Let g be a fixed normalized Hecke-Maass cusp form for SL(2, Z) associated to the Laplace eigenvalue 1 4 + ν 2 . We show that g is uniquely determined by the central values of the family {L(s, f ⊗ g): g ∈ H k (1)} for k sufficiently large, where H k (1) denotes a Hecke basis of the space of holomorphic cusp forms for SL(2, Z).

Let H 2 k ± ( N 3 ) H^{\pm }_{2k} (N^3) denote the set of modular newforms of cubic level N 3 N^3 , weight 2 k 2 k , and root number ± 1 \pm 1 . For N > 1 N > 1 squarefree and k > 1 k>1 , we use an analytic method to establish neat and explicit formulas for the difference | H 2 k + ( N 3 ) | − | H 2 k − ( N 3 ) | |H^{+}_{2k} (N^3)| - |H^{-}_{2k} (N^3)| as a multiple of the product of φ ( N ) \varphi (N) and the class number of Q ( − N ) \mathbb {Q}(\sqrt {- N}) . In particular, the formulas exhibit a strict bias towards the root number + 1 +1 . Our main tool is a root-number weighted simple Petersson formula for such newforms.

Let f (z) = ∞ n=1 λ f (n)n (κ−1)/2 e(nz) be a holomorphic cusp form of weight κ for the full modular group SL 2 (Z). In this paper we study the cancellation of the coefficients λ f (n) over primes in exponential sums.

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Let π be an irreducible unitary cuspidal representation of GLm(A Q ), m ≥ 2. Assume that π is self-contragredient. The author gets upper and lower bounds of the same order for fractional moments of automorphic L-function L(s, π) on the critical line under Generalized Ramanujan Conjecture; the upper bound being conditionally subject to the truth of Generalized Riemann Hypothesis.