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A family of newforms. (English) Zbl 0595.10019

For each \(r\in {\mathbb{N}}\) the author defines a newform \(g_ r\) by \(g_ r(z)=\sum^{\infty}_{n=1}a_ r(n) e^{2\pi inz}\) (Im z\(>0)\), where the coefficient \(a_ r(n)\) is determined by a Grössencharacter belonging to the field of Gaussian integers. The cusp form \(g_ r\) has weight \(r+1\) and level 64. He is primarily concerned with obtaining estimates for sums of the type \[ A(\beta,r; x)=\sum_{n\leq x}| \alpha_ r(n)|^{2\beta}, \] as \(x\to \infty\), where \(\beta >0\), \(r\in {\mathbb{N}}\) and \(\alpha_ r(n)=a_ r(n)n^{-r/2}.\)
Upper and lower bounds are provided for A(\(\beta\),r; x) when x is large. He also provides a precise asymptotic formula for A(\(\beta\),r; x) as \(x\to \infty\), for the case where \(\beta\in {\mathbb{N}}\). Of particular interest is the formula for A(2,r; x). C. J. Moreno and F. Shahidi [Math. Ann. 266, 233-239 (1983; Zbl 0508.10014)] have obtained an asymptotic formula for the sum \(S(x)=\sum_{n\leq x}\tau^ 4(n) n^{-22}\), as \(x\to \infty\), where \(\tau\) (n) is Ramanujan’s function. For each \(r\in {\mathbb{N}}\) the asymptotic behaviour of A(2,r; x) differs from that of S(x).
The author states that the methods of this paper can be applied to cusp forms associated in a similar way with Grössencharacters belonging to other imaginary quadratic fields.
Reviewer: A.W.Mason

MSC:

11F11 Holomorphic modular forms of integral weight
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