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**Trace formula of twisting operators of half-integral weight in the case of even conductors.**
*(English)*
Zbl 1053.11044

From the text: Let \(k\) and \(N\) be positive integers with \(4\mid N\). Let \(\chi\) be an even Dirichlet character defined modulo \(N\) with \(\chi^2= 1\). Denote the space of cusp forms of weight \(k+1/2\), level \(N\), and character \(\chi\) by \(S(k+1/2,N,\chi)\).

In the previous paper M. Ueda [The trace formulae of twisting operators on the spaces of cup forms of half-integral weight and some trace relations. Jap. J. Math., New Ser. 17, 83–135 (1991; Zbl 0742.11031)], the author calculated an explicit trace formula of the twisting operator \(R_\psi\) on \(S(k+1/2, N,\chi)\) for a quadratic primitive character \(\psi\) of odd conductor. Here the author now gives an explicit trace formula of the twisting operator \(R_\psi\) for a quadratic primitive character \(\psi\) of even conductor. Details will appear in Part II of the above cited paper.

In the previous paper M. Ueda [The trace formulae of twisting operators on the spaces of cup forms of half-integral weight and some trace relations. Jap. J. Math., New Ser. 17, 83–135 (1991; Zbl 0742.11031)], the author calculated an explicit trace formula of the twisting operator \(R_\psi\) on \(S(k+1/2, N,\chi)\) for a quadratic primitive character \(\psi\) of odd conductor. Here the author now gives an explicit trace formula of the twisting operator \(R_\psi\) for a quadratic primitive character \(\psi\) of even conductor. Details will appear in Part II of the above cited paper.

### MSC:

11F37 | Forms of half-integer weight; nonholomorphic modular forms |

11F25 | Hecke-Petersson operators, differential operators (one variable) |

### Citations:

Zbl 0742.11031
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\textit{M. Ueda}, Proc. Japan Acad., Ser. A 79, No. 4, 85--88 (2003; Zbl 1053.11044)

### References:

[1] | Miyake, T.: Modular Forms. Springer, Berlin (1989). · Zbl 0701.11014 |

[2] | Shimura, G.: On modular forms of half integral weight. Ann. of Math., 97 , 440-481 (1973). · Zbl 0266.10022 |

[3] | Ueda, M.: The trace formulae of twisting operators on the spaces of cusp forms of half-integral weight and some trace relations. Japan. J. of Math., 17 , 83-135 (1991). · Zbl 0742.11031 |

[4] | Ueda, M.: The trace formulae of twisting operators on the spaces of cusp forms of half-integral weight and trace identities II. (In preparation). · Zbl 0742.11031 |

[5] | Ueda, M.: Some trace relations of twisting operators on the spaces of cusp forms of half-integral weight. Proc. Japan Acad., 66A , 169-172 (1990). · Zbl 0726.11027 |

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