/* 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([2, 12, -11, -2, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([4, 2, 2/9*w^3 - 1/3*w^2 - 28/9*w + 28/9]) primes_array = [ [2, 2, 4/9*w^3 - 2/3*w^2 - 47/9*w + 20/9],\ [2, 2, 5/9*w^3 - 4/3*w^2 - 52/9*w + 79/9],\ [9, 3, 1/3*w^3 - 11/3*w - 1/3],\ [9, 3, w^3 - w^2 - 12*w - 1],\ [17, 17, -1/9*w^3 - 1/3*w^2 + 14/9*w + 49/9],\ [17, 17, -1/9*w^3 + 2/3*w^2 + 5/9*w - 59/9],\ [25, 5, -7/9*w^3 + 2/3*w^2 + 89/9*w + 19/9],\ [25, 5, -5/9*w^3 + 1/3*w^2 + 52/9*w + 11/9],\ [47, 47, -1/9*w^3 + 2/3*w^2 + 5/9*w - 41/9],\ [47, 47, 1/9*w^3 + 1/3*w^2 - 14/9*w - 13/9],\ [47, 47, -1/9*w^3 + 2/3*w^2 + 5/9*w - 23/9],\ [47, 47, 1/9*w^3 + 1/3*w^2 - 14/9*w - 31/9],\ [49, 7, -4/9*w^3 + 2/3*w^2 + 38/9*w - 29/9],\ [49, 7, 4/9*w^3 - 2/3*w^2 - 38/9*w + 11/9],\ [89, 89, 20/9*w^3 - 16/3*w^2 - 208/9*w + 325/9],\ [89, 89, 2/3*w^3 - w^2 - 25/3*w + 7/3],\ [89, 89, -2/3*w^3 + w^2 + 25/3*w - 19/3],\ [89, 89, -2*w^3 + 3*w^2 + 23*w - 11],\ [103, 103, 8/9*w^3 - 4/3*w^2 - 94/9*w + 67/9],\ [103, 103, -38/9*w^3 + 28/3*w^2 + 406/9*w - 541/9],\ [103, 103, 20/9*w^3 - 7/3*w^2 - 235/9*w + 19/9],\ [103, 103, -8/9*w^3 + 4/3*w^2 + 94/9*w - 31/9],\ [121, 11, -7/3*w^3 + 2*w^2 + 83/3*w + 7/3],\ [121, 11, -31/9*w^3 + 23/3*w^2 + 326/9*w - 443/9],\ [127, 127, -8/9*w^3 + 7/3*w^2 + 67/9*w - 103/9],\ [127, 127, -2/9*w^3 + 1/3*w^2 + 37/9*w - 55/9],\ [127, 127, -8/9*w^3 + 7/3*w^2 + 67/9*w - 85/9],\ [127, 127, -2/3*w^3 + w^2 + 25/3*w - 25/3],\ [137, 137, 7/9*w^3 - 2/3*w^2 - 89/9*w + 17/9],\ [137, 137, 1/9*w^3 - 5/3*w^2 + 4/9*w + 167/9],\ [137, 137, -29/9*w^3 + 19/3*w^2 + 316/9*w - 325/9],\ [137, 137, -7/9*w^3 + 5/3*w^2 + 80/9*w - 71/9],\ [151, 151, 2/9*w^3 + 2/3*w^2 - 28/9*w - 89/9],\ [151, 151, 8/9*w^3 - 7/3*w^2 - 85/9*w + 157/9],\ [151, 151, -4/3*w^3 + 2*w^2 + 44/3*w - 29/3],\ [151, 151, -20/9*w^3 + 13/3*w^2 + 217/9*w - 235/9],\ [169, 13, -4/9*w^3 + 2/3*w^2 + 56/9*w - 11/9],\ [169, 13, -4/9*w^3 + 2/3*w^2 + 56/9*w - 47/9],\ [191, 191, 7/9*w^3 - 2/3*w^2 - 71/9*w - 1/9],\ [191, 191, 7/9*w^3 - 5/3*w^2 - 62/9*w + 53/9],\ [191, 191, -7/9*w^3 + 2/3*w^2 + 71/9*w - 17/9],\ [191, 191, 7/9*w^3 - 5/3*w^2 - 62/9*w + 71/9],\ [223, 223, -59/9*w^3 + 43/3*w^2 + 628/9*w - 835/9],\ [223, 223, 7/9*w^3 - 5/3*w^2 - 80/9*w + 125/9],\ [223, 223, -7/9*w^3 + 2/3*w^2 + 89/9*w + 37/9],\ [223, 223, -41/9*w^3 + 31/3*w^2 + 430/9*w - 601/9],\ [239, 239, 1/9*w^3 + 1/3*w^2 - 14/9*w - 67/9],\ [239, 239, 1/9*w^3 - 2/3*w^2 - 5/9*w - 13/9],\ [239, 239, -1/9*w^3 - 1/3*w^2 + 14/9*w - 23/9],\ [239, 239, 1/9*w^3 - 2/3*w^2 - 5/9*w + 77/9],\ [257, 257, -13/9*w^3 + 11/3*w^2 + 110/9*w - 137/9],\ [257, 257, -13/9*w^3 + 2/3*w^2 + 137/9*w + 25/9],\ [257, 257, -13/9*w^3 + 11/3*w^2 + 110/9*w - 155/9],\ [257, 257, -5/9*w^3 + 4/3*w^2 + 61/9*w - 97/9],\ [263, 263, 22/9*w^3 - 17/3*w^2 - 227/9*w + 353/9],\ [263, 263, -10/9*w^3 + 5/3*w^2 + 113/9*w - 41/9],\ [263, 263, 14/9*w^3 - 10/3*w^2 - 142/9*w + 205/9],\ [263, 263, -20/9*w^3 + 10/3*w^2 + 226/9*w - 109/9],\ [271, 271, -17/9*w^3 + 13/3*w^2 + 184/9*w - 283/9],\ [271, 271, 23/9*w^3 - 7/3*w^2 - 268/9*w - 11/9],\ [271, 271, -49/9*w^3 + 35/3*w^2 + 524/9*w - 677/9],\ [271, 271, -17/9*w^3 + 7/3*w^2 + 184/9*w - 85/9],\ [281, 281, -5/9*w^3 + 4/3*w^2 + 43/9*w - 79/9],\ [281, 281, 5/9*w^3 - 4/3*w^2 - 43/9*w + 25/9],\ [281, 281, 1/3*w^3 - 17/3*w - 7/3],\ [281, 281, 5/9*w^3 - 4/3*w^2 - 61/9*w + 43/9],\ [353, 353, -13/9*w^3 + 5/3*w^2 + 146/9*w - 47/9],\ [353, 353, 1/3*w^3 + w^2 - 14/3*w - 43/3],\ [353, 353, -29/9*w^3 + 19/3*w^2 + 316/9*w - 343/9],\ [353, 353, 17/9*w^3 - 13/3*w^2 - 184/9*w + 265/9],\ [359, 359, -5/9*w^3 + 4/3*w^2 + 43/9*w - 115/9],\ [359, 359, -23/9*w^3 + 13/3*w^2 + 250/9*w - 205/9],\ [359, 359, 41/9*w^3 - 28/3*w^2 - 439/9*w + 511/9],\ [359, 359, -7/3*w^3 + 5*w^2 + 74/3*w - 101/3],\ [361, 19, -8/9*w^3 + 4/3*w^2 + 112/9*w - 121/9],\ [361, 19, -4/9*w^3 + 2/3*w^2 + 56/9*w - 83/9],\ [383, 383, 32/9*w^3 - 25/3*w^2 - 331/9*w + 493/9],\ [383, 383, 14/9*w^3 - 4/3*w^2 - 178/9*w - 47/9],\ [383, 383, -40/9*w^3 + 29/3*w^2 + 425/9*w - 533/9],\ [383, 383, -38/9*w^3 + 25/3*w^2 + 415/9*w - 451/9],\ [409, 409, -5/9*w^3 + 4/3*w^2 + 43/9*w - 7/9],\ [409, 409, -w^3 + 2*w^2 + 11*w - 9],\ [409, 409, -5/9*w^3 + 4/3*w^2 + 43/9*w - 97/9],\ [409, 409, -5/9*w^3 + 1/3*w^2 + 52/9*w - 43/9],\ [433, 433, -8/9*w^3 + 7/3*w^2 + 85/9*w - 175/9],\ [433, 433, 40/9*w^3 - 29/3*w^2 - 425/9*w + 569/9],\ [433, 433, 22/9*w^3 - 8/3*w^2 - 254/9*w + 29/9],\ [433, 433, -16/9*w^3 + 8/3*w^2 + 170/9*w - 125/9],\ [457, 457, -28/9*w^3 + 17/3*w^2 + 311/9*w - 275/9],\ [457, 457, 2/9*w^3 + 2/3*w^2 - 28/9*w - 71/9],\ [457, 457, 2/9*w^3 - 4/3*w^2 - 10/9*w + 91/9],\ [457, 457, -8/9*w^3 + 10/3*w^2 + 76/9*w - 247/9],\ [463, 463, 8/9*w^3 - 4/3*w^2 - 94/9*w + 85/9],\ [463, 463, -2*w + 5],\ [463, 463, 2*w + 3],\ [463, 463, 8/9*w^3 - 4/3*w^2 - 94/9*w + 13/9],\ [529, 23, 2/9*w^3 - 1/3*w^2 - 19/9*w - 35/9],\ [529, 23, -2/9*w^3 + 1/3*w^2 + 19/9*w - 55/9],\ [569, 569, -14/9*w^3 + 7/3*w^2 + 151/9*w - 97/9],\ [569, 569, -20/9*w^3 + 16/3*w^2 + 208/9*w - 343/9],\ [569, 569, -22/9*w^3 + 11/3*w^2 + 245/9*w - 137/9],\ [569, 569, -10/9*w^3 + 8/3*w^2 + 104/9*w - 185/9],\ [577, 577, 41/9*w^3 - 31/3*w^2 - 430/9*w + 619/9],\ [577, 577, -11/9*w^3 + 7/3*w^2 + 118/9*w - 91/9],\ [577, 577, -11/9*w^3 + 4/3*w^2 + 127/9*w - 37/9],\ [577, 577, 29/9*w^3 - 13/3*w^2 - 334/9*w + 109/9],\ [593, 593, 4/9*w^3 + 1/3*w^2 - 47/9*w - 25/9],\ [593, 593, -4/9*w^3 + 5/3*w^2 + 29/9*w - 83/9],\ [593, 593, 4/9*w^3 + 1/3*w^2 - 47/9*w - 43/9],\ [593, 593, -4/9*w^3 + 5/3*w^2 + 29/9*w - 65/9],\ [599, 599, 2/3*w^3 - 22/3*w - 17/3],\ [599, 599, 2/3*w^3 - 2*w^2 - 16/3*w + 19/3],\ [599, 599, -2/3*w^3 + 22/3*w - 1/3],\ [599, 599, -2/3*w^3 + 2*w^2 + 16/3*w - 37/3],\ [631, 631, -3*w^3 + 6*w^2 + 33*w - 35],\ [631, 631, 17/9*w^3 - 10/3*w^2 - 193/9*w + 157/9],\ [631, 631, -7/9*w^3 + 8/3*w^2 + 71/9*w - 179/9],\ [631, 631, 1/9*w^3 + 4/3*w^2 - 23/9*w - 139/9],\ [647, 647, -29/9*w^3 + 22/3*w^2 + 307/9*w - 451/9],\ [647, 647, -19/9*w^3 + 14/3*w^2 + 203/9*w - 293/9],\ [647, 647, -5/3*w^3 + 2*w^2 + 55/3*w - 19/3],\ [647, 647, -23/9*w^3 + 10/3*w^2 + 259/9*w - 97/9],\ [727, 727, -5/3*w^3 + 3*w^2 + 58/3*w - 55/3],\ [727, 727, -67/9*w^3 + 47/3*w^2 + 722/9*w - 875/9],\ [727, 727, 7/3*w^3 - w^2 - 86/3*w - 43/3],\ [727, 727, -1/9*w^3 + 5/3*w^2 + 14/9*w - 113/9],\ [761, 761, -13/3*w^3 + 9*w^2 + 140/3*w - 161/3],\ [761, 761, -13/9*w^3 + 11/3*w^2 + 128/9*w - 227/9],\ [761, 761, -5/3*w^3 + 3*w^2 + 58/3*w - 37/3],\ [761, 761, 7/9*w^3 + 1/3*w^2 - 98/9*w - 127/9],\ [769, 769, -7/9*w^3 + 2/3*w^2 + 89/9*w - 35/9],\ [769, 769, 1/9*w^3 + 1/3*w^2 - 32/9*w - 31/9],\ [769, 769, -1/9*w^3 + 2/3*w^2 + 23/9*w - 59/9],\ [769, 769, -7/9*w^3 + 5/3*w^2 + 80/9*w - 53/9],\ [841, 29, 5/3*w^3 - 2*w^2 - 61/3*w - 5/3],\ [841, 29, -5/3*w^3 + w^2 + 58/3*w + 5/3],\ [863, 863, 2/3*w^3 - 28/3*w + 1/3],\ [863, 863, 2/9*w^3 + 2/3*w^2 - 46/9*w - 53/9],\ [863, 863, 2/9*w^3 - 4/3*w^2 - 28/9*w + 91/9],\ [863, 863, -2/3*w^3 + 2*w^2 + 22/3*w - 25/3],\ [919, 919, -5/3*w^3 + 4*w^2 + 43/3*w - 49/3],\ [919, 919, 5/9*w^3 + 2/3*w^2 - 43/9*w - 29/9],\ [919, 919, 5/9*w^3 - 7/3*w^2 - 16/9*w + 61/9],\ [919, 919, -13/9*w^3 + 11/3*w^2 + 128/9*w - 191/9],\ [937, 937, -5/9*w^3 + 1/3*w^2 + 70/9*w - 61/9],\ [937, 937, -1/3*w^3 + w^2 + 14/3*w - 35/3],\ [937, 937, 1/3*w^3 - 17/3*w - 19/3],\ [937, 937, -5/9*w^3 + 4/3*w^2 + 61/9*w - 7/9],\ [953, 953, -2/9*w^3 + 4/3*w^2 + 10/9*w - 73/9],\ [953, 953, -2/9*w^3 + 4/3*w^2 + 10/9*w - 55/9],\ [953, 953, 2/9*w^3 + 2/3*w^2 - 28/9*w - 35/9],\ [953, 953, 2/9*w^3 + 2/3*w^2 - 28/9*w - 53/9],\ [961, 31, -8/9*w^3 + 4/3*w^2 + 76/9*w - 31/9],\ [961, 31, -8/9*w^3 + 4/3*w^2 + 76/9*w - 49/9],\ [967, 967, -25/9*w^3 + 8/3*w^2 + 305/9*w + 19/9],\ [967, 967, -7/9*w^3 - 1/3*w^2 + 80/9*w + 19/9],\ [967, 967, 7/9*w^3 - 5/3*w^2 - 98/9*w - 19/9],\ [967, 967, -35/9*w^3 + 10/3*w^2 + 409/9*w + 41/9],\ [977, 977, -8/9*w^3 + 4/3*w^2 + 94/9*w - 121/9],\ [977, 977, 14/9*w^3 - 1/3*w^2 - 151/9*w - 47/9],\ [977, 977, 14/9*w^3 - 13/3*w^2 - 115/9*w + 187/9],\ [977, 977, -14/9*w^3 + 13/3*w^2 + 115/9*w - 169/9],\ [1033, 1033, -2/9*w^3 + 1/3*w^2 + 55/9*w - 55/9],\ [1033, 1033, 14/9*w^3 - 7/3*w^2 - 169/9*w + 61/9],\ [1033, 1033, -14/9*w^3 + 7/3*w^2 + 169/9*w - 115/9],\ [1033, 1033, -2/9*w^3 + 1/3*w^2 + 55/9*w - 1/9],\ [1039, 1039, -4/9*w^3 + 5/3*w^2 + 29/9*w - 137/9],\ [1039, 1039, 4/3*w^3 - 4*w^2 - 38/3*w + 95/3],\ [1039, 1039, -28/9*w^3 + 17/3*w^2 + 311/9*w - 257/9],\ [1039, 1039, -4/9*w^3 - 1/3*w^2 + 47/9*w + 97/9],\ [1063, 1063, -w^3 + 3*w^2 + 8*w - 15],\ [1063, 1063, w^3 - 3*w^2 - 8*w + 13],\ [1063, 1063, w^3 - 11*w - 3],\ [1063, 1063, 19/9*w^3 - 14/3*w^2 - 185/9*w + 221/9],\ [1087, 1087, -4/9*w^3 - 1/3*w^2 + 65/9*w + 61/9],\ [1087, 1087, -4/9*w^3 + 5/3*w^2 + 47/9*w - 47/9],\ [1087, 1087, 4/9*w^3 + 1/3*w^2 - 65/9*w + 11/9],\ [1087, 1087, -4/9*w^3 + 5/3*w^2 + 47/9*w - 119/9],\ [1097, 1097, -26/9*w^3 + 16/3*w^2 + 292/9*w - 247/9],\ [1097, 1097, -16/9*w^3 + 8/3*w^2 + 188/9*w - 107/9],\ [1097, 1097, 14/9*w^3 - 13/3*w^2 - 151/9*w + 259/9],\ [1097, 1097, 10/9*w^3 - 11/3*w^2 - 95/9*w + 257/9],\ [1103, 1103, 8/9*w^3 - 1/3*w^2 - 85/9*w + 13/9],\ [1103, 1103, 8/9*w^3 - 1/3*w^2 - 85/9*w - 41/9],\ [1103, 1103, 8/9*w^3 - 7/3*w^2 - 67/9*w + 121/9],\ [1103, 1103, 8/9*w^3 - 7/3*w^2 - 67/9*w + 67/9],\ [1223, 1223, 10/9*w^3 - 8/3*w^2 - 104/9*w + 113/9],\ [1223, 1223, -10/9*w^3 + 5/3*w^2 + 113/9*w - 113/9],\ [1223, 1223, 10/9*w^3 - 5/3*w^2 - 113/9*w + 5/9],\ [1223, 1223, -10/9*w^3 + 2/3*w^2 + 122/9*w - 5/9],\ [1249, 1249, -23/9*w^3 + 13/3*w^2 + 268/9*w - 151/9],\ [1249, 1249, 5/3*w^3 - w^2 - 64/3*w - 29/3],\ [1249, 1249, -49/9*w^3 + 35/3*w^2 + 524/9*w - 641/9],\ [1249, 1249, 23/3*w^3 - 16*w^2 - 247/3*w + 301/3],\ [1279, 1279, 22/3*w^3 - 16*w^2 - 236/3*w + 311/3],\ [1279, 1279, 10/9*w^3 + 1/3*w^2 - 113/9*w - 103/9],\ [1279, 1279, -4*w^3 + 8*w^2 + 44*w - 51],\ [1279, 1279, 34/9*w^3 - 11/3*w^2 - 395/9*w + 17/9],\ [1327, 1327, -28/9*w^3 + 17/3*w^2 + 311/9*w - 293/9],\ [1327, 1327, 4*w^3 - 9*w^2 - 43*w + 59],\ [1327, 1327, -22/9*w^3 + 8/3*w^2 + 254/9*w - 47/9],\ [1327, 1327, -4/9*w^3 + 8/3*w^2 + 38/9*w - 209/9],\ [1361, 1361, -14/3*w^3 + 11*w^2 + 145/3*w - 217/3],\ [1361, 1361, 14/9*w^3 - 4/3*w^2 - 178/9*w - 65/9],\ [1361, 1361, 14/9*w^3 - 10/3*w^2 - 160/9*w + 241/9],\ [1361, 1361, -2/9*w^3 - 2/3*w^2 + 64/9*w - 73/9],\ [1369, 37, 13/9*w^3 - 2/3*w^2 - 173/9*w - 115/9],\ [1369, 37, -43/9*w^3 + 32/3*w^2 + 449/9*w - 611/9],\ [1409, 1409, -71/9*w^3 + 52/3*w^2 + 751/9*w - 1003/9],\ [1409, 1409, -61/9*w^3 + 44/3*w^2 + 647/9*w - 809/9],\ [1409, 1409, 7/3*w^3 - 2*w^2 - 89/3*w - 25/3],\ [1409, 1409, -37/9*w^3 + 14/3*w^2 + 437/9*w - 41/9],\ [1447, 1447, -4/3*w^3 + 2*w^2 + 50/3*w - 47/3],\ [1447, 1447, 2/3*w^3 + 2*w^2 - 34/3*w - 101/3],\ [1447, 1447, -70/9*w^3 + 47/3*w^2 + 755/9*w - 827/9],\ [1447, 1447, -20/9*w^3 + 16/3*w^2 + 190/9*w - 271/9],\ [1471, 1471, -5/3*w^3 + 3*w^2 + 52/3*w - 43/3],\ [1471, 1471, 7/9*w^3 - 5/3*w^2 - 62/9*w - 19/9],\ [1471, 1471, -61/9*w^3 + 41/3*w^2 + 656/9*w - 737/9],\ [1471, 1471, -7/9*w^3 + 5/3*w^2 + 62/9*w - 143/9],\ [1481, 1481, 10/9*w^3 - 5/3*w^2 - 131/9*w + 23/9],\ [1481, 1481, 46/9*w^3 - 23/3*w^2 - 527/9*w + 257/9],\ [1481, 1481, -50/9*w^3 + 40/3*w^2 + 520/9*w - 817/9],\ [1481, 1481, -10/9*w^3 + 5/3*w^2 + 131/9*w - 113/9],\ [1487, 1487, 11/9*w^3 - 4/3*w^2 - 145/9*w + 1/9],\ [1487, 1487, -11/9*w^3 + 1/3*w^2 + 136/9*w + 35/9],\ [1487, 1487, -11/9*w^3 + 10/3*w^2 + 109/9*w - 163/9],\ [1487, 1487, -7/9*w^3 + 5/3*w^2 + 116/9*w + 37/9],\ [1511, 1511, -5/3*w^3 + 3*w^2 + 58/3*w - 49/3],\ [1511, 1511, 1/9*w^3 + 7/3*w^2 - 32/9*w - 247/9],\ [1511, 1511, 1/3*w^3 - 2*w^2 - 11/3*w + 23/3],\ [1511, 1511, -5/9*w^3 + 7/3*w^2 + 52/9*w - 151/9],\ [1543, 1543, -1/9*w^3 + 2/3*w^2 - 13/9*w + 49/9],\ [1543, 1543, w^3 - w^2 - 12*w - 5],\ [1543, 1543, -w^3 + 2*w^2 + 11*w - 17],\ [1543, 1543, -1/9*w^3 - 1/3*w^2 - 4/9*w - 41/9],\ [1583, 1583, 1/9*w^3 + 1/3*w^2 - 14/9*w - 85/9],\ [1583, 1583, 1/9*w^3 - 2/3*w^2 - 5/9*w - 31/9],\ [1583, 1583, -1/9*w^3 - 1/3*w^2 + 14/9*w - 41/9],\ [1583, 1583, -1/9*w^3 + 2/3*w^2 + 5/9*w - 95/9],\ [1607, 1607, -20/9*w^3 + 13/3*w^2 + 217/9*w - 199/9],\ [1607, 1607, -32/9*w^3 + 25/3*w^2 + 331/9*w - 511/9],\ [1607, 1607, -28/9*w^3 + 14/3*w^2 + 320/9*w - 149/9],\ [1607, 1607, -2/3*w^3 + 2*w^2 + 16/3*w - 55/3],\ [1657, 1657, 1/9*w^3 + 1/3*w^2 - 32/9*w - 49/9],\ [1657, 1657, -7/9*w^3 + 5/3*w^2 + 80/9*w - 35/9],\ [1657, 1657, -7/9*w^3 + 2/3*w^2 + 89/9*w - 53/9],\ [1657, 1657, -1/9*w^3 + 2/3*w^2 + 23/9*w - 77/9],\ [1681, 41, 10/9*w^3 - 5/3*w^2 - 95/9*w + 23/9],\ [1681, 41, -10/9*w^3 + 5/3*w^2 + 95/9*w - 77/9],\ [1721, 1721, 7/9*w^3 - 8/3*w^2 - 53/9*w + 71/9],\ [1721, 1721, -7/9*w^3 + 8/3*w^2 + 53/9*w - 161/9],\ [1721, 1721, 7/9*w^3 + 1/3*w^2 - 80/9*w - 91/9],\ [1721, 1721, -7/9*w^3 - 1/3*w^2 + 80/9*w + 1/9],\ [1753, 1753, -4/3*w^3 + 4*w^2 + 38/3*w - 101/3],\ [1753, 1753, 2/9*w^3 - 4/3*w^2 - 10/9*w + 145/9],\ [1753, 1753, 2/9*w^3 + 2/3*w^2 - 28/9*w - 125/9],\ [1753, 1753, -32/9*w^3 + 19/3*w^2 + 349/9*w - 295/9],\ [1759, 1759, -19/9*w^3 + 14/3*w^2 + 221/9*w - 221/9],\ [1759, 1759, 37/9*w^3 - 14/3*w^2 - 437/9*w + 59/9],\ [1759, 1759, 67/9*w^3 - 50/3*w^2 - 713/9*w + 965/9],\ [1759, 1759, 1/3*w^3 - 2*w^2 - 5/3*w + 47/3],\ [1777, 1777, -13/9*w^3 + 8/3*w^2 + 155/9*w - 83/9],\ [1777, 1777, 7/9*w^3 + 4/3*w^2 - 107/9*w - 235/9],\ [1777, 1777, 1/3*w^3 - w^2 - 20/3*w + 29/3],\ [1777, 1777, 13/9*w^3 - 5/3*w^2 - 164/9*w + 83/9],\ [1783, 1783, -1/9*w^3 + 5/3*w^2 - 4/9*w - 185/9],\ [1783, 1783, -13/9*w^3 + 11/3*w^2 + 128/9*w - 263/9],\ [1783, 1783, -23/9*w^3 + 13/3*w^2 + 250/9*w - 187/9],\ [1783, 1783, -11/3*w^3 + 7*w^2 + 118/3*w - 121/3],\ [1801, 1801, 14/9*w^3 - 4/3*w^2 - 142/9*w - 11/9],\ [1801, 1801, -14/9*w^3 + 7/3*w^2 + 187/9*w - 187/9],\ [1801, 1801, 14/9*w^3 - 10/3*w^2 - 124/9*w + 97/9],\ [1801, 1801, -14/9*w^3 + 10/3*w^2 + 124/9*w - 151/9],\ [1823, 1823, 7/9*w^3 - 2/3*w^2 - 89/9*w + 107/9],\ [1823, 1823, -1/9*w^3 + 2/3*w^2 + 23/9*w - 131/9],\ [1823, 1823, 1/9*w^3 + 1/3*w^2 - 32/9*w - 103/9],\ [1823, 1823, -7/9*w^3 + 5/3*w^2 + 80/9*w + 19/9],\ [1849, 43, 8/9*w^3 - 4/3*w^2 - 112/9*w + 13/9],\ [1849, 43, -4/3*w^3 + 2*w^2 + 56/3*w + 13/3],\ [1871, 1871, -58/9*w^3 + 38/3*w^2 + 632/9*w - 677/9],\ [1871, 1871, -16/9*w^3 + 11/3*w^2 + 161/9*w - 143/9],\ [1871, 1871, 16/9*w^3 - 5/3*w^2 - 179/9*w + 35/9],\ [1871, 1871, 4/9*w^3 + 7/3*w^2 - 65/9*w - 277/9],\ [1879, 1879, 14/9*w^3 - 1/3*w^2 - 133/9*w - 29/9],\ [1879, 1879, -22/9*w^3 + 5/3*w^2 + 245/9*w + 7/9],\ [1879, 1879, -22/9*w^3 + 17/3*w^2 + 209/9*w - 245/9],\ [1879, 1879, -14/9*w^3 + 13/3*w^2 + 97/9*w - 151/9],\ [1889, 1889, 2/9*w^3 - 1/3*w^2 - 37/9*w + 91/9],\ [1889, 1889, 2/3*w^3 - w^2 - 25/3*w - 11/3],\ [1889, 1889, 2/3*w^3 - w^2 - 25/3*w + 37/3],\ [1889, 1889, 2/9*w^3 - 1/3*w^2 - 37/9*w - 53/9],\ [1913, 1913, 11/9*w^3 - 7/3*w^2 - 118/9*w + 73/9],\ [1913, 1913, 1/3*w^3 - 5/3*w - 13/3],\ [1913, 1913, -1/3*w^3 + w^2 + 2/3*w - 17/3],\ [1913, 1913, -11/9*w^3 + 4/3*w^2 + 127/9*w - 55/9],\ [1951, 1951, -4/3*w^3 + 3*w^2 + 41/3*w - 41/3],\ [1951, 1951, -4/9*w^3 - 1/3*w^2 + 29/9*w + 43/9],\ [1951, 1951, 4/9*w^3 - 5/3*w^2 - 11/9*w + 65/9],\ [1951, 1951, -4/3*w^3 + w^2 + 47/3*w - 5/3],\ [1993, 1993, 38/9*w^3 - 16/3*w^2 - 424/9*w + 163/9],\ [1993, 1993, -58/9*w^3 + 41/3*w^2 + 623/9*w - 803/9],\ [1993, 1993, 44/9*w^3 - 31/3*w^2 - 481/9*w + 589/9],\ [1993, 1993, -46/9*w^3 + 35/3*w^2 + 491/9*w - 725/9]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x K = QQ e = 1 hecke_eigenvalues_array = [0, -2, -2, 2, -4, 4, 8, -8, 2, -8, 2, -8, -2, -2, -6, -10, -6, -10, -8, 8, -12, 12, -10, 10, -16, -4, 16, 4, -18, -2, 18, 2, 16, 4, -16, -4, -22, -10, -18, -20, -18, -20, 10, 10, -20, -20, 14, -4, 14, -4, 10, -6, -10, 6, 0, 0, -4, 4, -10, 0, -10, 0, 24, -24, 0, 0, 16, -16, 32, -32, -8, -8, -14, -14, -2, 22, -24, 4, 24, -4, 4, 4, -4, -4, 38, 38, -14, -14, 26, 2, 26, 2, -32, 32, -8, 8, -38, -38, 22, 6, 22, 6, -10, 10, -2, 2, -2, 34, -2, 34, 48, -16, -48, 16, 4, 4, 6, 6, 24, 24, 10, 10, 30, 30, 12, 12, -42, 42, -42, 42, 18, -50, -18, 50, 48, -48, 16, 32, -16, -32, 46, 46, -16, -16, 4, -4, -28, 28, 22, 22, 30, 30, 42, 42, -36, -36, -2, -2, -14, -14, 2, 2, -26, 2, -26, 2, -32, 0, 32, 0, -12, -12, 38, 38, 8, -8, 20, -20, -14, 30, -14, 30, 48, -48, 32, -32, -56, 56, 16, -16, 58, -58, 18, -18, -40, 20, 40, -20, -8, 8, 16, -16, -38, -38, 54, 54, 64, -64, -60, -44, 60, 44, 72, -8, -72, 8, 24, 18, 24, 18, 10, 10, 6, 6, -20, -20, 18, 18, 38, 0, 38, 0, -60, -38, -60, -38, -6, -76, -6, -76, 56, -56, 20, -20, 18, -18, 34, -34, -10, -10, -12, 12, -60, 60, -74, -74, 10, 10, -40, -40, -50, -50, 60, -60, 60, -60, -70, -70, 72, 72, -26, -26, 50, 50, -36, -36, 6, 6, 70, -70, -48, 48, 0, 0, -64, 64, 16, -16, -18, 70, -18, 70, 26, 54, -26, -54, 56, -28, -56, 28, 54, -82, 54, -82] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([2, 2, 4/9*w^3 - 2/3*w^2 - 47/9*w + 20/9])] = -1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]