A modularity criterion for Klein forms, with an application to modular forms of level 13

Abstract

We find some modularity criterion for a product of Klein forms of the congruence subgroup Γ 1 (N) (Theorem 2.6) and, as its application, construct a basis of the space of modular forms for Γ 1 (13) of weight 2 (Example 3.4). In the process we face with an interesting property about the coefficients of certain theta function from a quadratic form and prove it conditionally by applying Hecke operators (Proposition 4.3).

Ramanujan type congruences for quotients of level 7 Klein forms

“…The transformation formulas for Parts (1) and (2) follow from (2.1) and Lemma 2.1 (2). A detailed proof of the modular properties of more general quotients subsuming the cases considered here appear in [7]. The required holomorphicity follows from (2.1) and Lemma 2.1 (2) and (3).…”

“…In this section, we define certain notation and state preliminary results on the graded algebra over C for the holomorphic modular forms for Γ (7). In the remainder of the present work, modular forms are referred to holomorphic modular forms.…”

“…In the remainder of the present work, modular forms are referred to holomorphic modular forms. The following lemma allows us to verify that the reciprocal of the Klein forms K 1/7 (7τ ), K 2/7 (7τ ), K 3/7 (7τ ), are modular forms for Γ (7). We subsequently use these properties to construct modular forms of weight 1 for Γ(7) and Γ 1 (7).…”

“…Then T r are holomorphic modular forms of weight 1 for Γ(7) (cf. [7]). Combinatorial interpretations for quotients of the Klein forms may be more readily deduced from the representation…”

“…In this work, we provide a unified derivation of Ramanujan type congruences modulo powers of 7 for a family of colored-weighted partitions P a 0 ,a 1 ,a 2 ,a 3 (n) with a 0 = 4, 6 and a 1 + a 2 + a 3 = −2, −3 that includes the 7-core partition function a 7 (n) = P 6,−1,−1,−1 (n). We use Klein forms to construct generators for the graded algebra of holomorphic modular forms for Γ (7) and prove a wealth of congruences. Since Klein forms are building blocks for more general modular forms, the analysis serves as a prototype for more general constructions at the higher weights and levels considered in [12].…”