Poincaré series for modular graph forms at depth two. Part II. Iterated integrals of cusp forms

We continue the analysis of modular invariant functions, subject to inhomogeneous Laplace eigenvalue equations, that were determined in terms of Poincaré series in a companion paper. The source term of the Laplace equation is a product of (derivatives of) two non-holomorphic Eisenstein series whence the modular invariants are assigned depth two. These modular invariant functions can sometimes be expressed in terms of single-valued iterated integrals of holomorphic Eisenstein series as they appear in generating series of modular graph forms. We show that the set of iterated integrals of Eisenstein series has to be extended to include also iterated integrals of holomorphic cusp forms to find expressions for all modular invariant functions of depth two. The coefficients of these cusp forms are identified as ratios of their L-values inside and outside the critical strip.


Introduction
In the companion Part I [1] to this paper we introduced the Laplace equations with integers s ≥ 2 and 2 ≤ m ≤ k, and where E k are non-holomorphic Eisenstein series E k = (Im τ ) k π k (m,n) =(0,0) 1 |mτ + n| 2k . (1. 2) The modular parameter τ is in the upper half-plane, and E k is invariant under the modular transformations The Cauchy-Riemann derivatives ∇ = 2i(Im τ ) 2 ∂ τ of E k are modular forms of weight (0, −2) and the Laplacian ∆ = 4(Im τ ) 2 ∂ τ ∂τ = ∇ (Im τ ) −2 ∇ is modular invariant. The superscripts ± on F ±(s) m,k indicate that these functions are required to be even/odd under the involution τ → −τ of the upper half-plane, in line with the respective right-hand sides of (1.1). The spectrum of eigenvalues appearing in (1.1) is

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In Part I, we constructed solutions to (1.1) in terms of absolutely convergent 1 Poincaré series F ±(s) m,k (τ ) = γ∈B(Z)\SL (2,Z) f ±(s) m,k (γ · τ ) , (1.5) where the seed functions f ±(s) m,k are invariant under shifts τ → τ +n for n ∈ Z which form the stabiliser of the cusp τ → i∞ For the convenience of the reader, appendix A recaps the explicit form for our choice of representatives of these seeds f ±(s) m,k . Even though the solution (1.5) is fully explicit and has interesting structures analysed in Part I, extracting the complete Fourier expansion of F ±(s) m,k from the Poincaré-series representations is fairly involved. For the Fourier zero mode one can use the methods of [2][3][4][5] but the non-zero modes with respect to τ → τ +1 are hard to obtain. For this reason it is desirable to find alternative expressions for the modular invariants F ±(s) m,k . A family of functions with well-defined modular transformation properties is provided by modular graph forms (MGFs) [6][7][8]. These arise in the α -expansion of configurationspace integrals of genus-one closed-string amplitudes and have been studied from a physical perspective in [2,3, and a mathematical perspective in [35][36][37][38][39][40][41][42][43]. As they arise from string amplitudes, MGFs possess a lattice-sum description over discrete momenta of Feynman graphs drawn on the genus-one string world-sheet.
In particular, generating functions of closed-string integrals and their associated differential equations [28,30]  where ∈ N is called the depth of β sv . 2 Depth serves as a filtration, and the highest-depth terms in the complex-conjugation and modular properties of the β sv take a simple form. The β sv are constructed from (single-valued) iterated integrals over holomorphic Eisenstein series and should be closely related to Brown's non-holomorphic modular forms [37,38], although a precise dictionary between the two formalisms is still missing beyond depth one. Together with certain antiholomorphic integration constants determined in Part I, the complete Fourier expansion of the β sv at depths one and two is known. Therefore it seems desirable to express the F ±(s) m,k in terms of the β sv . A further advantage of using such a representation of a modular-invariant function in terms of iterated integrals is that it is unique [44], unlike lattice-sum representations that are more frequent for MGFs.
In many cases, the Poincaré-series representations in this work may be viewed as interpolating between double sums over lattice momenta and double integrals over holomorphic 1 Absolute convergence is guaranteed for m < k and for m = k a suitable regularisation was described in Part I. 2 A more detailed review of the construction and properties of the β sv can be found in section 2.

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Eisenstein series: the seeds f ±(s) m,k in (1.5) are constructed from depth-one integrals, and the sum over SL (2, Z) transformations is comparable to a single lattice momentum. However, the Poincaré sums in (1.5) often produce MGFs that require three and more lattice momenta (see Part I for details) or modular invariant functions without any known lattice-sum representation.
In Part I, we have presented a procedure for obtaining linear combinations q F ±(s) m,k of β sv of depths two and one, together with Laurent polynomial terms in y = π Im τ (that can be thought of as depth zero). These linear combinations were constructed by starting from depth-two terms that solve the Laplace equation (1.1) modulo terms of lower depth. The latter were fixed from certain requirements on the desired solutions concerning their Cauchy-Riemann derivatives and asymptotics at the cusp, see Part I for further details. However, this procedure was tailored towards solving the Laplace system in terms of the building blocks β sv of MGFs and does not guarantee that the resulting expression is modular invariant.
By comparing the dimensions of the space of solutions to (1.1) and the space of MGFs at depth two we have seen in Part I that the MGFs do not suffice to span the space of F ±(s) m,k . This is reflected in the fact that certain q F ±(s) m,k fail to be modular invariant exactly in those cases when the dimensions of the function spaces differ. In the present paper, we shall discuss how to augment the q F ±(s) m,k so that they become modular invariant and therefore equal the corresponding Poincaré series F ±(s) m,k in (1.5). In other words, we illustrate through a variety of examples that MGFs do not exhaust the modular-invariant combinations of iterated integrals of holomorphic modular forms and their complex conjugates.
As we shall see, the missing ingredients beyond the β sv are (real and imaginary parts of) iterated integrals of holomorphic cusp forms. From the Eichler-Shimura theorem [45,46] and the work of Brown [35,37,38,47] on iterated integrals of general holomorphic modular forms, it is not surprising that restricting to the β sv , that only involve iterated integrals of holomorphic Eisenstein series, is insufficient to describe the full space of modular-invariant solutions to (1.1). The first discrepancy in the dimensions of the function spaces F ±(s) m,k and MGFs appears for eigenvalues s = 6, 8, 9, 10, . . . which coincide exactly with half the modular weight of the first holomorphic cusp forms of SL (2, Z). This can be seen as a hint that cusp forms are the missing piece of the puzzle.
A further indication for the relevance of holomorphic cusp forms stems from the appearance of conjectural matrix representations of Tsunogai's derivation algebra [48] in the generating series of MGFs [28,30]. Relations in the derivation algebra are also tied to holomorphic cusp forms [49] and imply that, starting from depth two, there are combinations of the β sv that are not contained in the generating series of MGFs. Completing these to modular invariants requires holomorphic cusp forms as we shall see. This follows from the S-modular transformations of the various β sv which contain interesting so-called multiple modular values [35] that involve the values of completed L-functions of cusp forms at integers [50] and extend the set of single-valued multiple zeta values. In order to cancel these L-values from S-modular transformations in general one has to combine the β sv with iterated integrals of cusp forms. We shall work out these ideas in detail in this paper and spell out a variety of examples.

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As a byproduct of our analysis we derive that the series expansions in q = exp(2πiτ ) andq = exp(−2πiτ ) of these non-holomorphic modular objects F ±(s) m,k display very interesting structures. Firstly, the leading terms in the expansion of the even functions F +(s) m,k around the cusp Im τ 1 are Laurent polynomials in y = π Im τ that will also be referred to as "perturbative". These Laurent polynomials have a single term with a rational coefficient, a single term with a Q-multiple of the product ζ 2m−1 ζ 2k−1 , while all other coefficients are Qmultiples of odd zeta values ζ 2m−1 , ζ 2k−1 , ζ m+k+s−1 , see Part I for further details. Secondly, the infinite tower of exponentially suppressed, non-perturbative terms of the form q nqm , with both of n, m > 0, have Laurent polynomials in y with rational coefficients for both the even and odd F ±(s) m,k . Finally, and perhaps more interestingly, due to the presence of iterated integrals of holomorphic cusp forms we find that the exponentially suppressed terms of the form q nq0 (and their complex conjugates q 0qn ) with n > 0 are multiplied by particularly rich Laurent polynomials in y: their coefficients are either rationals, or Q-multiples of single odd zeta values or surprisingly rationals (or more general number-field extensions of Q) times special ratios of completed L-values associated to whichever cusp form is at play.
These results allow us to make novel predictions regarding the non-zero Fourier-mode decomposition of the Poincaré series (1.5). In particular in Part I, we have thoroughly explained how, for all the constructed seed functions f ±(s) m,k (τ ), one can exploit the results of [3] to obtain the purely perturbative Laurent polynomials in y. To pass from the seed function f ±(s) m,k (τ ) to the actual associated Poincaré series (1.5) one needs to use a particular integral transform detailed for instance in appendix A of Part I. Such a mapping between seed and modular function can also be used to formally obtain the non-zero modes for the modular invariant Poincaré series. However, the computation of non-zero modes from this integral transform of the seed involves very complicated Kloosterman sums and the analogue of the analysis in [3] to this case is currently unknown. Despite this lack of full control over Kloosterman sums, our results imply that these Kloosterman sums must contain completed L-values of holomorphic cusp forms. It would be extremely interesting to extend the results of [3] to the non-zero Fourier mode sectors, thus deriving directly from the seed functions the exponentially suppressed terms q nq0 and q 0qn with n > 0 including their Laurent polynomials.
Outlook. The results of Part I and this work raise a variety of follow-up questions of relevance to string perturbation theory, algebraic geometry and number theory. Most obviously, the Fourier expansion of depth-two MGFs and their extension by iterated integrals of holomorphic cusp forms call for generalisations to higher depth. Among other things, (single-valued) multiple zeta values beyond depth one, iterated integrals that mix holomorphic Eisenstein series with cusp forms and generalisations of L-values [51] are expected to play a key role starting from depth three. The respective seed should have one unit of depth less than its modular invariant Poincaré sum, and it will be rewarding to study this kind of recursive structure at general depth.
Furthermore, a detailed connection with the recent mathematics literature promises powerful synergies. Various important properties of the β sv at general depth will follow once their precise relation to Brown's non-holomorphic modular forms is established. More-JHEP01(2022)134 over, we note that iterated integrals of cusp forms and their Poincaré sums have featured prominently in recent work [52] that also relates to so-called higher-order modular forms. Certain Laplace systems similar to (1.1) but at depth three have also been studied recently in [53]. These references can provide useful guidance when generalising our work.
Outline. In section 2, we review the basic properties of iterated integrals of holomorphic modular forms, with particular emphasis on their modular properties and certain SL(2, Z) group cocycles that arise. In section 3, we then use these results in the analysis of the modular invariant solutions to the Laplace equations. We further show how to combine the β sv with iterated integrals of cusp forms based on the vanishing of the cocycles thus restoring modularity. We explain the relation between Tsunogai's derivation algebra and the modular invariant Laplace eigenfunctions in section 4. Further properties of the solutions to (1.1), such as connections to Kloosterman sums, are discussed in section 5. An ancillary file that accompanies the arXiv submission and the supplementary material of the journal publication of this work contains many examples and explicit expressions related to the functions F ±(s) m,k .

Basics of iterated integrals
This section is dedicated to the central aspects of iterated integrals as well as their differential and modular properties as they enter our analysis. Frequent use will be made of the Cauchy-Riemann derivatives and the Laplace operator As in the equation above, we often use the symbol y = π Im τ , and powers of y satisfy

Iterated integrals of Eisenstein series
In the present work, we shall only require the depth-one and depth-two versions of the single-valued iterated Eisenstein integrals (1.7). These are defined by the integrals [30] 3 3 We shall often suppress the argument τ of various functions to simplify the notation.

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with 0 ≤ j ≤ k−2 and The holomorphic Eisenstein series are normalised as with divisor sum σ s (n) = d|n d s . The integrals (2.4) have to be understood with tangential base-point regularisation [35] and satisfy the shuffle relations as well as the differential equations [30] −4π∇β sv j The objects α[ j 1 j 2 k 1 k 2 ] appearing in (2.4b) are purely antiholomorphic functions and constrained by the shuffle relation (2.6). They are not fixed by the differential equation (2.7) and therefore referred to as integration constants -see [30] for a detailed discussion. A method to determine them from the reality properties and Laplace equations of F ±(s) m,k is discussed in Part I, and a large number of examples can be found in the supplementary material.

Fourier expansions of iterated Eisenstein integrals
The compact definition (2.4) of the β sv can be unpackaged to yield expressions in terms of other iterated integrals of the form [20,54]

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and their complex conjugates with integers k, k 1 , k 2 = 4, 6, 8, . . . and p, p 1 , p 2 ≥ 0. In the above expressions, 0 p is a placeholder for p successive zeroes (reminiscent of integration kernels G 0 0 = −1 [54]), and the G 0 k are obtained from the holomorphic Eisenstein series G k by removing the zero mode The removal of the zero mode destroys the good modular transformations of G k but renders the integrals convergent without regularisation. Moreover, the integrals E 0 (. . .) have fully explicit q-expansions, e.g.
see [20, eq. (2.21)] for arbitrary depth. This can be used to obtain the full Fourier expansions of the β sv . The rewriting of the β sv in terms of the E 0 (. . .) requires a number of steps that are well-understood and whose precise form can be found in section 3.3 and appendix D of [20] as well as appendix G of [30].
The non-holomorphic Eisenstein series E k defined in (1.2) can be decomposed in terms of iterated Eisenstein integrals as follows [7,55]: where +c.c. instructs us to add the complex conjugate, making E k real-analytic and even under τ → −τ . We also used y = π Im τ as we shall do frequently. This relation between JHEP01(2022)134 E k and the depth-one β sv , already present in [30], comes directly from (2.4a) when using the relation between G k and G 0 k as well as tangential base-point regularisation. From both the lattice-sum representation (1.2) and the final form of (2.12), one can show the well-known formula for the k-th Cauchy-Riemann derivative of E k [8]: where we have used Euler's formula relating the even Bernoulli numbers to the even Riemann zeta values 2ζ 2n = (−1) n+1 4 n π 2n (2n)! B 2n , n = 1, 2, 3, . . . . (2.14) We also record the following general formula for any integer s > 0 and holomorphic function f (τ ) irrespective of its modular properties. If f (τ ) has a q-expansion in terms of positive powers of q only, the integral in (2.15) is well-defined without tangential base-point regularisation. With (2.15) and (2.3) it is easy to demonstrate (2.13). For the Laplacian there is a similar lemma given by Besides direct evaluation of the Laplacian on the integral, we can also consider (2.16) by Fourier expanding the integrand f (τ ). Specialising to the case of a single Fourier mode f (τ ) = e 2πinτ with n > 0, the integral can be evaluated in terms of Bessel functions K s−1/2 giving since the Laurent monomials y k and y 1−k in (2.12) are in the kernel of (∆ − k(k−1)).

Multiple modular values
Besides the version of iterated Eisenstein integrals in (2.8), we shall also make use of that are, up to normalisation conventions, Brown's holomorphic iterated Eisenstein integrals and require tangential base-point regularisation [35]. At depth one, this regularisation means treating the zero mode of (2.5) differently, while the depth-two generalisation can be found in [35, eq. (4.13)]. A more general translation of (2.19) into the integrals (2.8) can be found in [20], and the depth-one instance of the dictionary is The extra term proportional to ζ k is due to E 0 (k, 0 p ) being defined in terms of G 0 k , thus lacking the zero mode ζ k when compared to G k appearing in (2.19), see (2.9).
The virtue of the definition (2.19) is that it is easier to describe the behaviour under S-modular transformations [20,35]: The objects m[ ··· ··· ] appearing in this equation do not depend on τ -they are examples of multiple modular values [35] and correspond to period integrals which are obtained formally as limits τ → 0 of (2.19). This limit is divergent and has to be treated again with tangential base-point regularisation. One way of doing this is to consider at depths one and two where we rewrote the S-modular behaviour (2.22) and evaluated this expression at the self-dual point τ = i. We recall that the integrals (2.19) are well-defined for any finite τ , using tangential base-point regularisation at the upper integration boundary τ → i∞. The choice of the self-dual point τ = i in (2.24) is arbitrary (any pair of S-dual points would do) but convenient for numerical evaluations.

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For depth one we can work out the multiple modular values explicitly as and these correspond to periods of the holomorphic Eisenstein series [56]. The j = 0 case can also be obtained as a limit after using the functional relations of the zeta function.
Since k ≥ 4 is an even integer, the multiple modular values of depth one vanish for even 0 < j < k−2 as they involve the zeta function evaluated at a negative even integer. For depth two, numbers beyond (multiple) zeta values can occur [35,50]. We will discuss further properties of multiple modular values and how they arise directly in the S-modular transformation of the β sv in section 2.3.

Iterated integrals of cusp forms
We now let ∆ 2s (τ ) denote a holomorphic cusp form of weight 2s ∈ {12, 16, 18, . . .}. Then we define the analogue of (2.8a) as Since ∆ 2s is a cusp form, this integral is well-defined for any p ≥ 0; however, in everything that follows we shall only encounter the usual range of values 0 ≤ p ≤ 2s−2. The cusp forms in this definition are Hecke normalised with ∆ 2s (τ ) = q + O(q 2 ) such that the transcendentality of the iterated integral (2.26) is given by p+1, just like for (2.8a). 4 The objects that are on a similar footing are ∆ 2s and G k (2πi) k since both have algebraic Fourier coefficients, for instance If the cusp form has Fourier expansion ∆ 2s (τ ) = ∞ n=1 a(n)q n then (2.28)

Real-analytic integrals of holomorphic cusp forms
From the fourth line of (2.12), we see that one can define even and odd analogues of the non-holomorphic Eisenstein series by trading G k in the integration kernel for holomorphic cusp forms ∆ 2s , where we have fixed a convenient normalisation. This function satisfies from (2.15) and (2.16) Clearly, the even function H + ∆ 2s is obtained from the cusp form ∆ 2s in the same way as E s is obtained from G 2s . Moreover H − ∆ 2s is its odd cousin, and the appearance of an odd analogue of E k (denoted by E m,k is discussed in section 5.5 of Part I. Variants of (2.29) with more general exponents (τ −τ 1 ) j (τ −τ 1 ) 2s−2−j , j = 0, 1, . . . , 2s−2 arise from Cauchy-Riemann derivatives of H ± ∆ 2s and have been studied in [52]. Following our discussion around (2.17) we expose the q-expansion of H ± ∆ 2s by rewriting them as a finite sum over the E 0 (∆ 2s , 0 p ; τ ) in (2.26) and (2.28), which is the direct analogue of the second line in (2.12) for E s and the definition of E (−) s in section 5.5 of Part I. Given that the transcendental weight of iterated integrals E 0 (∆ 2s , 0 p ; τ ) is p+1 in our conventions, their combinations with y = π Im τ of weight 1 in (2.31) assign transcendentality s to H ± ∆ 2s . The sum over Fourier modes in the first line of (2.31) can also be recast in terms of Bessel functions using (2.17) as Even though the functions H ± ∆ 2s and the modular invariant functions defined in [47] are both real-analytic and are both obtained from iterated integrals of holomorphic cusp forms ∆ 2s , they differ crucially in their modular properties. In particular, as we shall show next, the H ± ∆ 2s are not invariant under S-modular transformations. 5 The seed functions of F +(s) m,k constructed in Part I additionally involve Q-multiples of y m+k and ζ2m−1y 1−m+k .

Modular properties
For studying modular transformations of functions F (τ ) on the upper half-plane, it is convenient to introduce the following cocycles under the generating T-and S-transformations of SL(2, Z): When both of them vanish, F is invariant under modular transformations and in general there is a connection to the group cohomology of SL(2, Z) [57].
However, the functions H ± ∆ 2s have a non-trivial cocycle under the S-transformation where in the last equality the upper line is for H + ∆ 2s and the lower line for H − ∆ 2s . The function Λ(∆ 2s , t) appearing in the above expressions is real-valued for real t and corresponds to the completed L-function of the cusp form ∆ 2s (τ ) = ∞ n=1 a(n)q n of weight 2s defined by where the sum converges absolutely for Re(t)>s+ 1 2 , using [58] and the improved growth of the Fourier coefficients of cusp forms following from [59]. 6 From its integral Mellinform representation, it is well-known that the completed L-function enjoys an analytic continuation to the complex plane and satisfies the functional relation (2.37) The interval t ∈ (0, 2s) is called the critical strip, and (2.35) implies that the failure of modularity of H ± ∆ 2s involves the completed L-function evaluated at integers inside the critical strip, with only odd integer arguments t contributing to H + ∆ 2s and only even integers for H − ∆ 2s .

Integrals of Hecke normalised holomorphic cusp forms
If ∆ 2s is a normalised eigenform of all Hecke operators T n and of weight 2s, i.e., for all n > 0, implying Hecke normalisation a(1) = 1, then it is moreover known that the values Λ(∆ 2s , t) for all even t inside the critical strip are related, as are all the values for odd t [45,46,60]. The ratios between the even (or the odd) values must belong to the number-field extension of Q defined by the Fourier coefficients {a(n), n ∈ N} of ∆ 2s . The first time a non-trivial extension arises is for cusp forms of weight 2s = 24 where there are two linearly independent cusp forms and the number field is Q( √ 144169). The non-trivial Galois automorphism of the number field exchanges the two independent Hecke eigenforms.
Therefore, for (normalised) Hecke eigenforms ∆ 2s of weight 2s we find that we can rearrange (2.35) to with coefficients c ± from the number field associated with ∆ 2s defined by for = 0, 1, . . . , 2s−2. The polynomials arising in (2.39) have also appeared in [35,50] and are related to period polynomials as will be also discussed in section 4.1.1. We had argued above that H ± ∆ 2s should be assigned transcendentality s and this is consistent with the fact that there is no transcendentality carried by L-values inside the critical strip (0, 2s) like Λ(∆ 2s , 2s−1) and Λ(∆ 2s , 2s−2). As we will see later on, the value Λ(∆ 2s , 2s+m) has transcendentality (m+1), so that just at the end of the critical strip we have Λ(∆ 2s , 2s) of transcendentality one. This is analogous to the Riemann zeta function whose transcendental weight grows in the same way from the upper end of its critical strip (0, 1). The transcendentality of the cocycles (2.39) is therefore determined by the prefactor, including y = π Im τ , and is also given by s, consistent with that of H ± ∆ 2s itself. Note that with our definition (2.29) for the iterated integral of a cusp form, the Scocycles in (2.35) or (2.39) are two-variable generalisations of the classic period polynomial associated to the same cusp form [56], with δ S H ± ∆ 2s (τ ) playing the roles of the even/odd

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part of said period polynomial. Furthermore, along the same lines as Manin's original work [60], we have that δ S H ± ∆ 2s (τ ) satisfies the two cocycle conditions [57] δ S H ± ∆ 2s (τ ) where | denotes the SL(2, Z)-action on τ and U = TS is an order 3 generator of SL(2, Z). When applied to (2.35), the first cocycle condition (2.41a) is equivalent to the reflection formula (2.37) for the even/odd values inside the critical strip. Similarly the second condition (2.41b), together with the Hecke condition (2.38), is equivalent to the statement that all the ratios between the even (or odd) critical values must be in the number field generated by the Fourier coefficients. In (2.39) we chose to factorise out Λ(∆ 2s , 2s−1) and Λ(∆ 2s , 2s−2), respectively, thus making a particular choice for what are usually called the (holomorphic) periods of the cusp form ∆ 2s , sometimes denoted by ω ± ∆ 2s [50].

Example with s = 6
As a concrete example we can study the cusp form of lowest weight 2s = 12, i.e. the Ramanujan cusp form ∆ 2s = ∆ 12 = ∞ n=1 τ (n)q n with τ (1) = 1. Since the vector space of cusp forms at weight 2s = 12 is one-dimensional we trivially have that ∆ 12 is a normalised Hecke eigenform and obviously the associated number field is simply Q, i.e. τ (n) ∈ Q for all n > 0. Following [60] we have the following number-field relations amongst the completed L-values, even and odd, inside the critical strip: where the remaining values can be obtained via the reflection formula Λ(∆ 12 , 12 − t) = Λ(∆ 12 , t).
If we compute the cocycles (2.39) defined above we obtain Similar expressions for the cocycles of H ± ∆ 16 and H ± ∆ 18 can be found in appendix B, while in the supplementary material the expressions are given up to modular weight 2s = 26.
In summary, we can construct even and odd solutions H ± ∆ 2s of the homogeneous Laplace equation (1.1) whenever we have a holomorphic cusp form ∆ 2s of weight 2s. These homogeneous solutions are expressible through iterated integrals of ∆ 2s of depth one, see (2.29).

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They are not modular invariant but their failure of modularity is characterised by a single number that is a value of the completed L-function inside the critical strip. This number is expected to be independent over Q from the set of multiple zeta values.

Properties of multiple modular values and the β sv
We now study the multiple modular values defined in (2.23) in more detail and also present expressions for the S-modular transformation of the β sv introduced in (2.4).
From their definition (2.23) the multiple modular values inherit the shuffle relations of the (regularised) iterated integrals in (2.19) By applying another S-transformation to (2.22) one can show the reflection properties Under complex conjugation they satisfy The transcendentality of the multiple modular values as defined in (2.23) is given by i k i . For the depth-one case (2.25) this is evident from the fact that (2πi) k−j−1 has transcendentality k−j−1, ζ j+1 has transcendentality j+1 and ζ j+2−k has transcendentality zero. 7 In the general case, this follows from the definition (2.19) by realising that G k in our convention has transcendentality k, see (2.5). The iterated integrals (2.8a) and (2.8b) therefore have transcendentality p+1 and p 1 +p 2 +2, respectively.

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and (2.46), and they show that reduced multiple modular values vanish if both of j 1 and j 2 are odd, whereas cases with both j 1 and j 2 even yield the product 2m Moreover, reduced multiple modular values inherit the shuffle property and so satisfy

Depth one reduced multiple modular values and β sv modular transformations
At depth one, (2.25) leads to the following explicit expressions where the vanishing of all cases with j = 1, 2, . . . , k−3 is in agreement with their occurrence in certain coboundary polynomials [35,38,50]. They appear in the transformation of the depth-one β sv according to

Modular transformation of β sv at depth two
In the same way the depth-one reduced multiple modular values arise in the modular transformations of the depth-one β sv , the depth-two M[ j 1 j 2 k 1 k 2 ] arise in the modular transformation of the β sv at depth two. Performing the calculation based on the integral representation (2.4b) one can show that

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Here, the C[· · · ] are pure depth-zero terms, i.e. rational functions of τ andτ multiplied by rational combinations of odd zeta values and powers of π, that can be traced back to the modular transformation of the α[· · · ]. Their definition is most conveniently given in terms of the shorthand for the contributions of the antiholomorphic α[· · · ] to (2.4b), namely While the antisymmetry of β sv (α) in (j 1 , k 1 ) ↔ (j 2 , k 2 ) clearly propagates to it is not immediately obvious from the definition (2.53) that the depth-one terms cancel. In fact, one may view the dropout of E 0 (k, 0 p ) from (2.53) as a defining property of α[· · · ].

Examples at depth two expressible via zeta values
For reduced multiple modular values at depth two, no analogue of the closed formula (2.49) is known. We begin with a few illustrative examples. In the (G 4 , G 4 ) sector we have [50] M[ 0 0

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We note that all rational multiples of π 8 present in the individual terms such as [50] m[ 0 2 disappear in the combination (2.47).
More remarkable instances of such simplifications occur for higher weight, for instance in the reduced multiple modular values of the (G 4 , G 6 ) sector [50] M[ 0 0 Rational multiples of π 10 and π 2 ζ 3,5 drop out from all the reduced counterparts M[ j 1 j 2 4 6 ] even though they appear in individual multiple modular values such as [50] m[ 0 0 We expect that more generally, the double zeta values ζ n 1 , will drop out in the combination to their reduced counterparts at arbitrary weight.

Examples at depth two involving L-values
Individual m[ j 1 j 2 k 1 k 2 ] at weight k 1 +k 2 ≥ 14 involve certain "new numbers" [50] such as c(∆; 12) and L-values of holomorphic cusp forms outside the critical strip. However, the reduced combinations (2.47) are conjectured to feature only single zeta values, L-values of cusp forms and powers of π. This can be checked from the M[ j 1 j 2 k 1 k 2 ] provided in the supplementary material up to k 1 +k 2 ≤ 28 and the examples presented in this work.

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The simplest examples of reduced multiple modular values involving non-critical Lvalues occur in the (G 4 , G 10 ) and (G 6 , G 8 ) sectors [50] In the ancillary file accompanying the arXiv submission, the supplementary material in the journal publication of this work and Part I, we present the complete list of reduced

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multiple modular values at depth two up to k 1 +k 2 = 28. The values there were fixed by knowing on which numbers to expand the reduced multiple modular value [50] and fitting the rational coefficients via numerical evaluation. 10 Since we assign transcendental weight m+1 to Λ(∆ 2s , 2s+m), the explicit expressions are compatible with the transcendentality

Modular properties of solutions to the Laplace equations
The problem that motivates this work is to find modular invariant solutions F ±(s) m,k to the Laplace equations (1.1). In Part I, we showed how a leading-depth solution could be constructed in terms of the β sv and how to complete it by including lower-depth β sv terms. We already pointed out in Part I that the modular invariance of the resulting function q F ±(s) m,k is not guaranteed by the construction that was only tailored to produce an exact solution of (1.1) in terms of various β sv . Since the β sv have the more involved modular properties presented above, this does not necessarily entail modular invariance of q F ±(s) m,k . As we argued, failure of modular invariance can and will arise whenever the space of F ±(s) m,k is larger than the space of modular graph forms constructed from β sv at depth ≤ 2.
The explicit counting done in section 3.6 of Part I showed that this can happen at Laplace eigenvalue s(s−1) whenever there are holomorphic cusp forms at modular weight 2s. This is not surprising since the generating series of MGFs [30] only contains combinations of β sv that are compatible with the relations in Tsunogai's derivation algebra, see section 4 for a more detailed discussion of the corresponding 'dropouts' from the β sv . As the relations in the derivation algebra are triggered by holomorphic cusp forms [49] we have a consistent picture that iterated integrals of such cusp forms should arise. They also feature naturally in the space of real-analytic modular functions studied in [37,38,47,52,53].
For every holomorphic cusp form ∆ 2s of modular weight 2s, we have constructed even and odd homogeneous solutions H ± ∆ 2s to the Laplace equation in (2.29) and we have also shown that they are not modular invariant, see (2.39). Therefore, if the combination q F ±(s) m,k of β sv is not modular invariant but solves the correct inhomogeneous Laplace equation, we can consider To answer this question we have to determine the modular transformation of q F ±(s) m,k . As this is a combination of β sv , potentially multiplied by powers of y, we have to use the S-modular transformation of the β sv discussed in section 2.3. As is evident from (2.50) and (2.51), the modular transformation generates special combinations of multiple modular

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values and additional depth-zero terms C[· · · ], which can be derived from the relevant α[· · · ] appearing in (2.4b). Ultimately, we obtain the explicit S-modular transformation of q F ±(s) m,k in terms of these multiple modular values. In order for a failure of modularity of q F ±(s) m,k to be cancelled by that of H ± ∆ 2s , one must obtain very specific combinations of multiple modular values that ultimately are proportional to the same polynomial in τ and τ given in (2.39). This is true in all examples and should follow from the general analysis in [35,37]. We exemplify the mechanism for a variety of weights and Laplace eigenvalues. In all cases, their construction from L-values assigns transcendental weight m+k−s to the constants a ± ∆ 2s ,m,k in (3.1) such that their combination with H ± ∆ 2s of weight s matches the transcendentality m+k of F The occurrence of these extra terms H ± ∆ 2s was also argued for on general grounds from the Cauchy-Riemann equation in Part I. We shall initially focus on the case when S 2s is one-dimensional and defer more general cases to sections 3.3 and 3.4. We also note that the presence of H ± ∆ 2s in the modular-invariant solution F

Examples involving the Ramanujan cusp form
The cusp form of lowest weight is the Ramanujan cusp form ∆ 12 = ∞ n=1 τ (n)q n . Since it has holomorphic modular weight 12, its iterated integrals can arise as the modular completion of q F ±(s) m,k at eigenvalue s = 6. According to the spectrum (1.4) this happens first for m+k = 7 in the odd sector and for m+k = 8 in the even sector.  τ +τ (τ −τ ) 5 + . . . , Using these and similar results one can show that with the same polynomial in τ,τ in both cases These cocycles turn out to be proportional to those of H − ∆ 12 in (2.43b), We can therefore form the linear combinations Choosing a different normalisation for H ± ∆ 2s one could turn the quotient in (3.7) into a multiplication by an L-value only. Note that the transcendental weight six of H − ∆ 12 , weight zero of Λ(∆ 12 , 10) and weight one of Λ(∆ 12 , 12) ensure that both terms on the right-hand sides of (3.7) have the expected weight seven.
We note that the combination is an eigenfunction of the Laplacian with eigenvalue 30 which is perfectly modular invariant on its own without the need of adding any iterated integral of ∆ 12 . This is one of the examples of modular objects analysed in Part I that are expressible in terms of β sv , y and odd zeta values.

Even functions for (m, k) = (2, 6), (m, k) = (3, 5) and (m, k) = (4, 4)
In the even sector, the first occurrence of the eigenvalue s = 6 is for F   From this we also see that the following combinations are modular invariant without the inclusions of an iterated integral of a holomorphic cusp form, Similar to the odd case, the right-hand sides of (3.11) have uniform transcendental weight eight by combining weight six of H + ∆ 12 with weight zero of Λ(∆ 12 , 11) and weight two of Λ(∆ 12 , 13).

Examples with cusp forms of higher weight
We have performed the same analysis as in the previous section for all F

An example involving the two weight 24 cusp forms
As explained in Part I [1], as we increase the total transcendental weight w = m+k we encounter higher and higher eigenvalues s ≤ k+m−1 in the spectrum, see (1.4). This in turn means that the obstructions to finding modular solutions to the Laplace systems (1.1) are related to iterated integrals (2.29) of cusp forms ∆ 2s of higher and higher modular weight 2s. Denoting by S 2s the vector space of holomorphic cusp forms for SL(2, Z) with even integer modular weight 2s, we have the classic result [68] dim S 2s =    2s 12 − 1 2s ≡ 2 mod 12 ,  that the first instance for which the eigenvalue s = 12 appears is for the odd sector and with transcendental weight w = m+k = 13. We are then led to expect that a modular invariant solution to (1.1) in this sector must take the form with m+k = 13 and m, k ≥ 2, where we use a basis of Hecke eigenforms subject to (2.38) and denoted by ∆ 24,i , ∆ 24,ii .
Such a basis can be constructed by considering the linear combination α∆ 2 12 +β∆ 12 G 12 , i.e. the most general holomorphic cusp form of weight 2s = 24. Then, the real coefficients α, β for Hecke eigenforms are obtained by imposing the resulting Fourier coefficients to be multiplicative, i.e. a(m)a(n) = a(m·n) for m, n coprime: gcd(m, n) = 1. This procedure constructs the two Hecke eigenforms 11 whose Fourier coefficients lie in the number field Q( √ 144169): The number field generated by the above Fourier coefficients has a non-trivial Galois automorphism σ ∈ Aut Q Q( √ 144169) , which acts as σ : √ 144169 → − √ 144169 and under which the two basis elements are exchanged, i.e. σ(∆ 24,i ) = ∆ 24,ii .
In the basis of Hecke eigenforms we have that all the even/odd completed L-values inside the critical strip are Q( √ 144169) multiples of one another [60] and the S-cocycles for ∆ 24,i and ∆ 24,ii can be put in the form (2.39). The Galois automorphism exchanges the two cocycles, as well as the completed L-values.
Following the same types of arguments that led to (3.6), we see that in general the S-cocycle for q F where we have split the result into three factors: the ratio of L-values carries transcendental weight one, the middle factor is valued in the number field Q( √ 144169) and corresponds to the inverse of the Petersson-Haberland pairing [69][70][71] between two properly normalised polynomials associated with the cusp from ∆ 2s . The first factor is a rational number multiplying the vector in the number field and we refer to [35] for why this splitting occurs. The constants a − ∆ 24,ii ,m,k for the second cusp form can be directly obtained by the application of the Galois automorphism to a − ∆ 24,i ,m,k , i.e. a − ∆ 24,ii ,m,k = σ(a − ∆ 24,i ,m,k ) that acts on the number-field-valued middle factor by the Galois action and on the L-values by σ(Λ(∆ 24,i , t)) = Λ(∆ 24,ii , t).
With the above values we obtain that the combinations (3.17)  .
In section 4, we discuss the relation between linear combinations of this type and Tsunogai's derivation algebra.
The complete list of modular completions q F m,k with m+k ≤ 14 can be found in the supplementary material. From their general form given in (2.39) we know that the cocycles δ S H ± ∆ 2s can be normalised so that we have factorised out the completed L-values Λ(∆ 2s , 2s−1) in the even case and Λ(∆ 2s , 2s−2) in the odd case, times a rational function in τ,τ with coefficients in K ∆ 2s , the number field generated by the Fourier coefficients of ∆ 2s .

Structure for general weight
To understand the generic structure of the coefficients a ± ∆ 2s ,m,k we can analyse in more depth their transcendentality properties. Since the iterated-integral representation (2.31) assigns transcendentality s to H ± ∆ 2s , the coefficients a ± ∆ 2s ,m,k must have weight m+k−s in order to arrive at combinations F ±(s) m,k of weight w = m+k in (3.1). On these grounds, by the transcendentality of Λ(∆ 2s , 2s+ −1) outside the critical strip ≥ 1, we are led to conclude that where we have q ± ∆ 2s ,m,k ∈ Q, while κ ± ∆ 2s ,m,k ∈ K ∆ 2s is given by the inverse of the Petersson-Haberland pairing between the two cocycles δ S H + ∆ 2s (τ ) and δ S H − ∆ 2s (τ ) properly normalised, see [35]. Given that m+k+s is even (odd) for even functions F Note that, as discussed previously, to determine the number field K ∆ 2s , which contains the Fourier coefficient of ∆ 2s ∈ S 2s , one has to diagonalise the Hecke operators (2.38) in S 2s . The nature of this number field K ∆ 2s is clarified 12 by the Maeda conjecture [72], which states that the characteristic polynomials of the Hecke operator T n are irreducible JHEP01(2022)134 over Q. In a certain sense the number field K ∆ 2s is "maximal" in that the Maeda conjecture suggests that the associated Galois group is the full symmetric group S d with d = dim S 2s .
For example at weight 2s = 28 with d = dim S 28 = 2 we have K ∆ 28 = Q( √ 18209), for 2s = 30 we have once more d = 2 and now K ∆ 30 = Q( √ 51349), while moving to weight 2s = 36, the lowest weight for which d = 3, we have K ∆ 36 = Q[x]/(x 3 −12422194x−2645665785), which means that the algebraic number-field extension of Q contains all three roots of the polynomial. The Maeda conjecture, although still unproven, has been extensively tested for all modular weights up to 2s = 12000, see [73] where strong evidence is presented to support its validity. More examples of such number fields can be found on the very comprehensive LMFDB database [74] of L-functions and modular forms.
As a final comment we stress that the action of a non-trivial element, σ, of the Galois automorphism group Aut Q (K ∆ 2s ) allows us to relate the constants a ± ∆ 2s ,m,k for different Hecke eigenforms σ(a ± ∆ 2s ,m,k ) = a ± σ(∆ 2s ),m,k , hence if Maeda's conjecture were to be true it would be enough to find one such number a ± ∆ 2s ,m,k for a single Hecke cusp ∆ 2s to deduce all the others.

Selection rules on β sv from Tsunogai's derivation algebra
In this section, we study the interplay between the generating series of MGFs introduced in [28], the modular invariant functions F ±(s) m,k and an abstract algebra on generating derivations k introduced by Tsunogai [48] that is related to holomorphic cusp forms [49]. This connection will clarify why some instances of q The generating series of MGFs introduced in [28] captures the structure of the αexpansion of certain genus-one integrals in closed-string amplitudes that eventually comprise all MGFs when integrating over sufficiently many torus punctures. More specifically, the first-order differential equations in τ of this generating series is solved by with conjectural matrix representations of k acting on suitable initial values as τ → i∞ that are series in zeta values [30]. 13 In [28,30] the k were explicit finite-dimensional matrix operators that were checked at low orders to obey the relations of Tsuongai's derivation algebra, and this is conjectured to hold to all orders. Here, we think of the k as the abstract generators of Tsuongai's derivation algebra. In (4.1) and the following we make

Overview of k relations at depth two
Tsunogai's derivations satisfy a wealth of commutator relations. First of all, [ 2 , k ] = 0 with k = 0, 2, 4, . . . identifies 2 to be a central element which does not occur in the series (4.1). On the remaining derivations, ad 0 enjoys the nilpotency properties such that the j i = 0 terms in (4.1) exhaust the maximal non-vanishing nested commutators . Apart from the relations (4.3) that take a simple and universal form for all k, commutators of k 1 , k 2 , . . . at weight k 1 +k 2 + . . . ≥ 14 obey more involved identities starting with [49,54,75]  and already illustrates a generic feature: when referring to the number of k =0 in a nested commutator as its depth, 14 indecomposable relations involving more than two k usually mix terms of different depth. The depth-two terms in the first line of (4.5) affect the appearance of β sv of depth two in (4.1) while the depth-three terms in the second line are related to β sv of depth three that are part of the suppressed terms O( 3 k ) in (4.1) and beyond the scope of this work.
Relations in the derivation algebra are also assigned a notion of depth according to the maximal depth of the nested commutator therein, e.g. (4.5) is said to have depth three. As will be reviewed in the remainder of this subsection, Pollack determined the depth-two terms in indecomposable relations of arbitrary depth in closed form. More precisely, their rational coefficients are determined from the period polynomials of holomorphic cusp forms in [49].

Cusp forms and depth-two relations
In order to concisely relate the coefficients in relations like (4.4) or (4.5) to holomorphic cusp forms ∆ 2s of modular weight 2s, we follow the conventions of [49] for period polynomials where the arguments t of the L-function Λ(∆ 2s , t) are all within the critical strip t ∈ (0, 2s). Moreover, we introduce the even and odd parts of the period polynomials (4.6) via The rational coefficients r + ∆ 2s (X, Y ) X a Y b of the non-zero powers X a Y b with a, b > 0 are easily seen to match those in the depth-two relations (4.4) among [ k 1 , k 2 ] with k 1 +k 2 = 2s+2. More generally, the depth-two relations in the derivation algebra are given by in terms of the even parts of period polynomials (4.6) of cusp forms [49]. The extremal terms ∼ (X 2s−2 − Y 2s−2 ) in (4.8) are mapped to coefficients of the vanishing commutators [ 2 , * ] in (4.9), that is why their more involved coefficients 36 691 , 180 3617 and 2250 43867 do not enter the depth-two relations (4.4).

Cusp forms and higher-depth relations
Also for higher-depth relations such as (4.5), the coefficients of the depth-two terms [ are determined by the period polynomials (4.6). More specifically, relations of even (odd) depth are governed by the even part r + ∆ 2s (the odd part r − ∆ 2s ) in (4.7). At weight 2s = 12, 16, 18, the coefficients in the odd counterpart of (4.8) enter relations of depth three, five, . . . such as (4.5). By rewriting the first line of (4.5) as − 160 = 0 mod depth 3 , with d ≥ 2 subject to alternating symmetry properties t d p,q = (−1) d−1 t d q,p . The ratios of factorials in (4.12) are engineered such that t d p,q is firstly annhilated by p+q−2d+1 powers of ad 0 , i.e. ad p+q−2d+1 In the simplest instances of (4.12), , 15 We depart from Pollack's conventions for the commutators (4.12)  where t d 2,q = 0 since 2 commutes with all the k . Indecomposable relations at depth three due to (4.15)

Modular graph forms and k relations at depth two
Based on the relations among depth-two commutators [ k 2 ] reviewed above, we shall now describe the dropouts of iterated Eisenstein integrals of depth two from the generating series (4.1) of MGFs. We will be interested in the modular invariant cases where the entries of β sv [ j 1 j 2 k 1 k 2 ] obey 2j 1 +2j 2 +4 = k 1 +k 2 , see (2.51). These cases of β sv are related to the F k 2 ] relevant to modular invariant terms in (4.1) involve a total of j 1 +j 2 = 1 2 (k 1 +k 2 )−2 powers of ad 0 . In the following, we shall rewrite the shuffle-irreducible modular invariants with β sv at depth two in terms of F ±(s) m,k . In this way, the [ (4.12) and their images under ad N 0 with N ≤ p+q−2d, ] . (4.20) We will decompose the generating series (4.1) into depth-two sectors Φ τ (k 1 , k 2 ) associated with double integrals of given (G k 1 , G k 2 ). The modular invariant contributions are isolated by means of the delta function in     (1) The analogous expressions at weights m+k ≥ 7 will in the first place involve the combinations q F ±(s) m,k of β sv rather than the full modular invariants F ±(s) m,k : the iterated integrals (2.29) of cusp forms are consistently absent from the generating series (4.1). In the matrix representations of (4.1) relevant to closed-string genus-one integrals [30], the combinations of β sv j k contributing to MGFs at depth two can be recovered by the initial conditions τ → i∞ that the derivations in Φ τ act on.
However, the initial conditions do not allow us to retrieve the iterated integrals of cusp forms in (2.29): they do not have any known realisation in closed-string integrals over torus punctures since Cauchy-Riemann derivatives of MGFs [8] or their generating series [28] do not introduce any holomorphic cusp forms. Many of the subsequent equations will hold modulo lower depth and shuffles as in (4.24), and we will indicate by using ∼ = in the place of = that shuffles, β sv of depth one and depth-zero terms have been dropped while depth-one integrals of holomorphic cusp forms are still tracked.
We will exemplify in the following sections that the depth-two terms (4.15) of krelations are sufficient to effectively replace all the q
In this case, it is the depth-two terms in first line of (4.17) which imply the vanishing of ad 5 0 (4t 3 4,12 − 25t 3 6,10 + 21t 3 8,8 ) and therefore the dropout of H + ∆ 12 modulo higher-depth terms that are given in the second line of (4.5). The higher-depth commutators of k in the second line of (4.5) will be associated with higher-depth β sv , and their modular invariant completions must also contain the iterated integrals of H + ∆ 12 of depth one. Using the first line of (4.17) also for the coefficients of the functions F +(6) m,k in (4.28) reproduces the linear combinations appearing in (3.12) (modulo commutators of k of higher depth).

Weight 9
Similar to (4.26) at weight seven, the even functions q

Summary
In this section, we have demonstrated in detail how (specific linear combinations of) the modular functions F ±(s) m,k appear in the generating series of MGFs. On the one hand, the representations of MGFs as lattice sums or integrals over torus punctures manifest that they are modular forms; on the other hand, their differential equations [8,28] rule out iterated integrals of holomorphic cusp forms. These requirements have been explicitly confirmed for the β sv -contributions (4.1) to modular invariant MGFs at depth two and a wide range of weights.
By reorganising the shuffle irreducible β sv of depth two in terms of q F ±(s) m,k , the accompanying derivations in (4.1) conspire to specific combinations t d p,q of commutators defined in (4.12) that are singled out by representation theory of SL(2, Z). More importantly, these combinations t d p,q were identified by Pollack [49] to streamline relations in the derivation algebra. By rewriting parts of the generating series (4.1) in terms of q In this last section we want to highlight some of the "side-effects" due to the presence of iterated integrals of holomorphic cusp forms in the generic expression (3.1) for F ±(s) m,k . One of the consequences will be to provide a connection between L-values and Kloosterman sums that come out of the Poincaré-series representation of the F ±(s) m,k as anticipated in Part I. The addition of the iterated integrals over cusp forms has consequences for the Fourier expansion of the F ±(s) m,k : in the following, we will compare different approaches to determining the coefficients of q aqb in the expansion around the cusp The coefficients d a,b (y) are Laurent polynomials in y = π Im τ (with powers ranging from y m+k to y −m−k+2 ) which will be referred to as F m,k only introduces single-valued zeta values and rational numbers into d a,b (y). However, the novelty is that now the addition of H ± ∆ 2s also introduces L-values into the Fourier coefficients in (5.1).

Odd example
As a first example we assemble the order q 1q0 term in F Here, we can see clearly the separate contributions from q F −(6) 2,5 in the first two lines, containing only rational coefficients and single odd zetas, and H − ∆ 12 in the last line, which is instead multiplied by the ratio of L-values. Note that the complete Fourier mode e 2πi Re τ receives an infinite series of additional contributions beyond the q 1q0 term in (5.2): all the exponentially suppressed corrections q(qq) n for n > 0 share the factor of e 2πi Re τ and are multiplied by Laurent polynomials in y with rational coefficients, see section 7.1 of Part I for their precise form. where the SL(2, Z) element γ acts on the τ -dependence via both q,q and y.
The general formula for obtaining the Fourier series of a Poincaré sum from the Fourier series of its seed function [78,79], see also appendix A of Part I, then leads to the following identity where 0 ≤ r ≤ d is coprime to d, such that r has the multiplicative inverse r −1 in (Z/dZ) × . Note that the above expression (5.4) contains the full e 2πi Re τ Fourier mode sector, i.e. it contains both the q 1q0 term as well as the infinite tower of exponentially suppressed corrections q(qq) n for n > 0. Since H ± ∆ 2s does not have any (qq) n terms, one can restrict to q F ±(s) m,k , and the only sources of (qq) n are the depth-two β sv , for which the full q >0q>0 terms were given in section 7.1 of Part I. All of them are accompanied by Laurent polynomials with rational coefficients.

Even example
The same kind of analysis can be performed for the q,q-expansion (5. and strongly resembles its odd counterpart (5.2). These two examples (5.2) and (5.7) once more illustrate the fact that the ratio of L-values Λ(∆ 2s ,t 1 ) Λ(∆ 2s ,t 2 ) , appearing in the perturbative expansion of the non-zero Fourier modes, has t 1 , t 2 odd for F

General comments
As we discussed in Part I, the seed functions for the various F ±(s) m,k can all be written as finite, rational linear combinations of building blocks whose Fourier coefficients of e 2πi Re τ take the simple form c (Im τ ) ∼    | | −r σ 2m−1 (| |)(π Im τ ) m+k−r e −2π| | Im τ : > 0 , ±| | −r σ 2m−1 (| |)(π Im τ ) m+k−r e −2π| | Im τ : < 0 , (5.9) for integers r in the range m+1 ≤ r ≤ 2m−1 and where the sign ± for the negative Fourier modes < 0 is adapted to the modular function F ±(s) m,k considered. In [3] it was explained how to extract the asymptotic expansion for Im τ → ∞ for the Poincaré sum of a seed of the form (5.9) and how to derive its Laurent polynomial d 0,0 (y) for the zero Fourier mode, discussed in full detail in Part I. It is furthermore possible to exploit the asymptotic nature of such an expansion to obtain, via resurgence analysis, the exponentially suppressed terms in the same Fourier mode sector, i.e. the terms ∼ (qq) n with n > 0.

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In a similar spirit, we believe that it should be possible, starting from the Poincaré sum of the general seed (5.9), to extract its asymptotic expansion for Im τ → ∞ in any Fourier mode sector. Unlike for the zero-mode sector, no such general expression is at the present time known for (5.9). For example, it would be extremely interesting to start from the expression (5.4) for the first Fourier mode e 2πi Re τ of F −(6) 2,5 (τ ), or the analogous expression for F +(6) 2,6 starting from (5.8), and to derive their asymptotic expansions for Im τ → ∞. Similar to what was done in [3] the integral in (5.4) could be done term-wise after expanding the τ -dependent part of the exponential in an absolutely convergent series. This yields multiple, partly divergent, infinite sums over Kloosterman sums. The analytic continuation of these sums is left for future work.
Irrespective of the explicit result, we can still make some predictions. Firstly, due to the presence of this novel, and extremely non-trivial, Kloosterman sum S( , 1; d) in the first Fourier mode (5.4) of F −(6) 2,5 (or (5.6) for the generic Fourier mode), we should find that this asymptotic expansion truncates after finitely many terms. More importantly, this asymptotic expansion has either rational numbers, single odd zetas or L-values in its coefficients, reproducing all the q 1q0 terms given in (5.2) and similarly (5.7) for the even example F +(6) 2,6 . Secondly, the asymptotic nature of such an expansion should also hide and encode the presence of an infinite tower of exponentially suppressed corrections, i.e. the q(qq) n for n > 0, each one of them accompanied by a Laurent polynomial in y with rational coefficients.