Rational decomposition of modular forms

Abstract

We prove explicit formulas decomposing cusp forms of even weight for the modular group, in terms of generators having rational periods, and in terms of generators having rational Fourier coefficients. Using the Shimura correspondence, we also give a decomposition of Hecke cusp forms of half integral weight k + 1/2 with k even in terms of forms with rational Fourier coefficients, given by RankinCohen brackets of theta series with Eisenstein series.

Haberland’s formula and numerical computation of Petersson scalar products

“…In the first naïve approach, we have explained that nonetheless these integrals can be computed, somewhat slowly, by using doubly-exponential integration techniques. A remarkable fact however, discovered by Haberland [6] (see also [9]) some time ago, is that PSP's can be reduced to the computation of a reasonably small finite number of simple integrals, which can now be evaluated very rapidly using doubly-exponential integration.…”

Hecke eigenform and double Eisenstein series

“…Using (4), for a cusp form f ∈ S k one computes where for the last equality we have used (5) (the last equality holds only for a cusp form, but other equalities hold for f ∈ M k ). Finally, using Theorem 5.1 of [7] at m = 0, we see that the identity r+s=k r,s:even…”

“…This congruence may be related to the Ramanujan-type congruences a f (n) ≡ σ k−1 (n) mod p for a normalized Hecke eigenform f = n>0 a f (n)q n ∈ S k , where p is a prime such that p|B k and p > k (see e.g. [7,Section 5]). We may also expect a connection between (7) and Ihara's congruence [4, p.258] (taken from [1, Section 8]) {σ 3 , σ 9 } − 3{σ 5 , σ 7 } ≡ 0 mod 691 of generators σ 2i+1 of the motivic Lie algebra g m , since g m is one of dual structures of the algebra of multiple zeta values.…”

“…[7,Section 5]). We may also expect a connection between (7) and Ihara's congruence [4, p.258] (taken from [1, Section 8]) {σ 3 , σ 9 } − 3{σ 5 , σ 7 } ≡ 0 mod 691 of generators σ 2i+1 of the motivic Lie algebra g m , since g m is one of dual structures of the algebra of multiple zeta values.…”

Lacunary recurrences for Eisenstein series

“…Eisenstein series which have been classified in [2] and which are closely related to the theory of period polynomials (see [3,4]). In particular, we show the following.…”