In this paper, we propose an algorithm for computing mod $ \ell $ Galois representations associated to modular forms of weight $ k $ when $ \ell < k-1 $. We also present the corresponding results for the projective Galois representations. Moreover, we apply our algorithms to explicitly compute the mod $ \ell $ projective Galois representations associated to $ \Delta_{k} $ for $ k = 16, 20, 22, 26 $ and all the unexceptional primes $ \ell $, with $ \ell < k-1 $. As an application, for $ k = 16, 20, 22, 26 $, we obtain the new bounds $ B_k $ of $ n $ such that $ a_n(\Delta_k)\ne0 $ for all $ n < B_k $.
Citation: Peng Tian, Ha Thanh Nguyen Tran, Dung Hoang Duong. Computing mod $ \ell $ Galois representations associated to modular forms for small primes[J]. AIMS Mathematics, 2023, 8(12): 28766-28779. doi: 10.3934/math.20231473
In this paper, we propose an algorithm for computing mod $ \ell $ Galois representations associated to modular forms of weight $ k $ when $ \ell < k-1 $. We also present the corresponding results for the projective Galois representations. Moreover, we apply our algorithms to explicitly compute the mod $ \ell $ projective Galois representations associated to $ \Delta_{k} $ for $ k = 16, 20, 22, 26 $ and all the unexceptional primes $ \ell $, with $ \ell < k-1 $. As an application, for $ k = 16, 20, 22, 26 $, we obtain the new bounds $ B_k $ of $ n $ such that $ a_n(\Delta_k)\ne0 $ for all $ n < B_k $.
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Peng Tian, Ha Thanh Nguyen Tran, Dung Hoang Duong. Computing mod $ \ell $ Galois representations associated to modular forms for small primes[J]. AIMS Mathematics, 2023, 8(12): 28766-28779. doi: 10.3934/math.20231473
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