Hello.

I wonder whether there are hecke operators for modular forms for $\Gamma = \Gamma_1(N)$ with additive character $\chi : \mathbb{Z}_N \mapsto \mathbb{C}^{\times}$. There is a somewhat reasonable abstract Hecke algebra for $\Gamma_1(N)$, namely the free $\mathbb{Z}$-module generated by those double cosets $\Gamma \alpha \Gamma \in \Gamma \backslash \Delta/\Gamma$ where $\Delta = \{ \alpha = \begin{pmatrix}a & b \\ c & d\end{pmatrix} \in \mathbb{Z}^{2 \times 2} : c \equiv 0 \mod N, a \equiv 1 \mod N,$ $\det(\alpha) \in \mathbb{N}\}$. In order to let this algebra act on the space of modular forms one has to construct a continuation of the character $\chi$ to a semigroup homomorphism $\tilde{\chi} : \Delta \mapsto \mathbb{C}^\times$ such that $$\alpha \gamma \alpha^{-1} \in \Gamma \Rightarrow \tilde{\chi}(\alpha \gamma \alpha^{-1}) = \chi(\gamma)$$ (see e.g. Miyake, Modular Forms, formula (2.8.1)). Tried though i have, i have been unable even to find a continuation of the character. For example, for $N=3$ i think that i was able to show that there is no continuation at all that satisfies $$\alpha \equiv \beta \mod N \Rightarrow \tilde{\chi}(\alpha) = \tilde{\chi}(\beta)$$ (which is a reasonable assumption). I am sure that i am not the first person ever who tried this. Are there Hecke operators on modular forms for $\Gamma_1(N)$ with character? If so, do they arise as actions of an abstract Hecke algebra as above? Does one have to choose another $\Delta$ maybe?

Best regards,

Fabian Werner

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