Abstract
We explore the theta functions of nineteen positive-definite integral non-diagonal quaternary quadratic forms of discriminant 784 with levels 28 or 56. We express these theta functions in terms of Eisenstein series and cusp forms, which we then use to give explicit formulas for the representation number of a positive integer n by their corresponding non-diagonal quaternary quadratic forms. We also find the theta functions of the genera to which those non-diagonal quaternary quadratic forms belong. Finally, we express the theta function of each non-diagonal quadratic form in terms of the theta functions of certain diagonal quaternary quadratic forms and a cusp form.
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The research of the author is supported by the Discovery Grant RGPIN-418029-2013 from the Natural Sciences and Engineering Research Council of Canada.
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Communicated by Sanoli Gun.
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Alaca, A. Theta functions of nineteen non-diagonal positive-definite quaternary quadratic forms of discriminant 784 with levels 28 or 56. Indian J Pure Appl Math 54, 595–607 (2023). https://doi.org/10.1007/s13226-022-00278-x
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DOI: https://doi.org/10.1007/s13226-022-00278-x
Keywords
- Theta functions
- Modular forms
- Genus of a quaternary quadratic form
- Class of a quaternary quadratic form
- Representaion number
- Dedekind eta function
- Eta quotients