If $f \in M_k(N, \chi)$ is a normalized integer weight Hecke eigenform for all $T_p$, is there a way to obtain $g \in M_k \left(4N, \chi \left( \frac{-4}{\cdot} \right) \right)$ via the use of operators in the theory of modular forms?
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2$\begingroup$ Do you mean $g \in M_k(4N,\chi(\frac{-4}{\cdot}))$ rather than $f \in M_k(N,\chi(\frac{-4}{\cdot}))$? Otherwise this is obviously impossible. $\endgroup$Peter Humphries– Peter Humphries2025-03-21 22:01:19 +00:00Commented Mar 21 at 22:01
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$\begingroup$ Yes, I have edited the question. Sorry for the confusion. $\endgroup$user554145– user5541452025-03-21 22:06:06 +00:00Commented Mar 21 at 22:06
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$\begingroup$ Just to be sure: you want forms with the same weight? $\endgroup$Kimball– Kimball2025-03-22 09:15:53 +00:00Commented Mar 22 at 9:15
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$\begingroup$ Yes, the forms should have same weight $k \in \mathbb{Z}$. $\endgroup$user554145– user5541452025-03-22 16:18:47 +00:00Commented Mar 22 at 16:18
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3$\begingroup$ Impossible since $\chi(-1)=(-1)^k$ and muliplying by $(-4/.)$ changes parity. $\endgroup$Henri Cohen– Henri Cohen2025-03-22 17:05:00 +00:00Commented Mar 22 at 17:05
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