## Nagoya Mathematical Journal

### On the Kohnen-Zagier formula in the case of $4 \times$ general odd' level

Hiroshi Sakata

We study the Fourier coefficients of cusp forms of half integral weight and generalize the Kohnen-Zagier formula to the case of $4 \times$ general odd$' level by using results of Ueda. As an application, we obtain a generalization of the result of Luo-Ramakrishnan [11] to the case of arbitrary odd level. #### Article information Source Nagoya Math. J., Volume 190 (2008), 63-85. Dates First available in Project Euclid: 23 June 2008 Permanent link to this document https://projecteuclid.org/euclid.nmj/1214229078 Mathematical Reviews number (MathSciNet) MR2423829 Zentralblatt MATH identifier 1220.11066 #### Citation Sakata, Hiroshi. On the Kohnen-Zagier formula in the case of `$4 \times$general odd' level. Nagoya Math. J. 190 (2008), 63--85. https://projecteuclid.org/euclid.nmj/1214229078 #### References • E. M. Baruch and Z. Mao, Central value of automorphic$L$-functions, preprint (2003). • B. Gross, W. Kohnen and D. Zagier, Heegner Points and Derivatives of$L$-series, Math. Ann., 278 (1987), 497--562. • W. Kohnen, New forms of half-integral weight, J. reine und angew. Math., 333 (1982), 32--72. • W. Kohnen, Fourier coefficients of modular forms of half-integral weight, Math. Ann., 271 (1985), 237--268. • W. Kohnen, A Remark on the Shimura correspondence, Glasgow Math. J., 30 (1988), 285--291. • W. Kohnen and D. Zagier, Values of$L$-series of modular forms at the center of the critical strip, Invent. Math., 64 (1981), 175--198. • H. Kojima, Remark on Fourier coefficients of modular forms of half-integral weight belonging to Kohnen's spaces II, Kodai Math. J., 22 (1999), 99--115. • H. Kojima, On the Fourier coefficients of Maass wave forms of half integral weight over an imaginary quadratic field, J. reine und angew. Math., 526 (2000), 155--179. • H. Kojima, On the Fourier coefficients of Jacobi forms of index$N$over totally real number fields, preprint (2003). • H. Kojima and Y. Tokuno, On the Fourier coefficients of modular forms of half integral weight belonging to Kohnen's spaces and the critical values of zeta functions, Tohoku Math. J., 56 (2004), 125--145. • W. Luo and D. Ramakrishnan, Determination of modular forms by twists of critical$L$-values, Invent. Math., 130 (1997), 371--398. • H. Sakata, On the Kohnen-Zagier Formula in the case of level$4p^m\$, Math. Zeit., 250 (2005), 257--266.
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