On a property of Fourier coefficients of cusp forms of half-integral weight
It is well known that the Fourier coefficients \(A_ n\) of a primitive cusp form \[ F(z)= \sum_{n\geq 1} A_ n\exp 2\pi inz \] of integral weight \(k\) with level \(N\) and character \(\chi\) satisfy \(A_ n= \chi(n) \overline{A}_ n\) provided \((n,N )=1\). Using this and the main theorem of Waldspurger similar relations are achieved for cusp forms \(f(z)= \sum a_ n\exp 2\pi inz\) of half-integral weight \({k\over 2}\) with level \(N\) and character \(\chi\) which are common eigenfunctions of all Hecke operators \(T(p^ 2)\), \((p,N)=1\): \[ a_ m \overline{a}_ n \chi(n)= \overline{a}_ m a_ n \chi(m) \] provided the square free positive integers \(n\) and \(m\) are relatively prime to \(N\) and satisfy \(m/n\in (\mathbb{Q}_ p^ \times )^ 2\) for all prime numbers \(p\) dividing \(N\). Furthermore it is assumed that certain values of the \(L\)-functions of the primitive cusp-form \(F\) of weight \(k-1\) with level \(N\) and character \(\chi^ 2\) corresponding to \(f\) do not vanish. Finally Fourier coefficients of forms forming a basis of the space of cusp forms of weight 5/2, level 28 and with character \(\chi\) of order 3 defined modulo 7 are tabulated.
scientific article; zbMATH DE number 701494