/* 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([3, 6, -8, -1, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([3, 3, w]) primes_array = [ [3, 3, w],\ [7, 7, 1/3*w^3 - 2/3*w^2 - 2*w + 2],\ [9, 3, w + 1],\ [13, 13, 2/3*w^3 - 1/3*w^2 - 5*w],\ [16, 2, 2],\ [17, 17, 1/3*w^3 - 2/3*w^2 - 3*w + 5],\ [19, 19, -1/3*w^3 + 2/3*w^2 + 2*w - 5],\ [23, 23, -1/3*w^3 + 2/3*w^2 + 3*w - 2],\ [25, 5, 1/3*w^3 + 1/3*w^2 - 3*w],\ [25, 5, -2/3*w^3 + 1/3*w^2 + 5*w - 3],\ [29, 29, -2/3*w^3 + 1/3*w^2 + 4*w - 3],\ [29, 29, -1/3*w^3 + 2/3*w^2 + 2*w],\ [37, 37, 1/3*w^3 + 1/3*w^2 - 2*w - 3],\ [41, 41, -2/3*w^3 + 1/3*w^2 + 5*w - 4],\ [43, 43, 2/3*w^3 - 1/3*w^2 - 6*w],\ [47, 47, 2/3*w^3 - 1/3*w^2 - 4*w],\ [59, 59, 1/3*w^3 + 1/3*w^2 - 2*w - 4],\ [59, 59, 4/3*w^3 - 5/3*w^2 - 10*w + 9],\ [67, 67, -1/3*w^3 + 2/3*w^2 + 3*w],\ [71, 71, 2/3*w^3 - 1/3*w^2 - 4*w + 1],\ [73, 73, -1/3*w^3 + 2/3*w^2 + w - 3],\ [83, 83, -w^3 + w^2 + 8*w - 4],\ [83, 83, -2/3*w^3 + 1/3*w^2 + 6*w - 1],\ [97, 97, 5/3*w^3 - 4/3*w^2 - 13*w + 6],\ [101, 101, -1/3*w^3 + 2/3*w^2 + 4*w - 2],\ [127, 127, w^2 - w - 4],\ [131, 131, 7/3*w^3 - 8/3*w^2 - 18*w + 15],\ [131, 131, 4/3*w^3 - 8/3*w^2 - 10*w + 17],\ [137, 137, 1/3*w^3 + 1/3*w^2 - 4*w - 3],\ [139, 139, w^2 - 5],\ [151, 151, 1/3*w^3 - 2/3*w^2 - 4*w + 6],\ [163, 163, -2/3*w^3 + 4/3*w^2 + 6*w - 9],\ [163, 163, -2/3*w^3 + 1/3*w^2 + 3*w - 4],\ [179, 179, w^2 - 2*w - 4],\ [179, 179, w^2 - w - 7],\ [191, 191, 1/3*w^3 + 1/3*w^2 - w - 3],\ [191, 191, -2/3*w^3 + 1/3*w^2 + 4*w - 6],\ [193, 193, -w^3 + 9*w + 1],\ [193, 193, 4/3*w^3 - 2/3*w^2 - 9*w],\ [197, 197, 2/3*w^3 - 1/3*w^2 - 5*w - 3],\ [197, 197, w^3 - 8*w + 1],\ [197, 197, w^3 - 7*w + 2],\ [197, 197, -4/3*w^3 + 2/3*w^2 + 11*w - 3],\ [223, 223, 4/3*w^3 - 5/3*w^2 - 10*w + 8],\ [233, 233, -4/3*w^3 + 2/3*w^2 + 9*w - 5],\ [241, 241, 1/3*w^3 + 1/3*w^2 - 4*w - 4],\ [257, 257, -w^3 - w^2 + 6*w + 5],\ [269, 269, -1/3*w^3 - 1/3*w^2 + 5*w],\ [269, 269, -2/3*w^3 + 4/3*w^2 + 4*w - 3],\ [271, 271, -2/3*w^3 + 4/3*w^2 + 3*w - 6],\ [271, 271, w^3 - 8*w - 5],\ [277, 277, 2*w^3 - 2*w^2 - 16*w + 13],\ [281, 281, 2/3*w^3 + 2/3*w^2 - 5*w],\ [281, 281, -w^3 + w^2 + 7*w - 2],\ [331, 331, 2/3*w^3 - 4/3*w^2 - 5*w + 4],\ [331, 331, 8/3*w^3 - 7/3*w^2 - 20*w + 15],\ [343, 7, -4/3*w^3 + 2/3*w^2 + 11*w - 6],\ [347, 347, 2/3*w^3 + 2/3*w^2 - 4*w - 5],\ [349, 349, -1/3*w^3 + 2/3*w^2 + 3*w - 8],\ [349, 349, -1/3*w^3 + 2/3*w^2 - 3],\ [353, 353, -5/3*w^3 + 1/3*w^2 + 12*w],\ [353, 353, 1/3*w^3 - 5/3*w^2 - 2*w + 14],\ [367, 367, -1/3*w^3 + 5/3*w^2 + 4*w - 9],\ [367, 367, 1/3*w^3 + 1/3*w^2 + w - 3],\ [379, 379, 4/3*w^3 + 1/3*w^2 - 12*w],\ [379, 379, -5/3*w^3 + 7/3*w^2 + 13*w - 12],\ [383, 383, -5/3*w^3 + 10/3*w^2 + 12*w - 21],\ [397, 397, 1/3*w^3 + 4/3*w^2 - 5*w - 6],\ [397, 397, 4/3*w^3 + 1/3*w^2 - 8*w],\ [397, 397, -1/3*w^3 + 2/3*w^2 + w - 6],\ [397, 397, w^3 + w^2 - 8*w - 7],\ [401, 401, 5/3*w^3 - 1/3*w^2 - 14*w + 1],\ [419, 419, 1/3*w^3 + 4/3*w^2 - 4*w - 3],\ [419, 419, 1/3*w^3 - 5/3*w^2 - 3*w + 12],\ [433, 433, -w^3 + w^2 + 9*w - 4],\ [433, 433, 1/3*w^3 + 4/3*w^2 - 4*w - 9],\ [443, 443, 4/3*w^3 - 2/3*w^2 - 8*w - 1],\ [443, 443, -4/3*w^3 + 8/3*w^2 + 9*w - 17],\ [449, 449, -w^3 + w^2 + 8*w - 2],\ [449, 449, 5/3*w^3 - 7/3*w^2 - 13*w + 18],\ [457, 457, -w^3 + 3*w^2 + 7*w - 20],\ [457, 457, 2/3*w^3 + 2/3*w^2 - 5*w - 9],\ [461, 461, 4/3*w^3 - 2/3*w^2 - 9*w + 2],\ [461, 461, -2/3*w^3 + 7/3*w^2 + 4*w - 16],\ [463, 463, 4/3*w^3 + 1/3*w^2 - 11*w - 1],\ [467, 467, w^3 - 2*w^2 - 9*w + 8],\ [479, 479, 4/3*w^3 - 8/3*w^2 - 10*w + 15],\ [479, 479, 5/3*w^3 - 1/3*w^2 - 14*w],\ [491, 491, -1/3*w^3 + 2/3*w^2 + 2*w - 8],\ [509, 509, -11/3*w^3 + 13/3*w^2 + 27*w - 27],\ [523, 523, -w - 5],\ [523, 523, 4/3*w^3 - 2/3*w^2 - 10*w + 9],\ [541, 541, 8/3*w^3 - 7/3*w^2 - 21*w + 13],\ [547, 547, -2/3*w^3 + 1/3*w^2 + 5*w - 7],\ [557, 557, -2/3*w^3 + 1/3*w^2 + 7*w - 3],\ [563, 563, w^2 + w - 8],\ [563, 563, 1/3*w^3 + 1/3*w^2 - w - 6],\ [569, 569, 1/3*w^3 + 4/3*w^2 - 4*w - 6],\ [569, 569, -1/3*w^3 + 5/3*w^2 + 2*w - 6],\ [571, 571, 5/3*w^3 - 4/3*w^2 - 11*w + 1],\ [593, 593, -2/3*w^3 + 1/3*w^2 + 4*w - 7],\ [593, 593, -2*w^3 + w^2 + 15*w - 1],\ [599, 599, -1/3*w^3 + 2/3*w^2 + 5*w - 2],\ [601, 601, w^3 - 10*w - 2],\ [607, 607, 1/3*w^3 + 4/3*w^2 - 4*w - 4],\ [607, 607, 7/3*w^3 - 8/3*w^2 - 18*w + 14],\ [617, 617, -5/3*w^3 + 10/3*w^2 + 13*w - 24],\ [617, 617, 1/3*w^3 + 4/3*w^2 - 2*w - 6],\ [617, 617, w^2 + w - 10],\ [617, 617, 1/3*w^3 + 1/3*w^2 - 6*w - 1],\ [631, 631, 7/3*w^3 - 2/3*w^2 - 18*w + 3],\ [631, 631, -1/3*w^3 + 2/3*w^2 + 5*w - 3],\ [641, 641, 5/3*w^3 - 1/3*w^2 - 11*w - 2],\ [643, 643, -5/3*w^3 + 7/3*w^2 + 14*w - 13],\ [643, 643, -2/3*w^3 + 4/3*w^2 + 6*w - 3],\ [653, 653, w^3 - w^2 - 8*w + 1],\ [653, 653, 1/3*w^3 + 4/3*w^2 - 3*w - 13],\ [659, 659, -1/3*w^3 + 5/3*w^2 + 2*w - 8],\ [661, 661, 5/3*w^3 - 4/3*w^2 - 11*w + 9],\ [683, 683, 4/3*w^3 - 2/3*w^2 - 7*w + 8],\ [709, 709, -2*w^3 + 17*w + 2],\ [709, 709, 1/3*w^3 + 1/3*w^2 - 4],\ [733, 733, 1/3*w^3 - 2/3*w^2 - 2*w - 3],\ [733, 733, 5/3*w^3 - 4/3*w^2 - 14*w + 9],\ [733, 733, 2/3*w^3 - 7/3*w^2 - 2*w + 10],\ [733, 733, 4/3*w^3 - 8/3*w^2 - 9*w + 18],\ [743, 743, w^3 - w^2 - 8*w - 1],\ [751, 751, -4/3*w^3 - 1/3*w^2 + 10*w],\ [757, 757, 5/3*w^3 - 7/3*w^2 - 11*w + 15],\ [757, 757, 2/3*w^3 + 5/3*w^2 - 3*w - 5],\ [761, 761, 7/3*w^3 - 5/3*w^2 - 18*w + 11],\ [761, 761, 1/3*w^3 + 4/3*w^2 - 4*w - 13],\ [773, 773, 1/3*w^3 + 4/3*w^2 - 3*w - 7],\ [773, 773, 1/3*w^3 + 1/3*w^2 - 3*w - 7],\ [809, 809, w^3 - 8*w + 4],\ [811, 811, -2/3*w^3 + 4/3*w^2 + 5*w],\ [823, 823, 7/3*w^3 - 5/3*w^2 - 17*w + 9],\ [827, 827, -5/3*w^3 + 1/3*w^2 + 12*w - 4],\ [827, 827, 1/3*w^3 + 4/3*w^2 - 4*w - 10],\ [829, 829, -1/3*w^3 - 1/3*w^2 - w - 3],\ [841, 29, -1/3*w^3 - 4/3*w^2 + 2*w + 12],\ [857, 857, 4*w^3 - 4*w^2 - 31*w + 26],\ [859, 859, 11/3*w^3 - 10/3*w^2 - 27*w + 21],\ [863, 863, 7/3*w^3 - 5/3*w^2 - 17*w + 8],\ [877, 877, 1/3*w^3 + 4/3*w^2 - 3*w - 6],\ [877, 877, -4/3*w^3 + 5/3*w^2 + 11*w - 6],\ [881, 881, 5/3*w^3 - 4/3*w^2 - 11*w + 4],\ [881, 881, 4/3*w^3 - 11/3*w^2 - 10*w + 24],\ [883, 883, 10/3*w^3 - 17/3*w^2 - 25*w + 36],\ [883, 883, -4/3*w^3 + 2/3*w^2 + 11*w - 8],\ [887, 887, -1/3*w^3 + 2/3*w^2 + 3*w - 9],\ [887, 887, 3*w^3 - 2*w^2 - 23*w + 8],\ [911, 911, 4/3*w^3 + 1/3*w^2 - 7*w],\ [919, 919, -4/3*w^3 + 5/3*w^2 + 7*w - 6],\ [929, 929, 2*w^3 + w^2 - 14*w - 8],\ [937, 937, -7/3*w^3 + 5/3*w^2 + 15*w - 14],\ [941, 941, -1/3*w^3 - 1/3*w^2 + 6*w],\ [941, 941, -7/3*w^3 - 1/3*w^2 + 18*w + 3],\ [971, 971, -4/3*w^3 + 5/3*w^2 + 10*w - 6],\ [977, 977, -7/3*w^3 + 11/3*w^2 + 18*w - 27],\ [977, 977, 7/3*w^3 - 2/3*w^2 - 17*w],\ [997, 997, 7/3*w^3 - 2/3*w^2 - 17*w + 2],\ [1009, 1009, 4/3*w^3 + 1/3*w^2 - 13*w + 3],\ [1009, 1009, 2/3*w^3 - 7/3*w^2 - 6*w + 15],\ [1013, 1013, 4/3*w^3 + 4/3*w^2 - 12*w - 7],\ [1013, 1013, 4/3*w^3 + 1/3*w^2 - 9*w + 2],\ [1019, 1019, -1/3*w^3 + 2/3*w^2 + 6*w],\ [1021, 1021, w^2 - 11],\ [1021, 1021, 10/3*w^3 - 8/3*w^2 - 25*w + 17],\ [1021, 1021, -7/3*w^3 + 2/3*w^2 + 19*w + 4],\ [1021, 1021, -5/3*w^3 + 7/3*w^2 + 10*w - 13],\ [1031, 1031, -w^3 + 2*w^2 + 9*w - 14],\ [1031, 1031, -2/3*w^3 + 4/3*w^2 + 4*w - 13],\ [1031, 1031, -w^3 + 9*w - 5],\ [1031, 1031, w^2 + 2*w - 7],\ [1033, 1033, 4/3*w^3 - 11/3*w^2 - 9*w + 26],\ [1033, 1033, -4/3*w^3 + 2/3*w^2 + 7*w - 9],\ [1061, 1061, -1/3*w^3 + 2/3*w^2 - 5],\ [1061, 1061, -5/3*w^3 + 4/3*w^2 + 14*w - 7],\ [1069, 1069, 4/3*w^3 + 4/3*w^2 - 11*w - 3],\ [1069, 1069, -7/3*w^3 + 14/3*w^2 + 17*w - 30],\ [1087, 1087, 2*w^3 - w^2 - 14*w + 2],\ [1091, 1091, -w^3 + 10*w - 7],\ [1091, 1091, -4/3*w^3 + 5/3*w^2 + 8*w - 2],\ [1093, 1093, 2/3*w^3 + 2/3*w^2 - 7*w + 3],\ [1103, 1103, 1/3*w^3 - 2/3*w^2 - w - 4],\ [1109, 1109, -1/3*w^3 + 2/3*w^2 + 5*w - 9],\ [1109, 1109, w^3 + w^2 - 7*w - 11],\ [1117, 1117, -w^3 + 10*w - 1],\ [1129, 1129, -4/3*w^3 + 8/3*w^2 + 10*w - 11],\ [1163, 1163, -4/3*w^3 + 2/3*w^2 + 12*w - 5],\ [1163, 1163, -1/3*w^3 + 5/3*w^2 + 2*w - 15],\ [1181, 1181, -5/3*w^3 + 4/3*w^2 + 11*w - 7],\ [1187, 1187, 10/3*w^3 - 8/3*w^2 - 26*w + 17],\ [1193, 1193, -2/3*w^3 + 7/3*w^2 + 5*w - 13],\ [1193, 1193, 5/3*w^3 - 1/3*w^2 - 11*w],\ [1193, 1193, 2/3*w^3 - 1/3*w^2 - 6*w - 5],\ [1193, 1193, 4/3*w^3 + 1/3*w^2 - 10*w - 9],\ [1201, 1201, 2/3*w^3 + 5/3*w^2 - 6*w - 11],\ [1229, 1229, -2/3*w^3 + 7/3*w^2 + 7*w - 13],\ [1231, 1231, 5/3*w^3 - 4/3*w^2 - 11*w - 2],\ [1249, 1249, -1/3*w^3 - 1/3*w^2 + 4*w - 6],\ [1249, 1249, -5/3*w^3 + 4/3*w^2 + 11*w - 6],\ [1259, 1259, 5/3*w^3 - 7/3*w^2 - 13*w + 19],\ [1277, 1277, 4/3*w^3 + 1/3*w^2 - 13*w],\ [1291, 1291, 5/3*w^3 - 1/3*w^2 - 9*w - 2],\ [1297, 1297, -1/3*w^3 + 5/3*w^2 + w - 12],\ [1327, 1327, w^3 + 2*w^2 - 7*w - 13],\ [1327, 1327, 2/3*w^3 - 1/3*w^2 - 5*w - 5],\ [1361, 1361, -1/3*w^3 + 5/3*w^2 + 4*w - 11],\ [1367, 1367, 2/3*w^3 - 1/3*w^2 - 8*w],\ [1367, 1367, -3*w - 5],\ [1399, 1399, -2/3*w^3 + 4/3*w^2 + 3*w - 9],\ [1399, 1399, -5/3*w^3 + 4/3*w^2 + 13*w - 3],\ [1433, 1433, -5/3*w^3 + 4/3*w^2 + 13*w - 15],\ [1433, 1433, -7/3*w^3 + 8/3*w^2 + 19*w - 15],\ [1439, 1439, -w^3 + 2*w^2 + 6*w - 14],\ [1439, 1439, -1/3*w^3 + 2/3*w^2 - w + 7],\ [1447, 1447, 1/3*w^3 + 1/3*w^2 - 5*w - 6],\ [1447, 1447, -4/3*w^3 + 2/3*w^2 + 12*w - 3],\ [1451, 1451, 4/3*w^3 - 8/3*w^2 - 10*w + 21],\ [1451, 1451, w^2 - 3*w - 5],\ [1451, 1451, -2/3*w^3 + 4/3*w^2 + 5*w - 13],\ [1451, 1451, 10/3*w^3 - 14/3*w^2 - 25*w + 33],\ [1453, 1453, 2/3*w^3 - 7/3*w^2 - 5*w + 12],\ [1453, 1453, 5/3*w^3 - 1/3*w^2 - 15*w - 2],\ [1453, 1453, -4/3*w^3 - 1/3*w^2 + 9*w - 3],\ [1453, 1453, 4/3*w^3 + 1/3*w^2 - 7*w - 4],\ [1471, 1471, -1/3*w^3 + 2/3*w^2 - 6],\ [1489, 1489, -5/3*w^3 + 1/3*w^2 + 13*w - 4],\ [1489, 1489, -2/3*w^3 + 1/3*w^2 + 2*w - 6],\ [1493, 1493, 7/3*w^3 - 11/3*w^2 - 17*w + 27],\ [1493, 1493, -2*w^3 + 2*w^2 + 15*w - 8],\ [1511, 1511, 17/3*w^3 - 19/3*w^2 - 43*w + 37],\ [1523, 1523, -w^3 + 3*w^2 + 9*w - 11],\ [1523, 1523, 1/3*w^3 + 1/3*w^2 - 5*w - 7],\ [1523, 1523, -w^3 + 3*w^2 + 8*w - 22],\ [1523, 1523, 2/3*w^3 + 5/3*w^2 - 4*w - 5],\ [1543, 1543, 1/3*w^3 + 1/3*w^2 - 6*w - 4],\ [1543, 1543, -4/3*w^3 + 5/3*w^2 + 10*w - 5],\ [1553, 1553, -4/3*w^3 + 5/3*w^2 + 11*w - 5],\ [1559, 1559, 2*w^3 - 15*w + 1],\ [1559, 1559, -4/3*w^3 + 2/3*w^2 + 5*w - 3],\ [1567, 1567, 1/3*w^3 + 1/3*w^2 - 6],\ [1571, 1571, 1/3*w^3 + 1/3*w^2 + 3*w - 1],\ [1571, 1571, -4/3*w^3 + 5/3*w^2 + 8*w - 14],\ [1579, 1579, -10/3*w^3 + 11/3*w^2 + 24*w - 20],\ [1583, 1583, 1/3*w^3 - 2/3*w^2 - 2*w - 4],\ [1597, 1597, 11/3*w^3 - 19/3*w^2 - 28*w + 42],\ [1601, 1601, 10/3*w^3 - 14/3*w^2 - 25*w + 26],\ [1613, 1613, 8/3*w^3 - 16/3*w^2 - 19*w + 36],\ [1627, 1627, -2/3*w^3 + 1/3*w^2 + 3*w - 9],\ [1627, 1627, -2/3*w^3 + 1/3*w^2 + 6*w - 9],\ [1637, 1637, 2*w^3 + w^2 - 16*w - 4],\ [1657, 1657, -2/3*w^3 + 7/3*w^2 + 6*w - 16],\ [1657, 1657, 2/3*w^3 + 2/3*w^2 - 3*w - 6],\ [1663, 1663, -3*w + 10],\ [1663, 1663, 1/3*w^3 + 4/3*w^2 - w - 9],\ [1667, 1667, -2/3*w^3 + 1/3*w^2 + 2*w - 7],\ [1667, 1667, 8/3*w^3 - 4/3*w^2 - 22*w + 3],\ [1669, 1669, 3*w^3 - 2*w^2 - 22*w + 11],\ [1693, 1693, 1/3*w^3 - 2/3*w^2 - 3*w - 4],\ [1697, 1697, -4/3*w^3 + 2/3*w^2 + 7*w - 3],\ [1709, 1709, 3*w^3 - 3*w^2 - 23*w + 23],\ [1741, 1741, -3*w^3 + 2*w^2 + 24*w - 10],\ [1747, 1747, -2/3*w^3 + 7/3*w^2 + 4*w - 10],\ [1747, 1747, 1/3*w^3 - 8/3*w^2 - 4*w + 9],\ [1753, 1753, 4/3*w^3 - 2/3*w^2 - 7*w + 2],\ [1753, 1753, 8/3*w^3 - 16/3*w^2 - 19*w + 37],\ [1759, 1759, 2*w^2 - w - 5],\ [1759, 1759, -5/3*w^3 + 10/3*w^2 + 14*w - 21],\ [1777, 1777, -1/3*w^3 + 2/3*w^2 + 6*w - 2],\ [1783, 1783, 7/3*w^3 - 8/3*w^2 - 17*w + 12],\ [1783, 1783, -5/3*w^3 - 2/3*w^2 + 10*w - 3],\ [1787, 1787, -1/3*w^3 - 1/3*w^2 - 2*w - 3],\ [1787, 1787, 4/3*w^3 + 1/3*w^2 - 12*w + 2],\ [1811, 1811, 5/3*w^3 - 1/3*w^2 - 10*w - 6],\ [1811, 1811, -w^3 + w^2 + 5*w - 10],\ [1847, 1847, w^2 - 2*w - 10],\ [1861, 1861, 2*w^2 - w - 11],\ [1867, 1867, w^3 + 2*w^2 - 7*w - 4],\ [1871, 1871, -1/3*w^3 + 5/3*w^2 - 9],\ [1873, 1873, 2/3*w^3 + 5/3*w^2 - 7*w - 6],\ [1877, 1877, -2/3*w^3 + 1/3*w^2 + 8*w - 3],\ [1877, 1877, -1/3*w^3 + 2/3*w^2 + 6*w - 3],\ [1879, 1879, w^3 + 2*w^2 - 11*w - 8],\ [1879, 1879, 8/3*w^3 - 7/3*w^2 - 20*w + 18],\ [1901, 1901, -2/3*w^3 + 1/3*w^2 + 2*w + 9],\ [1901, 1901, 10/3*w^3 - 11/3*w^2 - 24*w + 21],\ [1907, 1907, w^3 + w^2 - 10*w - 8],\ [1907, 1907, 5*w^3 - 6*w^2 - 38*w + 35],\ [1933, 1933, 1/3*w^3 + 1/3*w^2 - 7],\ [1933, 1933, 3*w^3 - w^2 - 22*w + 4],\ [1949, 1949, 5/3*w^3 + 2/3*w^2 - 15*w - 9],\ [1951, 1951, w^3 + w^2 - 12*w - 1],\ [1979, 1979, -7/3*w^3 + 5/3*w^2 + 17*w - 15],\ [1993, 1993, -5/3*w^3 + 1/3*w^2 + 12*w - 6],\ [1999, 1999, -4/3*w^3 - 1/3*w^2 + 13*w + 7],\ [1999, 1999, 2/3*w^3 + 5/3*w^2 - 6*w - 5]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^3 - x^2 - 5*x - 1 K. = NumberField(heckePol) hecke_eigenvalues_array = [-1, e, -e^2 + e + 5, e + 2, 3*e^2 - 5*e - 9, 2*e^2 - 2*e - 8, -e^2 + 1, 2*e^2 - 4*e - 4, -e^2 - 2*e + 5, 2*e^2 - 4*e - 5, -e^2 + 4*e + 3, -2*e^2 + 6*e + 5, -3*e^2 + 4*e + 9, e^2 - 1, -e^2 + 10, e^2 + 2*e - 5, -e^2 + 2*e + 11, e^2 - 2*e + 2, e^2 - 4*e + 5, e^2 - 2*e - 15, e^2 - 6*e - 3, e^2 - 2*e + 7, -e^2 - 1, -e^2 - 4*e + 11, -2*e^2 + 2*e + 10, -7*e^2 + 6*e + 21, -3*e^2 + 6*e + 19, -e^2 + 8*e + 3, 3*e^2 - 3*e - 7, 6*e^2 - 2*e - 21, 3*e^2 - 2*e - 3, 4*e^2 - 14*e - 12, -5*e^2 + 14*e + 15, -5*e^2 + 9*e + 21, 4*e^2 - 3*e - 18, 2*e^2 - 6*e - 1, -4*e^2 + 8*e + 12, 6*e + 4, -8*e^2 + 8*e + 24, 6*e + 8, 2*e^2 - 2*e - 26, -4*e^2 + 8*e + 20, -8*e^2 + 12*e + 16, 3*e^2 - 2*e + 5, e^2 - 12*e + 5, -5*e^2 + 2*e + 17, 7*e^2 - 14*e - 31, 2*e^2 + e - 10, -7*e^2 + 5*e + 17, 2*e^2 - 9*e + 2, -e^2 - e + 19, 5*e^2 - 12*e - 16, 3*e^2 + 2*e - 27, -9*e^2 + 16*e + 26, 6*e^2 - 14*e - 4, -6*e^2 + 10*e + 4, -5*e^2 + 10*e + 37, -2*e^2 + 10*e - 2, -12*e^2 + 22*e + 32, 5*e^2 + 2*e - 26, 4*e^2 - 9*e - 28, -10*e + 9, 6*e^2 - 12*e - 2, -4*e^2 + 2*e + 13, -2*e^2 + 2*e - 2, -e^2 + 10*e + 5, 4*e^2 - 2*e + 6, -e^2 + 4*e + 25, 4*e^2 + 4*e - 22, 3*e^2 - 2*e - 29, -9*e^2 + 8*e + 27, -13*e^2 + 20*e + 41, -e^2 - 9*e + 11, e^2 + e + 15, 8*e^2 - 18*e - 22, 7*e^2 - 14*e - 13, e^2 + 2*e - 15, -6*e^2 + 18*e + 24, 7*e^2 - 6*e - 7, -7*e^2 + 18*e + 34, -14*e^2 + 26*e + 46, -3*e^2 - 7*e + 9, 3*e^2 - 12*e - 7, -e^2 + 6*e - 13, 9*e^2 - 12*e - 40, -8*e^2 + 18*e + 38, -2*e^2 + 10*e, 3*e^2 - 16*e - 2, -2*e^2 - 8*e + 13, 8*e^2 - 6*e - 22, 3*e^2 - 16*e + 3, e^2 - 18*e + 7, 7*e^2 - 19, 3*e^2 + 8*e - 17, 8*e^2 - 6*e - 8, 7*e^2 - 16*e - 43, -2*e^2 + 8*e - 18, 11*e^2 - 7*e - 47, -5*e^2 + 4*e + 11, -6*e^2 + 4*e, e^2 - 7*e - 5, 4*e^2 - 14*e - 24, -7*e^2 - 4*e + 27, e^2 + 2*e + 4, 8*e^2 - 6*e - 30, e^2 + 6*e - 25, 5*e^2 - 6*e + 5, 5*e^2 - 6*e - 39, -12*e^2 + 22*e + 41, e^2 + 2*e - 30, -7*e^2 + 8*e + 37, 2*e^2 - 7*e - 18, 5*e^2 - 22*e - 16, -2*e^2 - e - 8, -e^2 + 2*e + 29, -7*e^2 + 43, -4*e^2 + 20*e + 2, -13*e^2 + 29*e + 39, -e^2 - 8*e + 33, 5*e^2 - 21*e - 23, 8*e + 10, -e^2 + 6*e + 27, 10*e^2 - 10*e, -2*e^2 + 10*e - 15, 4*e^2 + 4*e - 28, 20*e^2 - 34*e - 60, -12*e^2 + 6*e + 34, 2*e^2 + 6*e - 16, -13*e^2 + 17*e + 31, -9*e^2 + 6*e + 21, -7*e^2 + 20*e + 27, -2*e^2 + 14*e + 10, 15*e^2 - 20*e - 31, -6*e^2 + 19*e + 18, -8*e^2 + 12*e + 46, -6*e^2 + 12*e + 12, -7*e^2 + 15, 14*e^2 - 33*e - 54, 12*e^2 - 30*e - 51, -9*e^2 + 8*e + 13, -9*e^2 + 22*e + 40, 3*e^2 + 4*e - 49, e^2 + 10*e - 21, -14*e^2 + 15*e + 48, 5*e^2 - 16*e - 29, -10*e + 20, 13*e^2 - 4*e - 45, -2*e^2 - 8*e - 11, 4*e - 29, 9*e^2 - 4*e - 42, -8*e^2 + 10*e - 8, -e^2 + 11, -2*e^2 - 6*e + 28, 6*e^2 - 20*e - 44, -8*e^2 + 12*e + 8, 9*e^2 - 14*e + 5, 6*e^2 + 2*e - 24, -5*e^2 + 2*e + 51, 11*e^2 - 20*e - 42, 18*e^2 - 23*e - 34, 12*e^2 - 20*e - 46, -2*e^2 - 2*e + 42, 9*e^2 - 11*e - 15, 2*e^2 - 6*e - 20, 4*e^2 + 8*e - 9, 7*e^2 - 12*e + 2, e^2 + 4*e - 33, -8*e^2 + 22*e + 42, 7*e^2 - 22*e - 2, -10*e^2 + 13*e + 40, -12*e^2 + 10*e + 54, -5*e^2 + 2*e + 15, 9*e^2 - 20*e - 51, 3*e^2 + 9*e - 7, 5*e^2 + 2*e - 22, 3*e - 36, -19*e^2 + 16*e + 64, 7*e^2 - 31*e - 9, -17*e - 10, -15*e^2 + 6*e + 61, 18*e^2 - 24*e - 53, -2*e^2 - 23, -15*e^2 + 30*e + 64, -17*e^2 + 8*e + 59, -14*e^2 + 14*e + 55, e^2 + 7*e + 19, -4*e^2 + 16*e + 30, -4*e^2 + 4*e - 22, 10*e^2 - 18*e - 22, -10*e^2 + 31*e + 32, 5*e^2 + 8*e - 48, e^2 + 12*e - 39, 10*e^2 - 8*e - 47, -3*e^2 + 2*e - 1, 7*e^2 - 16*e - 24, 4*e^2 + e - 18, 10*e^2 - 24*e - 36, 14*e^2 - 24*e - 35, 8*e^2 - 11*e - 18, 6*e^2 + 11*e - 16, 3*e^2 - 6*e - 7, 14*e^2 - 34*e - 30, -12*e^2 + 6*e + 36, 3*e^2 - 25*e - 9, 9*e^2 - 24*e - 25, 5*e^2 + 4*e - 52, -e^2 + 6*e + 42, -9*e^2 + 24*e + 35, -8*e^2 + 6*e + 5, -9*e^2 + 16*e + 21, 12*e^2 - 16*e - 37, -4*e^2 + 30*e + 7, 25*e^2 - 32*e - 57, -18*e^2 + 36*e + 52, -8*e^2 + 6*e + 20, 9*e^2 - 25*e - 11, 3*e^2 + 2, -17*e^2 + 29*e + 47, 4*e^2 - e - 28, -4*e^2 + 5*e + 56, 4*e^2 - 16*e + 22, -4*e^2 - 6*e + 33, 13*e^2 - 21*e - 51, 7*e^2 - 34*e - 23, 11*e^2 - 7*e - 69, 6*e^2 - 19*e - 10, 8*e^2 + 3*e - 34, -19*e^2 + 39*e + 51, 7*e^2 - 16*e - 23, 12*e^2 - 11*e - 30, -6*e^2 - 10*e + 66, -19*e^2 + 50*e + 63, -2*e^2 + 4*e + 36, -22*e + 11, 12*e^2 - 46*e - 44, 19*e^2 - 36*e - 61, 2*e^2 - 2*e + 12, 23*e^2 - 41*e - 75, 16*e^2 - 22*e - 50, 11*e^2 - 30*e - 37, -14*e^2 + 35*e + 72, 4*e^2 - 34, 14*e^2 - 10*e - 47, -6*e^2 + 14*e - 7, 4*e^2 - 4*e - 24, 12*e^2 + 4*e - 64, -11*e^2 + 4*e + 35, -e^2 + 6*e - 7, 11*e^2 - 2*e - 66, -11*e^2 + 26*e + 21, -3*e^2 - 15*e + 1, -8*e^2 + 10*e - 29, e^2 + 2*e - 59, -2*e^2 + 23*e + 4, -21*e^2 + 34*e + 89, -11*e^2 + 20*e + 11, -3*e^2 - 2*e - 21, 3*e^2 + 16*e - 48, 4*e^2 - 20*e + 11, -6*e + 26, 10*e^2 - 30*e - 38, 9*e^2 - 6*e - 13, 8*e^2 - 14*e - 26, -15*e^2 + 3*e + 69, -7*e^2 - 8*e + 33, -9*e^2 - 10*e + 22, 8*e + 17, -11*e^2 + 2*e + 41, -2*e^2 - 14*e - 4, -19*e^2 + 16*e + 63, -17*e^2 + 14*e + 31, -19*e^2 + 50*e + 55, -13*e^2 + 10*e + 59, 4*e^2 + 2*e + 6, e^2 + 6*e + 30, -14*e^2 + 30*e + 48, -6*e^2 + 8*e - 4, -6*e^2 + 12*e - 24, 23*e^2 - 41*e - 53, -5*e^2 + 6*e + 33, 7*e^2 - e - 9, -16*e^2 + 30*e + 68, -20*e^2 + 28*e + 75, 7*e^2 - 85, e^2 - 25, -15*e^2 + 40*e + 33, -7*e^2 + 20*e + 58, 17*e^2 - 35*e - 63, 12*e^2 - 32*e - 20, -2*e^2 + 16*e - 9, -8*e^2 + 4*e + 36, 9*e^2 + 12*e - 47, 2*e^2 + 3*e + 30, e^2 - 2*e - 17, -26*e^2 + 19*e + 72, -19*e^2 + 30*e + 87, -11*e^2 + 30*e + 17, -14*e^2 + 12*e + 30, 7*e^2 - 20*e + 7] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([3, 3, w])] = 1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]