Realization of modular Galois representations in the Jacobians of modular curves

In Tian (Acta Arith. 164:399-412, 2014), the author improved the algorithm proposed by Edixhoven and Couveignes for computing mod l Galois representations associated to eigenforms f for the cases that l >= k - 1 and f has level one, where k is the weight of f. In this paper, we generalize the results of Tian (Acta Arith. 164:399-412, 2014) and present a method to find the Jacobians of modular curves of minimal dimensions to realize the modular Galois representations. Our method works for the cases that l >= 5 may be any prime without the assumption l >= - 1 and the eigenforms f have arbitrary levels prime to l. Moreover, if k > 2, we give criteria for realizing the mod l Galois representations in the Jacobians J(0) of X-0.