This is a 66MB scan of Miyake, which is an extremely comprehensive book about modular forms.

According to Amazon.com, this book is "Out of Print--Limited Availability", so I've made it available online.

The author offers a complete collection of definitions, formulas, and proofs as required for modular forms in one variable, particularly for Hecke operators (where the trace formulas are painstakingly derived). This book serves as a valuable source and handy secondary reference for results. Indeed, the methods and terminology are of a (minimal) handbook style, without excessive abstraction. The chapters are on Fuchsian groups, Automorphic forms, $L$-functions, Modular groups and forms, Unit groups of quaternion algebras, Traces of Hecke operators, and Eisenstein series. This is a Japanese-to-English translation of part of a joint book with \n K. Doi\en \ref[ Automorphic forms and number theory (Japa-\break nese), Kinokuniya, Tokyo, 1976; per bibl.]. There are three accompanying tables. Table A lists the dimensions of cusp forms for $\Gamma_0(N)$ with even weights $2\leq k\leq 50$ and level $1\leq N\leq 50$ and prime $50\leq N\leq 100$. The same is done for newforms. A shorter supplementary table does the same for prime $N$ and with quadratic $\chi=(*/N)$. Table B has eigenvalues and characteristic equations of $T(p)$ for some primes $11\leq N\leq 71$ and weight $2$ (but $\chi=1$). Table C is a specialization of Table B to the cases $N=29,37$, with quadratic $\chi$. The tables, attributed to \n Y. Maeda\en, \n H. Wada\en, and \n N. Iwasaki\en, are covered by formulas in the text but are not further explained or examined for interesting entries through a post-mortem.
Reviewed by Harvey Cohn |