Eta Quotients of Level $18$ and Weight $1$: Classification and Applications
Résumé
We classify all eta quotients in the space $M_{1}\left(\Gamma_{0}(18), \left(\frac{-3}{*}\right)\right)$ of modular formsand explicitly compute their Fourier coefficients,
where $\left(\frac{d}{*}\right)$ denotes the Legendre–Jacobi–Kronecker symbol,
viewed as a Dirichlet character modulo $18$ taking values in $\mathbb{Q}$.
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Références
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{\it A first course in modular forms}. Grad. Texts in Math., Springer, New York, (2005).
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{\it Eta products and theta series identities}. Springer Monogr. Math., Springer, New York, (2011). https://doi.org/10.1007/978-3-642-16152-0
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{\it Courbes modulaires de genre $1$}. Bull. Soc. Math. France {\bf 43}, 5--80, (1975). http://eudml.org/doc/94716
\bibitem{Okamoto} A. Okamoto,
{\it On expressions of theta series by $\eta$-products}. Tokyo J. Math. {\bf 34}(2), 319--326, (2011). https://doi.org/10.3836/tjm/1327931388
\bibitem{Stein} W. A. Stein,
{\it Modular forms, a computational approach}. Grad. Stud. Math., Vol. 79, Amer. Math. Soc., Providence, RI, (2007). http://dx.doi.org/10.1090/gsm/079
\bibitem{bib7} W. A. Stein et al.,
{\it Sage Mathematics Software (Version 10.4)}. The Sage Development Team, (2023). http://www.sagemath.org/
\bibitem{Williams} K. S. Williams,
{\it Fourier series of a class of eta quotients}. Int. J. Number Theory {\bf 8}(4), 993--1004, (2012). https://doi.org/10.1142/S1793042112500595
{\it A first course in modular forms}. Grad. Texts in Math., Springer, New York, (2005).
\bibitem{Gordon} B. Gordon and D. Sinor,
{\it Multiplicative properties of $\eta$-products}. In Number Theory, Madras 1987: Proceedings of the International Ramanujan Centenary Conference held at Anna University, Madras, India, Dec. 21, 1987, Springer, Berlin, Heidelberg, 173--200, (2006). https://doi.org/10.1007/BFb0086404
\bibitem{Kilford} L. J. P. Kilford,
{\it Modular forms: A classical and computational introduction}. 2nd ed., Imperial College Press, London, (2015).
\bibitem{Kohler} G. K\"{o}hler,
{\it Eta products and theta series identities}. Springer Monogr. Math., Springer, New York, (2011). https://doi.org/10.1007/978-3-642-16152-0
\bibitem{Ligozat} G. Ligozat,
{\it Courbes modulaires de genre $1$}. Bull. Soc. Math. France {\bf 43}, 5--80, (1975). http://eudml.org/doc/94716
\bibitem{Okamoto} A. Okamoto,
{\it On expressions of theta series by $\eta$-products}. Tokyo J. Math. {\bf 34}(2), 319--326, (2011). https://doi.org/10.3836/tjm/1327931388
\bibitem{Stein} W. A. Stein,
{\it Modular forms, a computational approach}. Grad. Stud. Math., Vol. 79, Amer. Math. Soc., Providence, RI, (2007). http://dx.doi.org/10.1090/gsm/079
\bibitem{bib7} W. A. Stein et al.,
{\it Sage Mathematics Software (Version 10.4)}. The Sage Development Team, (2023). http://www.sagemath.org/
\bibitem{Williams} K. S. Williams,
{\it Fourier series of a class of eta quotients}. Int. J. Number Theory {\bf 8}(4), 993--1004, (2012). https://doi.org/10.1142/S1793042112500595
Publiée
2026-01-22
Rubrique
Advances in Algebra, Analysis, Optimization, and Modeling
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