A First Course in Modular Forms

Finiteness of simple holomorphic eta quotients of a given weight

“…The last inequality implies that the number of holomorphic eta quotients of weight k/2 on Γ 0 (N ) is less than (2kF (N )) d(N ) , where d(N ) denotes the number of divisors of N . But the dimension of the space of modular forms of any fixed even weight on Γ 0 (N ) becomes arbitrarily large as N → ∞ (see [7]). So, if we fix the number of divisors of N along with a k ∈ 4N, then except only finitely many possibilities for N , the space of modular forms of weight k/2 on Γ 0 (N ) never contains enough eta quotients to constitute a basis.…”

“…We identify P 1 (Q) with Q ∪ {∞} via the canonical bijection that maps [α : λ] to α/λ if λ = 0 and to ∞ if λ = 0. For s ∈ Q ∪ {∞} and a weakly holomorphic modular form f on Γ 0 (N ), the order of f at the cusp s of Γ 0 (N ) is the exponent of q 1/ws occurring with the first nonzero coefficient in the q-expansion of f at the cusp s, where w s is the width of the cusp s (see [10], [23]). The following is a minimal set of representatives of the cusps of Γ 0 (N ) (see [10], [20]): [20]).…”

Holomorphic eta quotients of weight 1/2

“…for some a ∈ Z which is coprime to gcd(N, b) (see [7]). We identify P 1 (Q) with Q ∪ {∞} via the canonical bijection that maps [α : λ] to α/λ if λ = 0 and to ∞ if λ = 0.…”

“…We identify P 1 (Q) with Q ∪ {∞} via the canonical bijection that maps [α : λ] to α/λ if λ = 0 and to ∞ if λ = 0. For s ∈ Q ∪ {∞} and a weakly holomorphic modular form f on Γ 0 (N ), the order of f at the cusp s of Γ 0 (N ) is the exponent of q 1/ws occurring with the first nonzero coefficient in the q-expansion of f at the cusp s, where w s is the width of the cusp s (see [7], [14]). The following is a minimal set of representatives of the cusps of Γ 0 (N ) (see [7], [12]): [12]).…”

“…We recall that such an eta quotient f is holomorphic if it does not have any poles at the cusps of Γ 0 (N ). Under the action of Γ 0 (N ) on P for some a ∈ Z which is coprime to gcd(N, b) (see [7]). We identify P 1 (Q) with Q ∪ {∞} via the canonical bijection that maps [α : λ] to α/λ if λ = 0 and to ∞ if λ = 0.…”