Suppose that f is a primitive Hecke eigenform or a Mass cusp form for Γ₀(N) with normalized eigenvalues λf (n) and let X > 1 be a real number. We consider the sum \({{\cal S}_k}(X): = \sum\limits_{n < X} {\sum\limits_{n = {n_1},{n_2}, \ldots ,{n_k}} {{\lambda _f}({n_1}){\lambda _f}({n_2}) \ldots {\lambda _f}({n_k})}}\) and show that \({{\cal S}_k}(X){\ll _{f,\varepsilon }}{X^{1 - 3/(2(k + 3)) + \varepsilon}}\) for every k ⩾ 1 and ε > 0. The same problem was considered for the case N = 1, that is for the full modular group in Lü (2012) and Kanemitsu et al. (2002). We consider the problem in a more general setting and obtain bounds which are better than those obtained by the classical result of Landau (1915) for k ⩾ 5. Since the result is valid for arbitrary level, we obtain, as a corollary, estimates on sums of the form \({{\cal S}_k}(X)\), where the sum involves restricted coefficients of some suitable half integral weight modular forms.

假设f是具有归一化特征值λf ( n ) 的Γ ₀ ( N )的原始 Hecke 特征形式或质量尖点形式，并让X > 1 为实数。我们考虑和\({{\cal S}_k}(X): = \sum\limits_{n < X} {\sum\limits_{n = {n_1},{n_2}, \ldots ,{n_k} } {{\lambda _f}({n_1}){\lambda _f}({n_2}) \ldots {\lambda _f}({n_k})}}\)并证明\({{\cal S}_k }(X){\ll _{f,\varepsilon }}{X^{1 - 3/(2(k + 3)) + \varepsilon}}\)对于每个k ⩾ 1 和ε > 0. 相同问题被考虑用于案例N= 1，即 Lü (2012) 和 Kanemitsu 等人的全模组。(2002)。我们在更一般的环境中考虑这个问题，并获得比 Landau (1915) 的经典结果获得的边界更好的边界，对于k ⩾ 5。由于结果对任意级别有效，因此我们获得作为推论的估计形式的总和\（{{\ CAL S} _K}（X）\） ，其中总和涉及限制的一些合适的半积分重量模形式系数。