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It is said we can use the operator $$ \pi_\chi=\frac{1}{\phi(N)}\sum_{d\in\mathbb{Z}_N^*}\chi(d)^{-1}\langle d\rangle $$

to project function in $\mathcal{M}_k(\Gamma_1(N))$ into the $\chi-$eigenspace $\mathcal{M}_k(N,\chi)$.

I just down know what the symbol $\langle d\rangle$ means here. Does it mean a cyclic group? If so, how does this operator work?

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$\langle d\rangle$ is the Diamond operator. See e.g. these notes of William Stein, or pretty much any other source on modular forms that covers Hecke operators.

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OK, thank you.. –  hxhxhx88 Jan 14 '13 at 14:22

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