/* 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([-25, -1, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([36, 6, 6]) primes_array = [ [4, 2, 2],\ [5, 5, -w + 5],\ [5, 5, -w - 4],\ [9, 3, 3],\ [13, 13, w + 3],\ [13, 13, w - 4],\ [17, 17, w + 6],\ [17, 17, -w + 7],\ [19, 19, w + 2],\ [19, 19, w - 3],\ [23, 23, w + 1],\ [23, 23, -w + 2],\ [31, 31, -w - 7],\ [31, 31, w - 8],\ [37, 37, 2*w - 9],\ [37, 37, 2*w + 7],\ [43, 43, 4*w + 17],\ [43, 43, 4*w - 21],\ [47, 47, -w - 8],\ [47, 47, w - 9],\ [49, 7, -7],\ [71, 71, 3*w - 14],\ [71, 71, -5*w + 29],\ [79, 79, 5*w + 21],\ [79, 79, 5*w - 26],\ [97, 97, 2*w - 3],\ [97, 97, -2*w - 1],\ [101, 101, 2*w - 1],\ [107, 107, -w - 11],\ [107, 107, w - 12],\ [121, 11, -11],\ [131, 131, -w - 12],\ [131, 131, w - 13],\ [137, 137, 3*w - 11],\ [137, 137, -3*w - 8],\ [157, 157, -w - 13],\ [157, 157, w - 14],\ [179, 179, -4*w - 13],\ [179, 179, 4*w - 17],\ [181, 181, 7*w + 29],\ [181, 181, 7*w - 36],\ [193, 193, 3*w - 22],\ [193, 193, -3*w - 19],\ [197, 197, -3*w - 4],\ [197, 197, 3*w - 7],\ [211, 211, -9*w + 52],\ [211, 211, 5*w - 23],\ [223, 223, 2*w - 19],\ [223, 223, -2*w - 17],\ [227, 227, 3*w - 2],\ [227, 227, 3*w - 1],\ [233, 233, 6*w - 29],\ [233, 233, 6*w + 23],\ [239, 239, 9*w - 47],\ [239, 239, 9*w + 38],\ [251, 251, -5*w + 22],\ [251, 251, 5*w + 17],\ [281, 281, -w - 17],\ [281, 281, w - 18],\ [283, 283, -4*w - 9],\ [283, 283, 4*w - 13],\ [307, 307, 7*w - 34],\ [307, 307, 7*w + 27],\ [317, 317, -w - 18],\ [317, 317, w - 19],\ [359, 359, 5*w - 19],\ [359, 359, -5*w - 14],\ [367, 367, -11*w + 64],\ [367, 367, 7*w - 33],\ [373, 373, -3*w - 23],\ [373, 373, 3*w - 26],\ [379, 379, 4*w - 7],\ [379, 379, -4*w - 3],\ [383, 383, 2*w - 23],\ [383, 383, -2*w - 21],\ [409, 409, 10*w + 41],\ [409, 409, 10*w - 51],\ [421, 421, 5*w - 17],\ [421, 421, -5*w - 12],\ [449, 449, -5*w - 11],\ [449, 449, 5*w - 16],\ [491, 491, 5*w - 36],\ [491, 491, -5*w - 31],\ [499, 499, 5*w - 14],\ [499, 499, -5*w - 9],\ [509, 509, 6*w - 23],\ [509, 509, -6*w - 17],\ [521, 521, -5*w - 8],\ [521, 521, 5*w - 13],\ [541, 541, 5*w - 12],\ [541, 541, -5*w - 7],\ [557, 557, 4*w - 33],\ [557, 557, -4*w - 29],\ [563, 563, 9*w + 34],\ [563, 563, 9*w - 43],\ [569, 569, 15*w + 64],\ [569, 569, 15*w - 79],\ [587, 587, -7*w - 22],\ [587, 587, 7*w - 29],\ [593, 593, 14*w + 59],\ [593, 593, 14*w - 73],\ [601, 601, -5*w - 3],\ [601, 601, 5*w - 8],\ [607, 607, 13*w - 67],\ [607, 607, 13*w + 54],\ [619, 619, -5*w - 1],\ [619, 619, 5*w - 6],\ [631, 631, 5*w - 2],\ [631, 631, 5*w - 3],\ [643, 643, 3*w - 31],\ [643, 643, -3*w - 28],\ [653, 653, -6*w - 13],\ [653, 653, 6*w - 19],\ [677, 677, -w - 26],\ [677, 677, w - 27],\ [683, 683, 2*w - 29],\ [683, 683, -2*w - 27],\ [691, 691, 6*w - 43],\ [691, 691, -6*w - 37],\ [701, 701, -5*w - 34],\ [701, 701, 5*w - 39],\ [727, 727, 13*w + 53],\ [727, 727, 13*w - 66],\ [743, 743, 7*w + 41],\ [743, 743, 19*w + 82],\ [761, 761, -16*w + 93],\ [761, 761, 10*w - 47],\ [787, 787, -w - 28],\ [787, 787, w - 29],\ [809, 809, -6*w - 7],\ [809, 809, 6*w - 13],\ [821, 821, 4*w - 37],\ [821, 821, -4*w - 33],\ [827, 827, 12*w - 59],\ [827, 827, 12*w + 47],\ [829, 829, 3*w - 34],\ [829, 829, -3*w - 31],\ [839, 839, 16*w - 83],\ [839, 839, 16*w + 67],\ [841, 29, -29],\ [853, 853, 14*w + 57],\ [853, 853, 14*w - 71],\ [857, 857, 7*w - 23],\ [857, 857, -7*w - 16],\ [887, 887, 8*w - 31],\ [887, 887, -8*w - 23],\ [929, 929, -5*w - 37],\ [929, 929, 5*w - 42],\ [967, 967, -w - 31],\ [967, 967, w - 32],\ [977, 977, 17*w + 71],\ [977, 977, 17*w - 88],\ [991, 991, 8*w - 29],\ [991, 991, -8*w - 21],\ [997, 997, -7*w - 12],\ [997, 997, 7*w - 19]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^4 + 4*x^3 - 2*x^2 - 18*x - 14 K. = NumberField(heckePol) hecke_eigenvalues_array = [1, e^3 + 2*e^2 - 7*e - 8, e, -1, -2*e^3 - 5*e^2 + 13*e + 22, e^3 + 3*e^2 - 7*e - 16, -2*e^3 - 5*e^2 + 13*e + 23, e^2 + e - 7, -6*e^3 - 15*e^2 + 35*e + 59, 3*e^3 + 7*e^2 - 17*e - 27, -2*e^3 - 5*e^2 + 11*e + 18, -e^3 - 3*e^2 + 3*e + 10, -3*e^3 - 8*e^2 + 17*e + 30, 6*e^3 + 14*e^2 - 35*e - 56, 8*e^3 + 21*e^2 - 47*e - 82, -5*e^3 - 13*e^2 + 27*e + 48, 4*e^3 + 12*e^2 - 21*e - 47, -7*e^3 - 18*e^2 + 37*e + 67, 6*e^3 + 17*e^2 - 35*e - 68, e^3 + 3*e^2 - 5*e - 16, 3*e^3 + 8*e^2 - 14*e - 33, -2*e^3 - 4*e^2 + 10*e + 14, -6*e^3 - 16*e^2 + 32*e + 58, 4*e^3 + 12*e^2 - 24*e - 52, -4*e^3 - 12*e^2 + 20*e + 36, -8*e^3 - 23*e^2 + 42*e + 83, e^2 - 2*e - 5, 6*e^3 + 16*e^2 - 28*e - 60, 12*e^3 + 30*e^2 - 62*e - 109, -5*e^3 - 14*e^2 + 28*e + 53, -e^2 + 4*e + 4, 10*e^3 + 26*e^2 - 56*e - 109, e^3 + 4*e^2 - 2*e - 7, -11*e^3 - 28*e^2 + 63*e + 107, 5*e^3 + 10*e^2 - 29*e - 35, -4*e^3 - 10*e^2 + 20*e + 22, -2*e^3 - 6*e^2 + 6*e + 22, 6*e^3 + 19*e^2 - 28*e - 69, -13*e^3 - 33*e^2 + 68*e + 121, -2*e^3 - 5*e^2 + 19*e + 20, -7*e^3 - 19*e^2 + 41*e + 90, 15*e^3 + 38*e^2 - 88*e - 155, 17*e^3 + 44*e^2 - 88*e - 171, -6*e^3 - 14*e^2 + 29*e + 40, 11*e^3 + 26*e^2 - 59*e - 96, -7*e^3 - 23*e^2 + 32*e + 84, -13*e^3 - 33*e^2 + 74*e + 120, -4*e^3 - 14*e^2 + 20*e + 60, -10*e^3 - 24*e^2 + 62*e + 96, -12*e^3 - 27*e^2 + 72*e + 108, -8*e^3 - 21*e^2 + 46*e + 88, e^3 + 2*e^2 - 9*e - 5, -17*e^3 - 46*e^2 + 95*e + 177, -e^3 - 3*e^2 + 5*e + 24, 16*e^3 + 41*e^2 - 99*e - 168, -8*e^3 - 23*e^2 + 38*e + 92, 2*e^3 + 9*e^2 - 16*e - 56, -19*e^3 - 49*e^2 + 107*e + 181, 11*e^3 + 27*e^2 - 61*e - 97, -4*e^3 - 14*e^2 + 20*e + 50, -6*e^3 - 16*e^2 + 32*e + 70, 22*e^3 + 60*e^2 - 119*e - 221, -9*e^3 - 24*e^2 + 51*e + 73, 2*e^3 + 10*e^2 - 4*e - 40, -20*e^3 - 52*e^2 + 118*e + 200, -13*e^3 - 36*e^2 + 70*e + 132, 19*e^3 + 46*e^2 - 112*e - 192, e^3 - e + 10, -14*e^3 - 34*e^2 + 75*e + 116, 2*e^3 + 11*e^2 - 11*e - 54, 17*e^3 + 41*e^2 - 101*e - 180, e^3 + e^2 - 7*e - 13, -16*e^3 - 43*e^2 + 85*e + 165, 12*e^3 + 31*e^2 - 65*e - 108, -19*e^3 - 49*e^2 + 109*e + 208, 18*e^3 + 47*e^2 - 112*e - 189, -4*e^3 - 11*e^2 + 20*e + 39, 33*e^3 + 84*e^2 - 178*e - 324, 9*e^3 + 24*e^2 - 54*e - 72, -2*e^3 - 6*e^2 + 11*e - 1, -20*e^3 - 52*e^2 + 115*e + 217, 16*e^3 + 38*e^2 - 88*e - 144, -8*e^3 - 20*e^2 + 48*e + 76, -21*e^3 - 56*e^2 + 125*e + 213, -16*e^3 - 42*e^2 + 83*e + 147, 17*e^3 + 40*e^2 - 91*e - 148, -2*e^3 - 4*e^2 + 15*e + 20, e + 9, 6*e^3 + 10*e^2 - 31*e - 29, 5*e^2 - 7*e - 38, 35*e^3 + 93*e^2 - 199*e - 356, 8*e^3 + 16*e^2 - 46*e - 58, -10*e^3 - 26*e^2 + 56*e + 134, 16*e^3 + 42*e^2 - 94*e - 174, 38*e^3 + 104*e^2 - 214*e - 398, -37*e^3 - 98*e^2 + 203*e + 377, -27*e^3 - 72*e^2 + 159*e + 283, -5*e^2 + 33, 5*e^3 + 11*e^2 - 28*e - 49, -4*e^3 - 5*e^2 + 33*e + 23, 32*e^3 + 83*e^2 - 187*e - 335, 10*e^3 + 26*e^2 - 58*e - 91, -18*e^3 - 40*e^2 + 110*e + 161, -8*e^3 - 18*e^2 + 47*e + 84, -3*e^3 - 2*e^2 + 19*e + 2, -2*e^3 - 8*e^2 + 3*e + 41, 13*e^3 + 32*e^2 - 75*e - 137, -8*e^3 - 22*e^2 + 43*e + 68, -15*e^3 - 40*e^2 + 83*e + 170, 22*e^3 + 56*e^2 - 130*e - 250, -10*e^3 - 28*e^2 + 58*e + 86, 25*e^3 + 66*e^2 - 145*e - 248, 14*e^3 + 42*e^2 - 75*e - 176, -5*e^3 - 18*e^2 + 27*e + 80, 12*e^3 + 32*e^2 - 75*e - 152, 10*e^3 + 26*e^2 - 54*e - 67, -3*e^3 - 10*e^2 + 24*e + 59, 5*e^3 + 12*e^2 - 31*e - 21, 2*e^3 + 10*e^2 - 13*e - 55, 27*e^3 + 70*e^2 - 149*e - 258, 6*e^3 + 20*e^2 - 23*e - 78, 8*e^3 + 16*e^2 - 59*e - 60, -7*e^3 - 22*e^2 + 31*e + 86, 25*e^3 + 58*e^2 - 142*e - 228, 5*e^3 + 4*e^2 - 46*e - 12, 26*e^3 + 72*e^2 - 145*e - 291, -6*e^3 - 12*e^2 + 35*e + 63, 6*e^3 + 8*e^2 - 35*e - 17, -5*e^3 - 12*e^2 + 35*e + 45, -30*e^3 - 72*e^2 + 182*e + 286, -20*e^3 - 50*e^2 + 118*e + 206, -28*e^3 - 68*e^2 + 156*e + 268, -12*e^3 - 36*e^2 + 68*e + 132, 6*e^3 + 17*e^2 - 38*e - 86, -42*e^3 - 107*e^2 + 240*e + 422, -7*e^3 - 13*e^2 + 41*e + 36, 28*e^3 + 69*e^2 - 159*e - 290, -23*e^3 - 63*e^2 + 117*e + 226, 4*e^3 + 13*e^2 - 13*e - 54, 30*e^3 + 78*e^2 - 166*e - 293, -e^3 + 12*e + 26, -19*e^3 - 56*e^2 + 104*e + 214, 28*e^3 + 68*e^2 - 160*e - 280, 12*e^3 + 22*e^2 - 76*e - 100, -3*e^3 - 11*e^2 + e + 40, 18*e^3 + 41*e^2 - 113*e - 172, 14*e^3 + 34*e^2 - 82*e - 112, -10*e^2 - 10*e + 48, 14*e^3 + 34*e^2 - 84*e - 118, 8*e^3 + 20*e^2 - 34*e - 90, -20*e^3 - 50*e^2 + 114*e + 196, 2*e^3 + 2*e^2 - 18*e + 24, 27*e^3 + 72*e^2 - 153*e - 306, 2*e^3 - 2*e^2 - 19*e, 9*e^3 + 30*e^2 - 34*e - 118, 23*e^3 + 66*e^2 - 124*e - 238] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([4, 2, 2])] = -1 AL_eigenvalues[ZF.ideal([9, 3, 3])] = 1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]