### Abstract

Let k be an even integer and S_{k} be the space of cusp forms of weight k on SL_{2}(Z). Let S=⊕_{k∈2Z}S_{k}. For f,g∈S, we let R(f,g)be the set of ratios of the Fourier coefficients of f and g defined by R(f,g):={x∈P^{1}(C)|x=[a_{f}(p):a_{g}(p)]for some primep}, where a_{f}(n)(resp. a_{g}(n))denotes the nth Fourier coefficient of f (resp. g). In this paper, we prove that if f and g are nonzero and R(f,g)is finite, then f=cg for some constant c. This result is extended to the space of weakly holomorphic modular forms on SL_{2}(Z). We apply it to study the number of representations of a positive integer by a quadratic form.

Original language | English |
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Pages (from-to) | 298-315 |

Number of pages | 18 |

Journal | Journal of Number Theory |

Volume | 202 |

DOIs | |

Publication status | Published - 2019 Sep 1 |

### Keywords

- Fourier coefficient
- Galois representation
- Modular form

### ASJC Scopus subject areas

- Algebra and Number Theory

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## Cite this

Choi, D., & Lim, S. (2019). Independence between coefficients of two modular forms.

*Journal of Number Theory*,*202*, 298-315. https://doi.org/10.1016/j.jnt.2019.01.005