## Twists of newforms and pseudo-eigenvalues of $$W$$-operators.(English)Zbl 0369.10016

Let $$F$$ be a normalized newform of level $$N$$ and character $$\varepsilon$$. If $$q$$ is a prime dividing $$N$$ and $$Q$$ the $$q$$-primary factor of $$N$$, then $$F| W_q=\lambda_q(F)G$$ where $$G$$ is a normalized newform. We call $$\lambda_q(F)$$ the pseudo-eigenvalue of $$W_q$$ at $$F$$. In this paper we study the twist $$F_\chi$$ of $$F$$ by characters $$\chi$$ whose conductors are powers of $$q$$. We also investigate $$\lambda_q(F_\chi)$$ whenever $$F_\chi$$ is a newform. Without loss of generality, we may assume $$F$$ $$q$$-primitive in the sense that $$F$$ is not equal to a twist of a newform of level less than $$N$$ by a character with $$q$$-primary conductor. If $$a(q)$$, the $$q$$th Fourier coefficient of $$F$$, is nonzero, then for all characters $$\chi$$ with $$q$$-primary conductors, with at most one exception, $$F_\chi$$ is a newform and one can express $$\lambda_q(F)$$, $$\lambda_q(F_\chi)$$ in terms of $$a(q)$$. If $$a(q)$$ is zero, then all $$F_\chi$$ with conductor $$\chi$$ $$q$$-primary are newforms. In this case we have an explicit formula for $$\lambda_q(F_\chi)$$ if $$\text{cond}\,\chi\geq Q$$ and obtain relations among $$\lambda_q(F_\chi)$$’s with $$\text{cond}\,\chi< Q$$. One remarkable fact is that $$F$$ is not $$q$$-primitive provided $$Q> \text{cond}\,\varepsilon_q>\sqrt Q$$, where $$\varepsilon=\varepsilon_q\varepsilon_{N/q}$$.
Reviewer: W.-C. W. Li

### MSC:

 11F11 Holomorphic modular forms of integral weight
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### References:

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