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The paper is concerned with Shimura sums related to modular forms with multiplicative coefficients which are products of Dedekind η-functions of various arguments. Several identities involving Shimura sums are established. The type of identity obtained depends on the splitting of primes in certain imaginary quadratic number fields.

The study of relations between finite groups and modular forms is an interesting topic of modern mathematical investigations. Different ways of assigning modular forms to the elements of a group are considered. One of these mappings is as follows: let G be a finite group, let 9 be an element of G, let be a unimodular representation of the group G in the space V whose dimension is a multiple of 24, and let It is an interesting problem to find the finite groups to may of whose elements we can assign a modular form that is an eigenvector of all tiecke operators.In [2] it is proved that there exist exactly 30 cusp forms expressible as l-It, qt~(akz), where at,, tk E N, that are eigenvectors of all Hecke operators. These cusp forms are characterized by the condition ~k aktt = 24, where for all k we must have ak [ ak+l. The complete list of such forms is presented below. Two of these cusp forms have half-integer weight. The authors called these 30 forms multiplicative q-productn (this term can be explained by the fact that the coefficients of expansions in powers of q are multiplicative). In what follows, we use this term as well, for brevity and convenience.On the other hand, the author of the present note proved [3] that these 30 functions and only these functions are cusp forms that are eigenvectors of all Hecke operators and have no zeros on the upper complex half-plane outside the parabolic cusps.Mason [4] proved that for the Mathieu group M24 and for its representation on the Leech lattice, all cusp forms associated with elements 9 E M~4 are multiplicative w-products.The author of this note described all finite subgroups of SL(5, C) to any of whose elements one can assign, by means of the adjoint representation, a multiplicative q-product [6]. In the following theorem we give another example of such a group.

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