## Journal of the Mathematical Society of Japan

### Remark on Fourier coefficients of modular forms of half integral weight belonging to Kohnen's spaces

Hisashi KOJIMA

#### Abstract

The purpose of this paper is to derive that the square of Fourier coefficients $a(n)$ at a square free positive integer $n$ of modular forms $f$ of half integral weight belonging to Kohnen's spaces of arbitrary odd level and of arbitrary primitive character is essentially equal to the critical value of the zeta function attached to the modular form $F$ of integral weight which is the image of $f$ under the Shimura correspondence. Previously, KohnenZagier had obtained an analogous result in the case of Kohnen's spaces of square free level and of trivial character. Our results give some generalizations of them of KohnenZagier. Our method of the proof is similar to that of Shimura's paper concerning Fourier coefficients of Hilbert modular forms of half integral weight over totally real fields.

#### Article information

Source
J. Math. Soc. Japan, Volume 51, Number 3 (1999), 715-730.

Dates
First available in Project Euclid: 10 June 2008

https://projecteuclid.org/euclid.jmsj/1213107913

Digital Object Identifier
doi:10.2969/jmsj/05130715

Mathematical Reviews number (MathSciNet)
MR1691461

Zentralblatt MATH identifier
1012.11035

#### Citation

KOJIMA, Hisashi. Remark on Fourier coefficients of modular forms of half integral weight belonging to Kohnen's spaces. J. Math. Soc. Japan 51 (1999), no. 3, 715--730. doi:10.2969/jmsj/05130715. https://projecteuclid.org/euclid.jmsj/1213107913