/* This code can be loaded, or copied and paste using cpaste, into Sage. It will load the data associated to the HMF, including the field, level, and Hecke and Atkin-Lehner eigenvalue data. */ P. = PolynomialRing(QQ) g = P([-1, 3, 2, -5, -1, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([53, 53, 2*w^4 - 2*w^3 - 9*w^2 + 2*w + 2]) primes_array = [ [7, 7, w^4 - 6*w^2 - 2*w + 4],\ [9, 3, -w^4 + w^3 + 5*w^2 - w - 4],\ [11, 11, -2*w^4 + w^3 + 10*w^2 + w - 3],\ [11, 11, w^4 - 6*w^2 - 3*w + 3],\ [17, 17, w^2 - 2],\ [23, 23, -w^3 + w^2 + 3*w],\ [27, 3, w^4 - w^3 - 4*w^2 + 2*w - 1],\ [29, 29, 2*w^4 - 2*w^3 - 9*w^2 + 2*w + 3],\ [32, 2, -2],\ [47, 47, w^4 - 2*w^3 - 3*w^2 + 5*w],\ [47, 47, w^3 - w^2 - 4*w - 1],\ [53, 53, -w^4 + 7*w^2 - 3],\ [53, 53, 2*w^4 - 2*w^3 - 9*w^2 + 2*w + 2],\ [53, 53, 3*w^4 - 2*w^3 - 16*w^2 + 8],\ [67, 67, -w^4 + 6*w^2 + 4*w - 3],\ [73, 73, 2*w^4 - 12*w^2 - 4*w + 5],\ [83, 83, w^4 - 5*w^2 - 3*w + 3],\ [97, 97, -w^4 + w^3 + 5*w^2 - 3*w - 3],\ [103, 103, 2*w^3 - 3*w^2 - 7*w + 3],\ [109, 109, -3*w^4 + 2*w^3 + 14*w^2 + w - 6],\ [113, 113, -2*w^4 + 2*w^3 + 8*w^2 - w - 2],\ [137, 137, -2*w^4 + 3*w^3 + 9*w^2 - 7*w - 4],\ [139, 139, 2*w^4 - 2*w^3 - 8*w^2 + 3*w + 1],\ [157, 157, -w^4 + 2*w^3 + 4*w^2 - 5*w - 2],\ [157, 157, w^4 - 6*w^2 - 4*w + 6],\ [163, 163, 2*w^4 - 2*w^3 - 8*w^2 + 2*w + 3],\ [169, 13, w^4 - w^3 - 6*w^2 + w + 6],\ [173, 173, -w^4 + w^3 + 4*w^2 - 2*w + 3],\ [173, 173, -3*w^4 + w^3 + 17*w^2 + 3*w - 10],\ [173, 173, -3*w^4 + 3*w^3 + 14*w^2 - 5*w - 7],\ [179, 179, -2*w^4 + w^3 + 10*w^2 + 2*w - 7],\ [179, 179, 2*w^4 - 12*w^2 - 3*w + 6],\ [181, 181, 3*w^4 - 2*w^3 - 14*w^2 - 2*w + 4],\ [181, 181, w^4 - 2*w^3 - 4*w^2 + 5*w + 3],\ [191, 191, 3*w^4 - w^3 - 16*w^2 - 5*w + 7],\ [191, 191, 4*w^4 - 3*w^3 - 20*w^2 + 2*w + 8],\ [191, 191, -2*w^4 + 2*w^3 + 10*w^2 - 3*w - 3],\ [193, 193, -2*w^4 + 2*w^3 + 9*w^2 - 3*w - 6],\ [197, 197, -2*w^4 + w^3 + 9*w^2 + 2*w - 4],\ [197, 197, -2*w^4 + 2*w^3 + 8*w^2 - w + 1],\ [197, 197, -w^4 + w^3 + 4*w^2 - 4],\ [199, 199, w^4 - 7*w^2 - 3*w + 5],\ [199, 199, -2*w^4 + 2*w^3 + 9*w^2 - 4*w - 5],\ [211, 211, w^2 - 5],\ [211, 211, 2*w^4 - 12*w^2 - 5*w + 7],\ [211, 211, w^4 - 2*w^3 - 4*w^2 + 7*w + 1],\ [227, 227, -3*w^4 + 3*w^3 + 14*w^2 - 3*w - 6],\ [227, 227, w^4 + w^3 - 6*w^2 - 7*w + 4],\ [229, 229, w^3 - 6*w - 2],\ [233, 233, w^4 - 5*w^2 - 4*w + 4],\ [233, 233, w^4 - 2*w^3 - 3*w^2 + 5*w + 1],\ [239, 239, -w^4 + w^3 + 5*w^2 - 5],\ [239, 239, w^4 - w^3 - 6*w^2 + 3*w + 3],\ [241, 241, -w^3 + w^2 + 6*w - 1],\ [251, 251, -3*w^4 + 2*w^3 + 16*w^2 - 2*w - 7],\ [269, 269, 3*w^4 - 3*w^3 - 13*w^2 + 4*w + 5],\ [271, 271, w^4 - 2*w^3 - 2*w^2 + 4*w - 3],\ [271, 271, w^2 - 2*w - 4],\ [277, 277, -w^4 + 2*w^3 + 4*w^2 - 4*w - 2],\ [277, 277, w^4 + w^3 - 7*w^2 - 5*w + 4],\ [281, 281, 2*w^3 - 3*w^2 - 8*w + 4],\ [281, 281, -3*w^4 + 2*w^3 + 15*w^2 - 4],\ [293, 293, -3*w^4 + 2*w^3 + 14*w^2 - w - 3],\ [307, 307, -w^4 - w^3 + 7*w^2 + 8*w - 5],\ [311, 311, 3*w^4 - 3*w^3 - 13*w^2 + 4*w + 3],\ [317, 317, -2*w^4 + 12*w^2 + 5*w - 5],\ [331, 331, -3*w^4 + 3*w^3 + 13*w^2 - 2*w - 4],\ [331, 331, -w^4 + 2*w^3 + 5*w^2 - 6*w - 4],\ [331, 331, w^4 + w^3 - 8*w^2 - 5*w + 6],\ [337, 337, -4*w^4 + 3*w^3 + 19*w^2 - 2*w - 7],\ [347, 347, -w^4 + 2*w^3 + 4*w^2 - 4*w - 3],\ [349, 349, -3*w^4 + 2*w^3 + 14*w^2 - 5],\ [367, 367, 2*w^4 - 11*w^2 - 5*w + 7],\ [367, 367, w^4 - 2*w^3 - 2*w^2 + 4*w - 4],\ [373, 373, -2*w^4 + 3*w^3 + 9*w^2 - 7*w - 5],\ [379, 379, 3*w^4 - w^3 - 16*w^2 - 5*w + 8],\ [383, 383, w^4 - w^3 - 3*w^2 - 2],\ [383, 383, 3*w^4 - w^3 - 16*w^2 - 5*w + 9],\ [389, 389, -2*w^4 + 3*w^3 + 8*w^2 - 7*w - 3],\ [397, 397, -w^4 + 3*w^3 + 3*w^2 - 7*w + 1],\ [409, 409, 2*w^4 - 12*w^2 - 3*w + 7],\ [421, 421, 4*w^4 - 3*w^3 - 20*w^2 + 4*w + 11],\ [439, 439, 2*w^4 + w^3 - 12*w^2 - 8*w + 9],\ [449, 449, -w^4 + w^3 + 5*w^2 - 4*w - 2],\ [457, 457, w^4 + w^3 - 7*w^2 - 5*w + 5],\ [461, 461, -2*w^4 + 2*w^3 + 10*w^2 - 3*w - 2],\ [461, 461, -w^4 - w^3 + 7*w^2 + 5*w - 6],\ [463, 463, w^2 + w - 4],\ [463, 463, 3*w^4 - 3*w^3 - 14*w^2 + 3*w + 5],\ [467, 467, w^4 - w^3 - 3*w^2 - 3],\ [467, 467, -4*w^4 + 3*w^3 + 19*w^2 - 7],\ [479, 479, 3*w^4 - 2*w^3 - 14*w^2 - 3*w + 6],\ [487, 487, w^4 - 2*w^3 - 3*w^2 + 6*w + 1],\ [487, 487, -2*w^4 + w^3 + 10*w^2 + 2*w],\ [491, 491, -w^4 + w^3 + 5*w^2 - 6],\ [491, 491, w - 4],\ [499, 499, 2*w^4 - w^3 - 11*w^2 + 2*w + 4],\ [503, 503, 5*w^4 - 3*w^3 - 24*w^2 - w + 7],\ [509, 509, -2*w^4 + 3*w^3 + 6*w^2 - 6*w + 2],\ [521, 521, 2*w^4 - w^3 - 11*w^2 - 3*w + 6],\ [557, 557, -2*w^4 + 2*w^3 + 7*w^2 + 4],\ [571, 571, w^2 - 2*w - 5],\ [577, 577, 5*w^4 - 4*w^3 - 24*w^2 + 3*w + 12],\ [577, 577, w^4 + w^3 - 8*w^2 - 6*w + 7],\ [587, 587, 3*w^4 - 3*w^3 - 13*w^2 + 2*w + 3],\ [587, 587, 3*w^4 - 2*w^3 - 16*w^2 + 3*w + 5],\ [587, 587, w^4 - 7*w^2 + 7],\ [599, 599, 2*w^4 - 13*w^2 - 3*w + 5],\ [617, 617, 2*w^4 - w^3 - 11*w^2 + 2*w + 3],\ [631, 631, 2*w^2 - w - 4],\ [641, 641, w^4 - 5*w^2 - w - 1],\ [653, 653, -4*w^4 + 3*w^3 + 19*w^2 - 4],\ [653, 653, 4*w^4 - w^3 - 23*w^2 - 4*w + 13],\ [659, 659, -2*w^4 + 2*w^3 + 10*w^2 - 5*w - 6],\ [661, 661, 3*w^3 - 3*w^2 - 13*w + 2],\ [683, 683, 5*w^4 - 3*w^3 - 25*w^2 - 3*w + 13],\ [683, 683, 2*w^4 - w^3 - 10*w^2 - 4*w + 8],\ [683, 683, 3*w^4 - 4*w^3 - 12*w^2 + 6*w + 2],\ [691, 691, 2*w^4 - 11*w^2 - 2*w + 4],\ [691, 691, -w^4 + w^3 + 5*w^2 - 4*w - 3],\ [691, 691, -5*w^4 + 3*w^3 + 25*w^2 + 2*w - 7],\ [709, 709, -w^3 + 2*w^2 + w - 4],\ [719, 719, -w^4 + 2*w^3 + 5*w^2 - 6*w - 3],\ [727, 727, -3*w^4 + 4*w^3 + 12*w^2 - 6*w - 6],\ [727, 727, w^4 + w^3 - 7*w^2 - 7*w],\ [727, 727, -4*w^4 + 2*w^3 + 22*w^2 + 3*w - 14],\ [733, 733, -3*w^4 + 2*w^3 + 15*w^2 - 2*w - 5],\ [743, 743, -4*w^4 + 3*w^3 + 21*w^2 - 2*w - 10],\ [743, 743, 4*w^4 - 3*w^3 - 18*w^2 - w + 7],\ [751, 751, 2*w^4 - 2*w^3 - 7*w^2 - 3],\ [751, 751, -3*w^4 + 4*w^3 + 14*w^2 - 9*w - 6],\ [751, 751, 4*w^4 - 3*w^3 - 20*w^2 + 2*w + 7],\ [757, 757, 4*w^4 - w^3 - 22*w^2 - 8*w + 10],\ [773, 773, -4*w^4 + 2*w^3 + 22*w^2 - w - 11],\ [787, 787, w^4 - w^3 - 4*w^2 - w - 2],\ [797, 797, 3*w^4 - w^3 - 15*w^2 - 2*w + 3],\ [809, 809, -2*w^4 + 2*w^3 + 11*w^2 - 4*w - 6],\ [811, 811, -4*w^4 + 3*w^3 + 19*w^2 - 6],\ [821, 821, 5*w^4 - 5*w^3 - 22*w^2 + 6*w + 5],\ [823, 823, 4*w^4 - 4*w^3 - 18*w^2 + 3*w + 8],\ [823, 823, 2*w^2 - w - 7],\ [829, 829, -2*w^4 + 13*w^2 + 3*w - 12],\ [853, 853, 2*w^3 - 2*w^2 - 8*w - 1],\ [853, 853, w^3 - 2*w^2 - 4*w - 2],\ [857, 857, -4*w^4 + w^3 + 21*w^2 + 5*w - 10],\ [859, 859, 4*w^4 - w^3 - 21*w^2 - 5*w + 7],\ [859, 859, w^4 - 3*w^3 - 2*w^2 + 7*w + 1],\ [863, 863, -5*w^4 + 3*w^3 + 26*w^2 - w - 12],\ [877, 877, 2*w^4 - 3*w^3 - 8*w^2 + 3*w + 2],\ [887, 887, -3*w^4 + w^3 + 15*w^2 + 5*w - 7],\ [907, 907, -w^4 - 2*w^3 + 9*w^2 + 9*w - 10],\ [929, 929, -2*w^4 + 3*w^3 + 10*w^2 - 7*w - 8],\ [929, 929, w^4 + w^3 - 5*w^2 - 8*w],\ [929, 929, 3*w^4 - w^3 - 15*w^2 - 3*w + 1],\ [941, 941, -4*w^4 + 3*w^3 + 18*w^2 + 2*w - 5],\ [941, 941, -w^4 - w^3 + 8*w^2 + 6*w - 6],\ [947, 947, -4*w^4 + w^3 + 22*w^2 + 5*w - 11],\ [967, 967, w^4 + 2*w^3 - 8*w^2 - 9*w + 5],\ [971, 971, 2*w^4 - w^3 - 11*w^2 - 4*w + 5],\ [971, 971, 4*w^4 - 2*w^3 - 21*w^2 - 3*w + 7],\ [977, 977, -3*w^4 + 2*w^3 + 16*w^2 - 4*w - 6],\ [991, 991, 2*w^3 - 4*w^2 - 7*w + 6]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x K = QQ e = 1 hecke_eigenvalues_array = [0, 1, 3, 1, -4, -4, -5, -4, -1, 12, -6, -13, -1, -11, -1, -12, 3, 2, -4, -1, -2, 15, -19, 14, -2, 2, -19, -1, -9, -8, 12, 13, 2, 15, -19, 3, 1, -5, 3, -12, 16, -7, 11, -14, 25, 8, -8, -20, -10, -26, 6, -16, 0, -7, 6, 0, 8, -20, -9, 7, -6, 17, 0, 17, 8, -9, 28, -20, 14, -2, 16, 19, 2, 19, 19, -16, -28, 24, -29, 10, 18, -17, -28, -27, 37, 12, -30, -32, -24, -7, 36, -12, 16, 24, -6, 32, -24, 29, 6, 1, 29, -32, 8, -20, -12, -18, 21, -33, 2, 12, 5, -10, -30, 24, 17, 5, -24, 14, 20, -39, -22, -38, 24, 9, 40, -14, -14, 10, -7, -16, 50, 0, -16, -42, 26, 9, -4, 8, 16, 43, -16, -6, -11, -28, -36, -20, -17, -2, -53, 15, 33, 27, 15, -26, 12, 42, 8, 11, -33, 48, 22, 19] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([53, 53, 2*w^4 - 2*w^3 - 9*w^2 + 2*w + 2])] = 1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]