We find some modularity criterion for a product of Klein forms of the congruence subgroup Γ 1 (N) (Theorem 2.6) and, as its application, construct a basis of the space of modular forms for Γ 1 (13) of weight 2 (Example 3.4). In the process we face with an interesting property about the coefficients of certain theta function from a quadratic form and prove it conditionally by applying Hecke operators (Proposition 4.3).

Let K be an imaginary quadratic field different from Q( √ −1) and Q( √ −3). For a positive integer N , let K n be the ray class field of K modulo n = N O K . By using the congruence subgroup ±Γ 1 (N ), we construct an extended form class group whose operation is basically the Dirichlet composition, and explicitly show that this group is isomorphic to the Galois group Gal(K n /K). We also present algorithms to find all form classes and show how to multiply two form classes. As an application, we describe Gal(K ab n /K) in terms of these extended form class groups for which K ab n is the maximal abelian extension of K unramified outside prime ideals dividing n.

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