Genus 2 paramodular Eisenstein congruences

We investigate certain Eisenstein congruences, as predicted by Harder, for level p paramodular forms of genus 2. We use algebraic modular forms to generate new evidence for the conjecture. In doing this, we see explicit computational algorithms that generate Hecke eigenvalues for such forms.


Introduction
Congruences between modular forms have been found and studied for many years. Perhaps, the first interesting example is found in the work of Ramanujan. He studied in great detail the Fourier coefficients τ (n) of the discriminant function (z) = q ∞ n=1 (1 − q n ) 24 (where q = e 2πiz ). The significance of is that it is the unique normalized cusp form of weight 12.
Here σ 11 (n) = d|n d 11 is a power divisor sum. Naturally, one wishes to explain the appearance of the modulus 691. The true incarnation of this is via the fact that the B Dan Fretwell daniel.fretwell@bristol.ac.uk prime 691 divides the numerator of the "rational part" of ζ (12), i.e., ζ (12) π 12 ∈ Q (a quantity that appears in the Fourier coefficients of the Eisenstein series E 12 ).
Since the work of Ramanujan, there have been many generalizations of his congruences. Indeed, by looking for big enough primes dividing numerators of normalized zeta values, one can provide similar congruences at level 1 between cusp forms and Eisenstein series for other weights. In fact, one can even give "local origin" congruences between level p cusp forms and level 1 Eisenstein series by extending the divisibility criterion to include single Euler factors of ζ(s) rather than the global values of ζ(s) (see [8] for results and examples).
There are also Eisenstein congruences predicted for Hecke eigenvalues of genus 2 Siegel cusp forms. One particular type was conjectured to exist by Harder [15]. There is only a small amount of evidence for this conjecture, and the literature only contains examples at levels 1 and 2 (using methods specific to these levels). The conjecture is also far from being proved. Only one specific level 1 example of the congruence has been proved [5, p. 386].
In this paper, we will see new evidence for a level p version of Harder's conjecture for various small primes (including p = 2 but not exclusively). The Siegel forms will be of paramodular type and the elliptic forms will be of 0 ( p) type. In doing this, we will make use of Jacquet-Langlands style conjectures due to Ibukiyama.

Harder's conjecture
Given k ≥ 0 and N ≥ 1, let S k ( 0 (N )) denote the space of elliptic cusp forms for 0 (N ). Also for j ≥ 0 let S j,k (K (N )) denote the space of genus 2, vectorvalued Siegel cusp forms for the paramodular group of level N , taking values in the representation space Symm j (V ) ⊗ det k (V ) of GL 2 (C) (where V is the standard representation).
Given f ∈ S k ( 0 (N )), we let alg ( f, j + k) = ( f, j+k) , where ( f, s) is the completed L-function attached to f and is a Deligne period attached to f . The choice of is unique up to scaling by Q × f but Harder shows how to construct a more canonical choice of that is determined up to scaling by O × Q f [16]. In this paper, we consider the following paramodular version of Harder's conjecture (when N = 1 this is the original conjecture found in [15]).
Then there exists a Hecke eigenform F ∈ S new j,k (K (N )) with eigenvalues b n ∈ Q F such that b q ≡ q k−2 + a q + q j+k−1 mod for all primes q N (where is some prime lying above λ in the compositum Q f Q F ).
It should be noted that Harder's conjecture has still not been proved for level 1 forms. However, the specific example with j = 4, k = 10, and l = 41 mentioned in Harder's paper has recently been proved in a paper by Chenevier and Lannes [5]. The proof uses the Niemeier classification of 24-dimensional lattices and is specific to this particular case. Following the release of the level 1 conjecture, Faber and Van der Geer were able to do computations when dim(S j,k (Sp 4 (Z))) = 1. They have now exhausted such spaces and in each case have verified the congruence for a significant number of Hecke eigenvalues. Ghitza, Ryan, and Sulon give extra evidence for the case j = 2 [12]. More recently, Cléry, Faber, and Van der Geer gave more examples for the cases j = 4, 6, 8, 10, 12, and 14 [6].
For the level p conjecture, a substantial amount of evidence has been provided by Bergström et al for level 2 forms [1]. Their methods are specific to this level. A small amount of evidence is known beyond level 2. In particular, a congruence has been found with ( j, k, p, l) = (0, 3, 61, 43) by Mellit [16, p. 99].
In this paper, we use the theory of algebraic modular forms to provide evidence for the conjecture at levels p = 2, 3, 5, 7. The methods discussed can be extended to work for other levels.

Algebraic modular forms
In general, it is quite tough to compute Hecke eigensystems for paramodular forms. Fortunately, for a restricted set of levels, there is a (conjectural) Jacquet-Langlands style correspondence for GSp 4 due to Ihara and Ibukiyama [19].
Explicitly, it is expected that there is a Hecke equivariant isomorphism between the spaces S new j,k (K ( p)) and certain spaces of algebraic modular forms. Bearing this in mind, we give the reader a brief overview of the general theory of such forms. For more details, see the introductory article of Loeffler in [26].

The spaces A(G, K f , V ) of algebraic forms
Let G/Q be a connected reductive group with the added condition that the Lie group G(R) is connected and compact modulo center. Fix an open compact subgroup K f ⊂ G(A f ). Also let V be (the space of) a finite dimensional algebraic representation of G, defined over a number field F.
There is a natural embedding:

Theorem 3.2 The map φ induces an isomorphism:
A pleasing feature of the theory is that the groups m are often finite. Gross gives many equivalent conditions for when this happens [14]. One such condition is the following.

Hecke operators
It is well known that finitely many representatives occur. Then T u acts on It is easy to see that this is independent of the choice of representatives u i since they are determined up to right multiplication by K f . We wish to find the Hecke representatives u i explicitly and efficiently. To this end, a useful observation can be made when the class number is one.

Proposition 3.5 If h = 1, then we may choose Hecke representatives that lie in G(Q).
Finally, we note that for G satisfying Proposition 3.4 there is a natural inner product on the space A(G, K , V ). This is given in Gross' paper [14] but we shall give the rough details here.
Taking adelic points, we have a character μ : where f : A × → Q × is the natural projection map coming from the decomposition A × = Q × R +Ẑ× . Proposition 3.7 Let G satisfy the property of Proposition 3.4. Then A(G, K , V ) has a natural inner product given by

Trace of Hecke operators
The underlying representation V of G is typically big in dimension and so the action of Hecke operators is, although explicit, quite tough to compute. Fortunately, there is a simple trace formula for Hecke operators on spaces of algebraic modular forms. The details of the formula can be found in [9] but we give brief details here.
More generally, Letting u = id, we recover the following: When h = 1, the situation becomes much simpler. In this case, we may choose z 1 = id and γ 1,i = u i ∈ G(Q) for each i (this is possible by Corollary 3.5).
The trace formula was introduced to test a U(2, 2) analogue of Harder's conjecture. In this paper, we will use it to test the level p paramodular version of Harder's conjecture given by Conjecture 2.1.

Eichler's correspondence
From now on, D will denote a quaternion algebra over Q ramified at { p, ∞} (for a fixed prime p) and O will be a fixed maximal order. Since D is definite, we have that D × ∞ = D × ⊗ R ∼ = H × is compact modulo center (and is also connected). Thus, we may consider algebraic modular forms for the group G = D × .
Also note that in this case each m will be finite since D × (Z) = O × is finite. Let D q := D ⊗ Q q be the local component at prime q (no restriction on q) and let D A f be the restricted direct product of D q 's with respect to the local maximal orders Thus, locally away from the ramified prime, D × behaves like GL 2 .
In fact more is true. It is the case that the reductive groups D × and GL 2 are inner forms of each other. So by the principle of Langlands functoriality we expect a transfer of automorphic forms between D × and GL 2 . Eichler gives an explicit description of this transfer.
Let V n = Symm n (C 2 ) (for n ≥ 0). Then V n gives a well-defined representation of SU(2)/{±I } if and only if n is even. Thus, we get a well-defined action on V n by D × through Then there is a Hecke equivariant isomorphism: For k = 2, the above holds if on the right we quotient out by the space of constant functions.
It remains to describe how the Hecke operators transfer over the isomorphism. Fix a prime q = p and an isomorphism ψ : (1, q) and is the identity at all other places. The corresponding Hecke operator corresponds to the classical T q operator under Eichler's correspondence.

Ibukiyama's correspondence
Ibukiyama's correspondence is a (conjectural) generalization of Eichler's correspondence to Siegel modular forms. The details can be found in [19] but we explain the main ideas.
Given the setup in the previous subsection, consider the unitary similitude group: Hereḡ means componentwise application of the standard involution of D. This group is the similitude group of the standard Hermitian form on D n .

Lemma 4.2 For any field K , there exists a similitude-preserving isomorphism
One consequence of this is that the group GU 2 (D) behaves like GSp 4 locally away from the ramified prime. It is indeed true that these groups are also inner forms of each other.
A simple argument also shows that is compact modulo center and connected. Thus, we may consider algebraic modular forms for this group. Once again, we are guaranteed that the m groups are finite by the following.
Proof Solving the equations gives

Lemma 4.4
For any q = p, there exists a similitude-preserving isomorphism ψ : Proof Choose an isomorphism of quaternion algebras D q ∼ = M 2 (Q q ) that preserves the norm, trace, and integrality. This induces an isomorphism with the required properties since Let V j,k−3 be the irreducible representation of USp(4) with Young diagram parameters ( j + k − 3, k − 3). This gives a well-defined representation of USp(4)/{±I } if and only if j is even. Thus, GU 2 (D) acts on this through The groups GU(D) and GSp 4 are inner forms. Thus (as with Eichler), one expects a transfer of automorphic forms. The following is found in Ibukiyama's paper [19].
, then we also get an isomorphism after taking the quotient by the constant functions on the right.
Since our eventual goal is to study Harder's conjecture for paramodular forms, we will neglect the first of these isomorphisms. However, it will turn out that the open compact subgroup U 1 will prove useful in later calculations.

The levels U 1 and U 2 .
In Eichler's correspondence, the "level 1" open compact subgroup A result of Shimura tells us the possibilities for L q (see [28]).

Theorem 4.6 Let D be a quaternion algebra over
If D is ramified at q, then there are exactly two possibilities for L q , up to right When D is ramified at { p, ∞}, it is clear from this result that there are only two possibilities for L, up to local equivalence. Definition 4.7 Let D be ramified at p, ∞ for some prime p: • If L p is locally right equivalent to O 2 p for all q, then we say that L lies in the principal genus.
• If L p is locally right inequivalent to O 2 p , then we say that L lies in the non-principal genus.
Given L, results of Ibukiyama [23] allow us to write L = O 2 g for some g ∈ GL 2 (D) and determine the genus of L based on g.

Theorem 4.8 (1) L lies in the principal genus if and only if gg T = mx for some
positive m ∈ Q and some x ∈ GL n (O) such that x = x T and such that x is positive definite, i.e., yx y T > 0 for all y ∈ D n with y = 0.
(2) L lies in the non-principal genus if and only if gg T = m ps r r pt , where m ∈ Q is positive, s, t ∈ N, r ∈ O lies in the two-sided ideal of O above p and is such that p 2 st − N (r ) = p (so that the matrix on the right has determinant p).
The lattice O 2 is clearly in the principal genus and corresponds to the choice g = I . Alternatively, fix a choice of g such that O 2 g is in the non-principal genus. Let U 1 , U 2 denote, respectively, the corresponding open compact subgroups of GU 2 (D A f ) (as described above).

Hecke operators
The transfer of Hecke operators in Ibukiyama's correspondence is similar to the Eichler correspondence but there are subtle differences. Fix a prime q = p and an isomorphism ψ as in Lemma 4.4. Then we may find Under Ibukiyama's correspondence, it is predicted that T u,q corresponds to the classical T q operator acting on S new j,k (K ( p))).

The new subspace
Our final task in defining Ibukiyama's correspondence is to explain what is meant by . We will not go into too much detail but will refer the reader to Ibukiyama's papers [21,22].
We start with the decomposition: Ibukiyama takes F ∈ A(G, U , W j,k−3 ). If F is an eigenform, then F = F 1 ⊗ F 2 for eigenforms F 1 , F 2 . He then associates an explicit theta series θ F to F. This is an elliptic modular form for SL 2 (Z) of weight j + 2k − 2 (if j + 2k − 6 = 0, then it is a cusp form). It is known that θ F is an eigenform for all Hecke operators if and only if θ F = 0.

Definition 4.10 The subspace of old forms
is generated by the eigenforms F 2 such that there exists an eigenform F 1 satisfying θ F 1 ⊗F 2 = 0.
The subspace of new forms A new j,k−3 (D) is the orthogonal complement of the old space with respect to the inner product in Proposition 3.7.
It should be noted that by Eichler's correspondence F 1 can be viewed as an elliptic modular form for 0 ( p) of weight j + 2. Further it will be a new cusp form precisely when j > 0. Thus, computationally it is not difficult to find the new and old subspaces.

Finding evidence for Harder's conjecture
Now that we have linked spaces of Siegel modular forms S new j,k (K ( p)) with spaces of algebraic modular forms , we can begin to generate evidence for Harder's conjecture.

Brief plan of the strategy
In this paper, we deal with cases where h = 1 and dim(A new j,k−3 (D)) = 1.

Strategy
(1) Find all primes p such that h = 1.
(3) Using Corollary 3.9 find all j, k such that dim(A new j,k (D)) = 1. (4) For each pair ( j, k) look in the space of elliptic forms S new j+2k−2 ( 0 ( p)) for normalized eigenforms f which have a "large prime" dividing alg ( f, j + k) ∈ Q f . (5) Find the Hecke representatives for the T u,q operator at a chosen prime q. (6) Use the trace formula to find tr(T u,q ) for T q acting on A j,k−3 (D). The above strategy can be modified to work for the case dim(A new j,k−3 (D)) = d > 1 but one must compute tr(T t u,q ) for 1 ≤ t ≤ d.

Lemma 5.1
The group (2) consists of the following set of matrices: where g ∈ GL 2 (D) satisfies the condition in Theorem 4.8.
Proof We know that A simple calculation shows that any such matrix has similitude 1.
Recall also the open compact subgroup . This is the stabilizer of a left O-lattice lying in the principal genus.
In this case, the analogue of the group (2) is the group (1) = GU 2 (D) ∩ U 1 . We can employ identical arguments to the above to show the following: We already have an explicit description of (1) (see Lemma 4.3). Computationally, it is not straightforward to find the elements of (2) due to the non-integrality of the entries of such matrices.
For θ ∈ Q × , consider the sets Then in particular Y 1 = (2) . Later, the sets Y q for prime q = p will appear when finding Hecke representatives.

Proposition 5.3 For each θ ∈ Q × , conjugation by g gives a bijection
To calculate the sets W θ we diagonalize A. Choose a matrix P ∈ GL 2 (D) such that P AP T = B, where B ∈ M 2 (D) is a diagonal matrix.

Proposition 5.4
For each θ ∈ Q × , conjugation by P gives a bijection: If we make an appropriate choice of g and P, then we can diagonalize A in such a way as to preserve one integral entry in Pν P −1 . Further P −1 λ,μ = P λ,μ . Proof A simple calculation shows that
The final claim follows from the fact that P λ,μ P λ,μ = I (which again uses the fact that tr(r ) = 0). It is in fact always possible to find some maximal order O of D where such λ, μ exist. For proof of this I refer to an online discussion with John Voight [31], of which the author is grateful. We fix such a choice from now on.

Corollary 5.6 Let ν ∈ M 2 (O). Then the bottom left entries of ν and P λ,μ ν P λ,μ are equal (in particular this entry remains in O).
Proof Let ν = α β γ δ with α, β, γ , δ ∈ O. Then a simple calculation shows that The matrix η = x y z w ∈ M 2 (D) belongs to Z θ if and only if Equivalently, Clearly, these equations can have no solutions for θ < 0 and so we only consider θ ≥ 0.
A quick calculation shows that N (x) = N (w) and N (z) = p 2 N (y) (a fact we will use soon).
The following algorithm allows us to compute W θ for θ ∈ N. Denote by X i the subset of O consisting of norm i elements.

Algorithm 1
Step 0: Set j := 0. For each integer 0 ≤ i ≤ θ p, generate the norm lists X i , X p(θ p−i) , X p 2 i .
Step 4: For each tuple from Step 3, check whether the entries satisfy the third equation of Corollary 5.7.
Step 5: Set j := j + 1 and repeat steps 1-4 until j > θp. Of course, once the elements of W θ have been found, it is straightforward to generate the elements of Y θ by inverting the bijection θ in Proposition 5.3.
It should be noted that if we run this algorithm for p = 2 with the following choices: then we get exactly the same elements for Y 1 = (2) as Ibukiyama does on [19, p. 592].

Finding h
We can use mass formulae to get information on class numbers h 1 and h 2 for U 1 and U 2 .
Define the mass of open compact U ⊂ GU 2 (D A f ) as follows: Ibukiyama provides the following formulae for M(U 1 ) and M(U 2 ) in [19].

Theorem 5.8 If D is ramified at p and ∞, then
This formula is analogous to the Eichler mass formula and is also a special case of the mass formula of Gan et al. [10]. Proof A quick calculation shows that the only primes to satisfy 5760 ( p−1)( p 2 +1) ∈ N are p = 2, 3. Recall | (1) | = 2|O × | 2 . For p = 2, 3, we have |O × | = 24, 12, respectively, and one checks that both values satisfy the equation.

Finding the Hecke representatives
Now that we have found an algorithm to generate the elements of (2) , we consider the same question for the Hecke representatives for the T u,q operator on A j,k−3 (D) (where q = p is a fixed prime).
Proof Consider an arbitrary decomposition: By Proposition 3.5, we may take x i ∈ GU 2 (D) for each i. For the rest of the proof, Note that for any prime l = q we have Thus, Conjugation by h q gives a bijection between U 2,q u q U 2,q and Gv q G, where G = GU 2 (D q ) ∩ GL 2 (O q ). If we fix an isomorphism as in Lemma 4.4 1, q, q)).

, then this is in bijection with
Since by definition Thus, u q ∈ GU 2 (D q ) ∩ g −1 q M 2 (O q ) × g q and the same can be said about x i . Also since both the conjugation and our chosen isomorphism respect similitude, we find that μ(u q ) = μ(M q ) = q and so μ(U 2,q u q U 2,q ) ⊆ qZ × q . In particular μ(x i ) ∈ qZ × q . Globally, we now see that However, in our case, the similitude is positive definite so that μ(x i ) = q.
Thus, x i can be taken to lie in Y q . It is clear that each such element lies in the double coset.
It remains to see which elements of Y q generate the same left coset. We have 2) . So equivalence of left cosets is up to right multiplication by (2) .
We have a nice formula for the degree of T u,q , found in the work of Ihara [24].
Employing similar arguments to Theorem 5.11, we get the following: Theorem 5.13 Let D be a quaternion algebra over Q ramified at p, ∞ for some p ∈ {2, 3}. Suppose u ∈ GU 2 (D A f ) is chosen as in Definition 4.9. Then (1) x i U 1 .
Since (1) is given explicitly, it is possible to write down explicit representatives in this case.
The following matrices are representatives for (GU 2 (D) n ∩ M 2 (O) × )/ (1) : The finite subset R m ⊂ R m is to be constructed in the proof.
In order for ν ∈ GU 2 (D) n to hold, we must satisfy the equations: In a similar vein to previous discussion, these equations imply that N (α) = N (δ) and N (β) = N (γ ). Note that the first equation implies that 0 ≤ N (α) ≤ n. We wish to study equivalence of these matrices under right multiplication by (1) . Letting k = N (α), choose x, y ∈ O × so that αx = x i,k and δy = x j,k for some Clearly the matrices of this form are inequivalent.
It is now clear that R k gives representatives for the particular subcase N (α) = k > n 2 .
Then x, y are uniquely determined as follows: Thus, each such matrix with v, w ∈ X m can only be equivalent to at most one other matrix: where v ∼ x s,m and w ∼ x t,m under the action of right unit multiplication. Let R m be a set consisting of a choice of matrix from each of these equivalence pairs (as x i,m and x j,m run through representatives for X m /O × and v, w run through elements of X m satisfying x i,m w + vx j,m = 0). Then it is now clear that R m is a set of representatives for this subcase.
In the subcase k = n − 1, it is often easier to use anti-diagonal equivalence (since X 1 /O × = {1}). In this case, we can identify When n = 2 exactly half of these will form a set of representatives. In fact, it is simple to see that the equivalent pairs would be Thus

Example 5.15
If we apply Corollary 5.14 to the choices we find that Hecke representatives for U 1 with ramified prime p = 2 and q = 3 are given by There are 40 representatives here as expected and they agree with the explicit representatives given by Ibukiyama on [19, p. 594].
So far we have not needed the open compact subgroup U 1 , but it is actually of use to us in studying U 2 .

Lemma 5.16
Let u ∈ GU 2 (D A f ) be chosen to form the T u,q operator with respect to both U 1 and U 2 (for prime q = p). Then the Hecke representatives for T u,q with respect to U 1 and U 2 can be taken to be the same.
Proof Recall that u has identity component away from q and u q / ∈ U 2,q so we need only to check the local condition that U 2,q = U 1,q . This is clear since This result is useful since we have seen that it is generally easier to generate Hecke representatives for T u,q with respect to U 1 .

Corollary 5.17
Let the ramified prime of D be p ∈ {2, 3}. Then we may use the representatives from Corollary 5.14 as Hecke representatives for T u,q with respect to U 2 (for q = p).
Proof Since p ∈ {2, 3}, we know that both the class numbers of U 1 , U 2 are 1. Hence both admit rational Hecke representatives.
We also know that given Hecke representatives for T u,q with respect to U 1 we may use them for U 2 . Thus, the rational representatives from Corollary 5.14 can be used for U 2 .

Implementing the trace formula
Now that we have algorithms that generate the data needed to use the trace formula, we discuss some of the finer details in its implementation, namely how to find character values. Denote by χ j,k−3 the character of the representation V j,k−3 .
We know that the image of GU 2 (H) 1  In order to find χ j,k−3 (B), we first find the eigenvalues of B. This is equivalent to conjugating into the maximal torus of diagonal matrices. Since B ∈ USp(4), these eigenvalues will come in two complex conjugate pairs z, z, w, w for z, w on the unit circle.
The Weyl character formula gives For any of the cases z 2 = 1, w 2 = 1, zw = 1, z = w, one must formally expand this concise formula into a polynomial expression (not an infinite sum since each factor on the denominator except zw divides the numerator). It is easy for a computer package to compute this expansion for a given j, k.

Finding the trace contribution for the new subspace
Let tr(T u,q ) new and tr(T u,q ) old be the traces of the action of T u,q on A new j,k−3 (D) and A old j,k−3 (D), respectively. Then tr(T u,q ) new = tr(T u,q ) − tr(T u,q ) old .
Recall that each eigenform in A old j,k−3 (D) is given by a special pair of eigenforms Let α n , β n , γ n be the Hecke eigenvalues of F 1 , F 2 , θ F 1 ⊗F 2 , respectively. Ibukiyama links the eigensystems as follows.

Theorem 5.18
For q = p, we have the following identity in C(t): .
Ibukiyama conjectures that there is a bijection between pairs of eigenforms (F 1 , θ F ) and eigenforms F 2 . With this in mind, it is now possible to calculate the oldform trace contribution.
Then, for q = p and j > 0,

Examples and summary
The following table highlights the choices for D, O, λ, μ that were used.
Using these choices along with the algorithms and results mentioned previously, one can calculate the groups (1) , (2) for each such p, hence generating tables of dimensions of the spaces A new j,k−3 (D). These tables are given in Appendix 1. From these tables, one isolates 1-dimensional spaces. For each possibility, the MAGMA command LRatio allows us to test for large primes dividing alg on the elliptic side. The cases that remained were ones where we expect to find examples of Harder's congruence.
Tables of the congruences observed can be found in Appendix 2. In particular, for p = 2 one observes congruences provided in Bergström [1]. We finish with some new examples for p = 3. Then j = 2 and k = 8 so that j + 2k − 2 = 16. Let F ∈ S new 2,8 (K (3)) be the unique normalized eigenform.
One easily checks that dim(S new 16 ( 0 (3))) = 2. This space is spanned by the two normalized eigenforms with q-expansions: Indeed, MAGMA informs us that ord 109 ( alg ( f 1 , 10)) = 1 and so we expect a congruence of the form: b q ≡ a q + q 9 + q 6 mod 109 for all q = 3, where b q are the Hecke eigenvalues of F and a q the Hecke eigenvalues of f 1 . As discussed earlier, we will only work with the case q = 2 for simplicity.
MAGMA informs us that ord 67 ( alg ( f 2 , 13)) = 1 and so we expect a congruence of the form b q ≡ a q + q 12 + q 3 mod 67, for all q = 3.
One easily checks that dim(S new 14 ( 0 (3))) = 3. This space is spanned by the three normalized newforms with q-expansions: