Eigenforms, Hecke operators
It is well known that newforms of integral weight are simultaneous eigenforms for all the Hecke operators, and that the converse is not true. In this paper, we give a characterization of all simultaneous Hecke eigenforms associated to a given newform, and provide several applications. These include determining the number of linearly independent simultaneous eigenforms in a fixed space which correspond to a given newform, and characterizing several situations in which the full space of cusp forms is spanned by a basis consisting of such eigenforms. Part of our results can be seen as a generalization of results of Choie-Kohnen who considered diagonalization of “bad” Hecke operators on spaces with square free level and trivial character. Of independent interest, but used herein, is a lower bound for the dimension of the space of newforms with arbitrary character.
International Journal of Number Theory
Required Publisher's Statement
Preprint of an article published in International Journal of Number Theory, Volume 06, Issue 05, 2010, 1117-1137, https://doi.org/10.1142/S1793042110003411, © copyright World Scientific Publishing Company,https://www.worldscientific.com/toc/ijnt/06/05
Shemanske, T.; Treneer, Stephanie; and Walling, Lynne H., "Constructing Simultaneous Hecke Eigenforms" (2010). Mathematics. 107.
Copying of this document in whole or in part is allowable only for scholarly purposes. It is understood, however, that any copying or publication of this document for commercial purposes, or for financial gain, shall not be allowed without the author’s written permission.