On Lower Bounds for sums of Fourier Coefficients of Twist-Inequivalent Newforms
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arXiv
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| Natura: | Preprint |
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2026
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| _version_ | 1866910113825030144 |
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| author | Kumari, Moni Mishra, Prabhat Kumar Sengupta, Jyotirmoy |
| author_facet | Kumari, Moni Mishra, Prabhat Kumar Sengupta, Jyotirmoy |
| contents | In this article, we address the lower bounds for the sums $a_f(p)+a_g(p)$ of the $p$-th Fourier coefficients of two twist-inequivalent, non-CM normalized newforms $f$ and $g$. Our main result shows that for such forms with integer Fourier coefficients, the largest prime factor of $a_f(p)+a_g(p)$ satisfies $P(a_f(p)+a_g(p)) > (\log p)^{1/14} (\log \log p)^{3/7-ε}$ for almost all primes $p$ and for any $ε> 0$. Beyond primes, we apply Brun's sieve to show that a similar phenomenon holds for a set of positive integers with natural density one. The main result is further strengthened under the Generalized Riemann Hypothesis, where we establish exponential growth for the absolute value of $a_f(p)+a_g(p)$ in terms of $p$.Additionally, we derive an interesting result related to the multiplicity one theorem, demonstrating that if the sum $a_f(p)+a_g(p)$ is small for a positive-density subset of primes, then $f$ and $g$ must be twist-equivalent by a quadratic character. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_07474 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On Lower Bounds for sums of Fourier Coefficients of Twist-Inequivalent Newforms Kumari, Moni Mishra, Prabhat Kumar Sengupta, Jyotirmoy Number Theory 11F11(Primary) 11F30, 11F80(Secondary) In this article, we address the lower bounds for the sums $a_f(p)+a_g(p)$ of the $p$-th Fourier coefficients of two twist-inequivalent, non-CM normalized newforms $f$ and $g$. Our main result shows that for such forms with integer Fourier coefficients, the largest prime factor of $a_f(p)+a_g(p)$ satisfies $P(a_f(p)+a_g(p)) > (\log p)^{1/14} (\log \log p)^{3/7-ε}$ for almost all primes $p$ and for any $ε> 0$. Beyond primes, we apply Brun's sieve to show that a similar phenomenon holds for a set of positive integers with natural density one. The main result is further strengthened under the Generalized Riemann Hypothesis, where we establish exponential growth for the absolute value of $a_f(p)+a_g(p)$ in terms of $p$.Additionally, we derive an interesting result related to the multiplicity one theorem, demonstrating that if the sum $a_f(p)+a_g(p)$ is small for a positive-density subset of primes, then $f$ and $g$ must be twist-equivalent by a quadratic character. |
| title | On Lower Bounds for sums of Fourier Coefficients of Twist-Inequivalent Newforms |
| topic | Number Theory 11F11(Primary) 11F30, 11F80(Secondary) |
| url | https://arxiv.org/abs/2604.07474 |