## Abstract

We consider the simultaneous sign changes of normalized Fourier coefficients associated to two distinct Hecke eigenforms supported at positive integers represented by a primitive integral binary quadratic form with negative discriminant whose class number is 1. We provide a quantitative result for the number of sign changes of such sequence in the interval \((x,2x]\) for sufficiently large \(x\).

This is a preview of subscription content, access via your institution.

## References

K. Aggarwal, A new subconvex bound for \(GL(3)\)

*L*-functions in the t-aspect, Int. J. Number Theory, 17 (2021), 1111–1138S. Banerjee and M. K. Pandey, Signs of Fourier coefficients of cusp form at sum of two squares, Proc. Indian Acad. Sci. Math. Sci., 130 (2020), Paper No. 2, 9 pp.

J. Bourgain, Decoupling, exponential sums and the Riemann zeta function, J. Amer. Math. Soc., 30 (2017), 205–224.

L. Clozel and J. A. Thorne, Level-raising and symmetric power functoriality. I, Compos. Math., 150 (2014), 729–748.

L. Clozel and J. A. Thorne, Level-raising and symmetric power functoriality. II, Ann. of Math., 181 (2015), 303–359.

L. Clozel and J. A. Thorne, Level-raising and symmetric power functoriality. III, Duke Math. J., 166 (2017), 325-402.

P. Deligne, La Conjecture de Weil. I, Inst. Hautes Études Sci. Pull. Math., 43 (1974), 273–307.

L. Dieulefait, Automorphy of Symm\(^{5}(GL(2))\) and base change, J. Math. Pures Appl., 104 (2015), 619–656.

S. Gelbart and H. Jacquet, A relation between automorphic representations of GL(2) and GL(3), Ann. Sci. École Norm. Sup., 11 (1978), 471–542.

S. Gun, W. Kohnen and P. Rath, Simultaneous sign change of Fourier-coefficients of two cusp forms, Arch. Math., 105 (2015), 413–424.

D. R. Heath-Brown, The growth rate of the Dedekind zeta-function on the critical line, Acta Arith., 49 (1988), 323–339.

H. Iwaniec, Topics in Classical Automorphic Forms, Grad. Stud. Math., vol. 17, Amer. Math. Soc. (Providence, RI, 1997).

H. Iwaniec and E. Kowalski, Analytic Number Theory, Amer. Math. Soc. Colloquium Publ., vol. 53, Amer. Math. Soc. (Providence, RI, 2004).

H. Jacquet and J. A. Shalika, On the Euler products and the classification of automorphic representations. I, Amer. J. Math., 103 (1981), 499–558.

H. Jacquet and J. A. Shalika, On the Euler products and the classification of automorphic forms. II, Amer. J. Math., 103 (1981), 777–815.

H. Kim and F. Shahidi, Functorial products for \(GL_2 \times GL_3\) and functorial symmetric cube for \(GL_2\), with an appendix by C. J. Bushnell and G. Heniart, Ann. of Math., 155 (2002), 837–893.

H. Kim and F. Shahidi, Cuspidality of symmetric power with applications, Duke Math. J., 112 (2002), 177–107.

H. Kim, Functoriality for the exterior square of \(GL_4\) and symmetric fourth of \(GL_2\), Appendix 1 by D. Ramakrishan, Appendix 2 by H. Kim and P. Sarnak, J. Amer. Math. Soc., 16 (2003), 139–183.

M. Knopp, W. Kohnen and W. Pribitkin, On the signs of Fourier coefficients of cusp forms, Ramanujan J., 7 (2003), 269–277.

W. Kohnen and J. Sengupta, Signs of Fourier coefficients of two cusp forms of different weights, Proc. Amer. Math. Soc., 137 (2009), 3563–3567.

W. Kohnen and Y. Martin, Sign changes of Fourier coefficients of cusp forms supported on prime power indices, Int. J. Number Theory, 10 (2014), 1921–1927.

M. Kumari and M. R. Murty, Simultaneous non-vanishing and sign changes of Fourier coefficients of modular forms, Int. J. Number Theory, 14 (2018), 2291–2301.

Y.-K. Lau and G. S. L¨u, Sums of Fourier coefficients of cusp forms, Quart. J. Math., 62 (2011), 687–716.

H. X. Lao and S. Luo, Sign changes and non-vanishing of Fourier coefficients of holomorphic cusp forms, Rocky Mountain J. Math., 51 (2021), 1701–1714.

M Ram Murty, Oscillations of the Fourier coefficients of modular forms, Math. Ann., 262 (1983), 431–446.

J. Meher, K. D. Shankhadhar and G. K. Viswanadham, A short note on sign changes, Proc. Indian Acad. Sci. (Math. Sci.), 123 (3) (2013), 315–320.

J. Newton, J. A. Thorne, Symmetric power functoriality for holomorphic modular forms, Publ. Math. Inst. Hautes ´ Etudes Sci., 134 (2021), 1–116.

J. Newton, J. A. Thorne, Symmetric power functoriality for holomorphic modular forms. II, Publ. Math. Inst. Hautes Études Sci., 134 (2021), 117–152.

A. Perelli, General L-functions, Ann. Mat. Pura Appl., 130 (1982), 287–306.

Z. Rudnick and P. Sarnak, Zeros of principal L-functions and random matrix theory, Duke Math. J., 81 (1996), 269–322.

C. L. Siegel, Berechnung von Zetafunctionen an ganzzanhligen Stellen, Nachr. Akad. Wiss. G¨ottingen Math. Phys. K1. II, 2 (1969), 87–102.

F. Shahidi, On certain L-functions, Amer. J. Math., 103 (1981), 297–355.

F. Shahidi, Third symmetric power L-functions for GL(2), Compos. Math., 70 (1989), 245–273.

L. Vaishya, Signs of Fourier coefficients of cusp forms at integers represented by an integral binary quadratic form, Proc. Indian Acad. Sci. Math. Sci., 131 (2021), Paper No. 41, 14 pp.

## Acknowledgements

The author would like to express his gratitude to Professor Guangshi Lü and Professor Bin Chen for their constant encouragement and valuable suggestions. The author is extremely grateful to the anonymous referee for meticulous checking, for thoroughly reporting countless typos and inaccuracies as well as for valuable comments. These corrections and additions have made the manuscript clearer and readable.

## Author information

### Authors and Affiliations

### Corresponding author

## Additional information

This work is supported in part by the National Key Research and Development Program of China (Grant No. 2021YFA1000700).

## Rights and permissions

## About this article

### Cite this article

HUA, G. ON THE SIMULTANEOUS SIGN CHANGES OF HECKE EIGENVALUES OVER AN INTEGRAL BINARY QUADRATIC FORM.
*Acta Math. Hungar.* **167**, 476–491 (2022). https://doi.org/10.1007/s10474-022-01252-z

Received:

Revised:

Accepted:

Published:

Issue Date:

DOI: https://doi.org/10.1007/s10474-022-01252-z

### Key words and phrases

- Hecke eigenform
- Fourier coefficient
- simultaneous sign change

### Mathematics Subject Classification

- 11F11
- 11F30
- 11N37