Delta form

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In mathematics, Delta is a modular form, arising from the discriminant of an elliptic curve. As a modular form it is a cusp form of weight 12 and level 1 for the full modular group. It is an eigenform for the Hecke algebra.

The q-expansion is

Δ / ( 2 π ) 12 = q ∏ n = 1 ∞ ( 1 − q n ) 24 = ∑ n τ ( n ) q n , {\displaystyle \Delta /(2\pi )^{12}=q\prod _{n=1}^{\infty }\left(1-q^{n}\right)^{24}=\sum _{n}\tau (n)q^{n},\,}

where τ is Ramanujan's tau function. Since Δ is a Hecke eigenform, the tau function is multiplicative.

Dedekind's eta function is a 24-th root of Δ.