Eta Quotients of Level $18$ and Weight $1$: Classification and Applications
Resumo
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 $\mathbb{Q}$.
Downloads
Não há dados estatísticos.
Referências
\bibitem{Diamond} F. Diamond and J. M. Shurman,
{\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
{\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
Publicado
2026-01-22
Seção
Advances in Algebra, Analysis, Optimization, and Modeling
Copyright (c) 2026 Boletim da Sociedade Paranaense de Matemática
This work is licensed under a Creative Commons Attribution 4.0 International License.
When the manuscript is accepted for publication, the authors agree automatically to transfer the copyright to the (SPM).
The journal utilize the Creative Common Attribution (CC-BY 4.0).
