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The purpose of this paper is to prove that the symmetric fourth power of a cusp form on GL(2), whose existence was proved earlier by the first author, is cuspidal unless the corresponding automorphic representation is of dihedral, tetrahedral, or octahedral type. As a consequence, we prove a number of results toward the Ramanujan-Petersson and Sato-Tate conjectures. In particular, we establish the bound q 1/9 v for unramified Hecke eigenvalues of cusp forms on GL(2). Over an arbitrary number field, this is the best bound available at present.

By using new techniques with the degenerate Whittaker functions found by Ikeda-Yamana, we construct higher level cusp form on E ad 7,3 (adjoint exceptional group of type E7), called Ikeda type lift, from any Hecke cusp form whose corresponding automorphic representation has no supercupidal local components. This generalizes the results in [16] on level one forms. But there are new phenomena in higher levels; first, we can handle any non-trivial central characters. Second, the lift does not depends on the choice of an irreducible cuspidal constituent of the restriction of the Hecke cusp form to SL2. Hence any twist of the cusp form gives rise to the same lift. We also compute the degree 133 adjoint L-function of the Ikeda type lift. 5.5. Automorphic forms on G(A Q ) 15 6. Automorphic forms on SL 2 (A Q ) 17 7. Fourier-Jacobi expansions 20 7.1. The non-archimedean case 20 7.2. The archimedean case 22

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