of contours and interchanging of orders of integration and summation, for the details may be found in the aforementioned references.
Assuming that max{ (s 1 ), (s 2 )} < 3/2, we may move the contour (s) = σ 0 to (s) = σ 1 such that −3 < σ 1 < −2 max{ (s 1 ), (s 2 )}; the Phragmén-Lindelöf convexity principle ensures that the ensuing integral converges. The only pole that we encounter along the way is at s = 2(1 − s 1 ), with the resulting residue being via Lemma A.30. For (s 2 ) > (s 1 ), the Voronoȋ L-series L(1 − s 1 + s 2 , g ψ 2 , −d/c) may be written as an absolutely convergent Dirichlet series, so that the sum over c and d is equal to The sum over d is a Gauss sum, which may be reexpressed as a sum over a | (c/D, m) via Lemma A.31. By making the change of variables c → acD and m → am, (7.13) becomes Applying Möbius inversion to (4.1), we see that (7.14) Making the change of variables a → ab and m → bm, (7.13) is rewritten as recalling that g ψ 2 being dihedral means that it is twist-invariant by χ D . So the residue (7.12) is N (s 1 , s 2 ; h)/L(2s 1 , χ D )L(2s 2 , χ D ), at least initially for (s 2 ) > (s 1 ), and this is also valid for 5/4 < (s 1 ), (s 2 ) < 3/2, since it is holomorphic in this region. Now we wish to reexpress (7.11), where σ 0 has been replaced by σ 1 , with −3 < σ 1 < −2 max{ (s 1 ), (s 2 )}. We apply the Voronoȋ summation formulae, Lemma A.30, to both Voronoȋ L-series. The resulting Voronoȋ L-series are absolutely convergent Dirichlet series; opening these up and interchanging the order of summation and integration then leads to the expression with O D as in (A.11) and H ± s1,s2,tg as in (7.10). As the Mellin transform of K − h defines a holomorphic function of s for −3 < (s) < 3, while the Mellin transform of J ± r has simple poles at s = 2(±ir − n) with n ∈ N 0 , the integrand is holomorphic in the strip −3 < (s) < 2(1 − max{ (s 1 ), (s 2 )}).
Finally, we apply Theorem A.20, the Kloosterman summation formula, in order to express this sum of Kloosterman sums in terms of Fourier coefficients of automorphic forms; the admissibility of h ensures that H ± s1,s2,tg satisfies the requisite conditions for this formula to be valid. We then interchange the order of summation and once again use Lemmas A.4 and A.8, making the change of variables m → v 1 m and n → v 2 n. In this way, we arrive at ± M ± s 2 , s 1 ; T ± s1,s2,tg h L(2s 1 , χ D )L(2s 2 , χ D ) .

Bounds for the Transform for the Short Initial Range
We take h = (h, 0) in Proposition 7.1 to be for some fixed large integer N ≥ 500 and T > 0, which is positive on R∪i(−1/2, 1/2) and bounded from below by a constant for t ∈ [−2T, −T ]∪[T, 2T ]. We wish to determine the asymptotic behaviour of the functions (L ± H ± T,tg )(t) and (L hol H + T,tg )(k) with uniformity in all variables T , t g , and t or k, where H ± tg = H ± T,tg is as in (7.4). Were we to consider t g as being fixed, then such asymptotic behaviour has been studied by Blomer, Li, and Miller [BLM19, Lemma 3]. As we are interested in the behaviour of T ± tg h as t g tends to infinity, a little additional work is required.
For s = σ + iτ with −N/2 < σ < 1, provided that additionally s is at least a bounded distance away from {2(±it − n) : n ∈ N 0 }, and for t ∈ R ∪ i(−1/2, 1/2) we have that and for t ∈ R, Res s=2(±it−n) For s = σ+iτ with −N/2 < σ < 1, provided that additionally s is at least a bounded distance away from {2(±it − n) : n ∈ N 0 }, and for k ∈ 2N, we have that and Res s=1−k−2n Proof. From [BLM19, Lemma 4], we have the bound for j ∈ {0, . . . , N}, and consequently the Mellin transform of K − h T is holomorphic in the strip −N/2 < (s) < N/2, in which it satisfies the bounds Next, we use Corollary A.27 to bound J hol k (s) and J ± t (s), as well as bound the residues at s = 1 − k − 2n and s = 2(±it − n) respectively, where n ∈ N 0 . Finally, Stirling's formula (2.4) shows that Combining these bounds yields the result.
Corollary 8.3. For fixed −N/2 < σ < 1, t 1/2 g T t g , t ∈ R ∪ i(−1/2, 1/2), and k ∈ 2N, we have that Proof. By Mellin inversion, for any 0 < σ 1 < 1. We break each of these integrals over s = σ 1 + iτ into different ranges of τ depending on the size of |t| or k relative to t g and use the bounds for the integrands obtained in Lemma 8.2 to bound each portion of the integrals. In most regimes, we have exponential decay of the integrands due to the presence of e − π 2 Ω ± (τ,t,tg) or e − π 2 Ω hol (τ,k,tg) ; it is predominantly the regimes for which Ω ± (τ, t, t g ) or Ω hol (τ, k, t g ) are zero that have nonnegligible contributions.
For (L + H + T,tg )(t), this is straightforward, noting that we can assume without loss of generality in this case that 0 < σ < 1 with σ 1 = σ; the dominant contribution comes from the section of the integral with 2|t| ≤ |τ | ≤ 4t g , as this is the regime for which Ω + (τ, t, t g ) is equal to zero.
Finally, we may again assume without loss of generality for (L hol H + T,tg )(k) that 0 ≤ σ < 1 for k ≤ t g T −1 and −N/2 < σ ≤ 0 for k > t g T −1 , since we may shift the contour with impunity in this vertical strip; once again, the dominant contribution comes from the section of the integral with |τ | bounded due to the polynomial decay of (1 + |τ |) −N −σ .

Proof of Proposition 1.21 (1): The Short Initial Range
Proof of Proposition 1.21 (1). For T < t δ/2(1+A) g , where δ, A > 0 are absolute constants arising from Theorem A.34, we use the subconvex bounds in Theorem A.34 to bound the terms L(1/2, f ⊗ g ψ 2 ) and |L(1/2 + it, g ψ 2 )| by O(T A t 1−δ g ), so that for We then use the Cauchy-Schwarz inequality, the approximate functional equation, Lemma A.5, and the large sieve, Theorem A.32, to bound the remaining moments of L(1/2, f)L(1/2, f ⊗ χ D ) and of |ζ(1/2 + it)L(1/2 + it, χ D )| 2 by O ε (T 2+ε ), and so in this range, with h T as in (6.3). Proposition 6.1 (1) then bounds M Maaß (h T ) + M Eis (h T ) by O ε (t 1+ε g ). So in this range, where h = (h T , 0) with h T as in (8.1). Noting that N (h) ε T 2+ε , Corollary 8.3 then shows that M ± (T ± tg h) are both O(T t 1−δ g ) via the Cauchy-Schwarz inequality together with the approximate functional equation and the large sieve, except in a select few ranges, namely the range , and the range k f t g /T in M hol (L hol H + T,tg ). The former two terms are then treated as we have just done for T < t δ/2(1+A) g and for t δ/2(1+A) g ≤ T < t 1/2 g , and the latter is treated via the same method, recalling that Proposition 6.1 (2) entails such bounds for holomorphic cusp forms.

Proof of Proposition 1.21 (2): The Bulk Range
The proof that we give of Proposition 1.21 (2) follows the approach of [DK18b], where an asymptotic formula is obtained for a similar expression pertaining instead to the regularised fourth moment of an Eisenstein series. As such, we shall be extremely brief, detailing only the minor ways in which our proof differs from that of [DK18b].

An application of the
. We may artificially insert the parity f into the spectral sum M Maaß (h) since L(1/2, f ⊗ χ D ) = L(1/2, f)L(1/2, f ⊗ g ψ 2 ) = 0 when f = −1; this allows us to use the opposite sign Kuznetsov formula, which greatly simplifies future calculations.
Akin to the proof of Lemma 6.4, we make use of the Kuznetsov formula associated to the pair of cusps (a, b) with a ∼ ∞ and b ∼ 1, which once again naturally introduces the root numbers of f f ⊗ χ D and of f ⊗ g ψ 2 in such a way to give approximate functional equations of the correct length for each level dividing D.
Lemma 10.2. With h as in (10.1), we have that noting that this requires Yoshida's extension of the Kuznetsov formula [Yos97, Theorem], since H(t) has poles at t = ± 1 2t g ± 2 i/2. We subsequently multiply through by λ χD,1 (n, 0)λ g ψ 2 (m)χ D (k ) √ mnk and sum over n, m, k, ∈ N. Via the explicit expression in Lemma A.8, the Maaß cusp form term is after making the change of variables m → v 1 m and n → v 2 n. We do the same with the opposite sign Kuznetsov formula associated to the (∞, 1) pair of cusps, Theorem A.16, for which the resulting Maaß cusp form term is via the explicit expression in Lemma A.9, after making the change of variables m → d 2 m/w 1 , n → v 2 n, and interchanging v 1 and w 1 . We also do the same but with m and n interchanged. We add twice the first expression to the second and the third. Using the approximate functional equations, Lemma A.5, with X = √ d 2 /v 1 and X = √ d 2 /v 2 respectively, and recalling Lemma 3.2, we obtain M Maaß (h) with h as in (10.1) as well as an error term arising from using V 1 1 in place of Following [DK18b, Section 2.3], we insert a smooth compactly supported function U (r/2t g ) as in [DK18b,(2.13)] into the integrand of the right-hand side of (10.3), absorb W (r) into U (r/2t g ), replace H(r) with its leading order term via Stirling's formula (2.4), and treat only the leading order terms V (nk 2 /D 3/2 r 2 ) and V (m 2 /D 3/2 (4t 2 g − r 2 )) of V 1 1 (nk 2 /D 3/2 , r) and V 1 2 (m 2 /D 3/2 , r) respectively, with respectively, as in [DK18b,(2.15)], at the cost of a negligible error. We are left with obtaining an asymptotic formula for We open up both Kloosterman sums and use the Voronoȋ summation formula, Lemma A.30, for the sum over n. In both sums over c, the corresponding Voronoȋ L-series has a pole at s = 1, which contributes a main term that we now calculate.

The main term.
Lemma 10.7. The pole at s = 1 in the Voronoȋ L-series contributes a main term equal to for (10.6) for some δ > 0.
Proof. For the first sum over c, the pole of the associated Voronoȋ L-series as in Lemma A.30 yields a residue equal to where 1/4 < σ 1 < σ 2 < 1/2. We use Lemma A.31 to reexpress the sum over d, a Gauss sum, as a sum over a | (c/D, m); next, we make the change of variables c → acD and m → am, then use (7.14) to separate λ g ψ 2 (am) as a sum over b | (a, m); finally, we make the change of variables a → ab and m → bm, yielding The sums over m, k, , c, a, and b in the second line simplify to L(1 + 2s 2 , χ D )L D (1 − s 1 + s 2 , g ψ 2 )L(1 + s 1 + s 2 , g ψ 2 ) ζ D (2 + 2s 2 ) .
We shift the contour in the integral over s 2 to the line (s 2 ) = σ 1 − 1/2; via the subconvex bounds in Theorem A.34, the resulting contour integral is bounded by a negative power of t g , so that the dominant contribution comes from the residue due to the simple pole at s 2 = 0, namely ds 1 s 1 .
Now we do the same with the second sum over c. We open up the Kloosterman sum, make the change of variables d → −Dd, and use the Voronoȋ summation formula, Lemma A.30, for the sum over n; the pole of the Voronoȋ L-series at s = 1 yields the term We make the change of variables x → c √ Dx 2 /2π √ m, extend the function U (r/2t g ) in the definition (10.5) of Q(r) to the endpoints 0 and 2t g at the cost of a negligible error, make the change of variables x → 2t g x, and use the definition (10.4) of V as a Mellin transform, yielding the asymptotic expression The sum over d is a Ramanujan sum, a|(m,c) aμ(c/a). We make the change of variables c → ac and m → am, then use (7.14) and make the change of variables a → ab and m → bm, leading to ( The sums over m, k, , c, a, and b in the second line simplify to L(1 + 2s 2 , χ D )L(1 − s 1 + s 2 , g ψ 2 )L D (1 + s 1 + s 2 , g ψ 2 ) ζ D (2 + 2s 2 ) .
Again, we shift the contour in the integral over s 2 to the line (s 2 ) = σ 1 − 1/2, with a main term coming from the residue at s 2 = 0 given by ds 1 s 1 .
We finish by adding together these two main contributions and observing that the resulting integrand is odd and hence equal to half its residue at s 1 = 0, namely 6 πν(D) = 2 vol(Γ 0 (D)\H) .

The Voronoȋ dual sums.
Having applied the Voronoȋ summation formula, Lemma A.30, to the sum over n in (10.6) and dealt with the terms arising from the pole of the Voronoȋ L-series, we now treat the terms arising from the Voronoȋ dual sums.
Lemma 10.8. The Voronoȋ dual sums are of size O(t −δ g ) for some δ > 0.
Proof. There are two dual sums associated to the two sums over c in (10.6). We prove this bound only for the former dual sum; the proof for the latter follows with minor modifications. The dual sum to the first term can be expressed as a dyadic sum over N ≤ t 2+ε with −3 < σ 1 < 1, where G ± 0 (s) is as in (7.5). We once again wish to determine the asymptotic behaviour of the functions with uniformity in all variables T , U , and t or k.
We briefly mention the fact that [IK04, Proposition 5.4] implies that the functions V (x, ·) appearing in Lemma A.5 are of rapid decay in x once x is much larger than the square root of the archimedean part of the analytic conductor of the associated L-function.