/* 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, -11, 10, 10, -6, -2, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([41, 41, -w^5 + w^4 + 4*w^3 - w^2 - 3*w - 1]) primes_array = [ [7, 7, w^4 - 2*w^3 - 3*w^2 + 6*w - 1],\ [7, 7, -w^4 + w^3 + 4*w^2 - w - 1],\ [13, 13, w^5 - 2*w^4 - 4*w^3 + 7*w^2 + 3*w - 3],\ [29, 29, -w^4 + w^3 + 4*w^2 - 3*w - 2],\ [29, 29, w^4 - 5*w^2 - 2*w + 4],\ [29, 29, -w^4 + w^3 + 3*w^2 - w + 2],\ [29, 29, -w^2 + w + 1],\ [41, 41, -w^5 + w^4 + 4*w^3 - w^2 - 3*w - 1],\ [43, 43, 2*w^3 - 2*w^2 - 7*w + 3],\ [43, 43, w^3 - w^2 - 2*w + 1],\ [64, 2, -2],\ [71, 71, w^5 - w^4 - 6*w^3 + 4*w^2 + 8*w - 3],\ [71, 71, w^5 - 2*w^4 - 2*w^3 + 5*w^2 - 3*w - 1],\ [83, 83, -2*w^4 + 3*w^3 + 6*w^2 - 7*w + 2],\ [83, 83, w^4 - w^3 - 5*w^2 + 2*w + 3],\ [83, 83, -w^5 + 3*w^4 + w^3 - 9*w^2 + 5*w + 3],\ [83, 83, w^5 - 7*w^3 + 10*w - 1],\ [97, 97, w^5 - 2*w^4 - 3*w^3 + 5*w^2 - w - 1],\ [97, 97, w^5 - 5*w^3 - w^2 + 3*w - 3],\ [113, 113, w^4 - w^3 - 4*w^2 + w + 4],\ [113, 113, w^4 - 2*w^3 - 3*w^2 + 6*w + 2],\ [127, 127, -w^4 + 2*w^3 + 4*w^2 - 5*w - 3],\ [127, 127, w^5 - 9*w^3 + w^2 + 18*w - 1],\ [127, 127, 2*w^4 - 4*w^3 - 5*w^2 + 10*w - 2],\ [127, 127, w^5 - w^4 - 6*w^3 + 3*w^2 + 9*w - 2],\ [127, 127, -3*w^4 + 5*w^3 + 9*w^2 - 11*w],\ [127, 127, -w^5 + 3*w^4 + w^3 - 8*w^2 + 4*w - 1],\ [139, 139, w^4 - 3*w^3 - 2*w^2 + 10*w - 3],\ [139, 139, -w^5 + 2*w^4 + 4*w^3 - 5*w^2 - 4*w - 3],\ [167, 167, -w^4 + w^3 + 5*w^2 - 4*w - 3],\ [167, 167, -w^5 + w^4 + 6*w^3 - 3*w^2 - 8*w + 3],\ [169, 13, w^4 - 3*w^3 - 2*w^2 + 8*w - 2],\ [169, 13, 2*w^4 - 3*w^3 - 7*w^2 + 7*w + 3],\ [181, 181, -w^5 + 4*w^3 + 3*w^2 - 2],\ [181, 181, w^5 - 3*w^4 - 3*w^3 + 11*w^2 - 4],\ [181, 181, w^5 - 3*w^4 - 3*w^3 + 10*w^2 + 2*w - 4],\ [181, 181, w^5 - 2*w^4 - 4*w^3 + 6*w^2 + 3*w - 2],\ [197, 197, -3*w^4 + 5*w^3 + 10*w^2 - 11*w - 3],\ [197, 197, -3*w^4 + 6*w^3 + 9*w^2 - 16*w - 1],\ [197, 197, 2*w^3 - 3*w^2 - 5*w + 4],\ [197, 197, -w^5 + 6*w^3 - 6*w + 1],\ [223, 223, 2*w^4 - 2*w^3 - 9*w^2 + 3*w + 6],\ [223, 223, -3*w^4 + 5*w^3 + 9*w^2 - 13*w + 1],\ [223, 223, w^5 - w^4 - 6*w^3 + 4*w^2 + 10*w - 4],\ [223, 223, w^4 - 4*w^3 - 2*w^2 + 12*w],\ [239, 239, 2*w^4 - w^3 - 8*w^2 + 3],\ [239, 239, -w^5 + 3*w^4 - 8*w^2 + 9*w + 3],\ [239, 239, -w^5 + 2*w^4 + 2*w^3 - 6*w^2 + 3*w + 4],\ [239, 239, w^4 - 6*w^2 + 6],\ [239, 239, 3*w^4 - 4*w^3 - 11*w^2 + 10*w + 3],\ [239, 239, -w^5 + 7*w^3 - 9*w - 1],\ [251, 251, w^5 - 2*w^4 - 2*w^3 + 5*w^2 - 3*w - 2],\ [251, 251, -w^5 + w^4 + 6*w^3 - 4*w^2 - 8*w + 2],\ [281, 281, -3*w^4 + 4*w^3 + 11*w^2 - 10*w - 4],\ [281, 281, 3*w^4 - 3*w^3 - 12*w^2 + 5*w + 6],\ [293, 293, -w^5 + 8*w^3 - 15*w + 1],\ [293, 293, w^5 - 4*w^4 + w^3 + 11*w^2 - 9*w + 1],\ [293, 293, -2*w^4 + 5*w^3 + 5*w^2 - 14*w + 3],\ [293, 293, -w^5 + 8*w^3 - 14*w + 2],\ [293, 293, -2*w^4 + 4*w^3 + 6*w^2 - 9*w + 1],\ [293, 293, w^5 - 4*w^4 + 12*w^2 - 5*w - 2],\ [307, 307, 2*w^4 - 3*w^3 - 7*w^2 + 9*w + 2],\ [307, 307, -w^5 + 2*w^4 + 5*w^3 - 8*w^2 - 8*w + 6],\ [307, 307, -w^3 + 6*w],\ [307, 307, w^5 - w^4 - 7*w^3 + 5*w^2 + 12*w - 3],\ [337, 337, -2*w^4 + 2*w^3 + 9*w^2 - 4*w - 8],\ [337, 337, 3*w^4 - 4*w^3 - 10*w^2 + 9*w + 3],\ [349, 349, 2*w^5 - 4*w^4 - 7*w^3 + 13*w^2 + 3*w - 6],\ [349, 349, 2*w^5 - 2*w^4 - 9*w^3 + 5*w^2 + 7*w - 2],\ [349, 349, w^5 - 3*w^4 + 8*w^2 - 7*w - 2],\ [349, 349, w^5 - 9*w^3 + 2*w^2 + 17*w - 4],\ [379, 379, w^5 - 3*w^4 + 6*w^2 - 8*w + 5],\ [379, 379, w^5 - 2*w^4 - 4*w^3 + 5*w^2 + 3*w + 1],\ [379, 379, w^5 - 4*w^4 + 12*w^2 - 8*w - 2],\ [379, 379, -w^5 + w^4 + 7*w^3 - 5*w^2 - 11*w + 5],\ [419, 419, w^5 - 4*w^4 + w^3 + 10*w^2 - 10*w + 3],\ [419, 419, 3*w^4 - 3*w^3 - 11*w^2 + 5*w + 4],\ [421, 421, w^5 - 2*w^4 - 5*w^3 + 8*w^2 + 8*w - 4],\ [421, 421, -w^5 + w^4 + 5*w^3 - 3*w^2 - 7*w + 4],\ [433, 433, -2*w^2 + w + 5],\ [433, 433, -2*w^4 + 3*w^3 + 5*w^2 - 6*w + 3],\ [461, 461, 2*w^3 - 2*w^2 - 5*w + 3],\ [461, 461, -w^5 + 5*w^3 + 2*w^2 - 4*w + 1],\ [461, 461, -w^5 + 3*w^4 + 2*w^3 - 10*w^2 + 3*w + 5],\ [461, 461, -w^5 + 3*w^4 + 3*w^3 - 11*w^2 - w + 3],\ [463, 463, -2*w^4 + 3*w^3 + 7*w^2 - 8*w - 4],\ [463, 463, -w^5 + 4*w^4 + w^3 - 12*w^2 + 3*w - 1],\ [491, 491, w^5 - 7*w^3 - w^2 + 10*w + 3],\ [491, 491, -w^5 + 4*w^4 - w^3 - 11*w^2 + 10*w + 2],\ [503, 503, -2*w^5 + 5*w^4 + 5*w^3 - 15*w^2 + w + 3],\ [503, 503, 2*w^4 - w^3 - 8*w^2 + w + 1],\ [503, 503, -2*w^5 + 3*w^4 + 7*w^3 - 7*w^2 - 3*w + 1],\ [503, 503, -2*w^5 + 2*w^4 + 10*w^3 - 5*w^2 - 11*w + 2],\ [547, 547, w^5 - 2*w^4 - 6*w^3 + 9*w^2 + 10*w - 5],\ [547, 547, -w^4 + 2*w^3 + 2*w^2 - 4*w + 5],\ [547, 547, -2*w^5 + 4*w^4 + 7*w^3 - 12*w^2 - 3*w + 4],\ [547, 547, -w^5 + w^4 + 3*w^3 - 2*w^2 + 3*w + 2],\ [587, 587, -w^5 - w^4 + 8*w^3 + 4*w^2 - 13*w + 1],\ [587, 587, w^5 - w^4 - 5*w^3 + 2*w^2 + 7*w - 2],\ [617, 617, w^5 - w^4 - 4*w^3 + 3*w^2 - 1],\ [617, 617, w^4 - 2*w^3 - 5*w^2 + 5*w + 6],\ [617, 617, -w^5 + w^4 + 4*w^3 - 3*w^2 + 4],\ [617, 617, -w^5 + 3*w^4 + 6*w^3 - 13*w^2 - 11*w + 8],\ [631, 631, -2*w^5 + 2*w^4 + 10*w^3 - 5*w^2 - 11*w + 1],\ [631, 631, -w^5 + w^4 + 4*w^3 - 2*w - 4],\ [631, 631, -2*w^4 + 5*w^3 + 4*w^2 - 12*w + 4],\ [631, 631, -2*w^5 + 5*w^4 + 5*w^3 - 15*w^2 + w + 4],\ [643, 643, w^5 - 2*w^4 - 5*w^3 + 7*w^2 + 6*w - 2],\ [643, 643, -w^5 + 2*w^4 + w^3 - 3*w^2 + 6*w - 3],\ [659, 659, -2*w^4 + 4*w^3 + 7*w^2 - 10*w - 2],\ [659, 659, w^4 - 3*w^3 - 3*w^2 + 10*w - 1],\ [659, 659, w^5 - 4*w^4 + 13*w^2 - 6*w - 6],\ [659, 659, w^5 - 2*w^4 - 5*w^3 + 7*w^2 + 7*w - 5],\ [673, 673, 2*w^5 - 3*w^4 - 7*w^3 + 8*w^2 + 2*w - 3],\ [673, 673, w^5 - 4*w^4 - w^3 + 12*w^2 - 3*w - 2],\ [673, 673, -2*w^5 + 3*w^4 + 9*w^3 - 10*w^2 - 8*w + 5],\ [673, 673, w^5 - w^4 - 4*w^3 + 3*w + 6],\ [701, 701, -2*w^4 + 4*w^3 + 6*w^2 - 11*w - 3],\ [701, 701, 2*w^4 - 3*w^3 - 7*w^2 + 6*w + 5],\ [727, 727, -w^5 + 2*w^4 + w^3 - 3*w^2 + 7*w - 2],\ [727, 727, 2*w^5 - 3*w^4 - 7*w^3 + 8*w^2 + 3*w - 4],\ [727, 727, w^5 - 2*w^4 - 4*w^3 + 6*w^2 + 2*w + 1],\ [727, 727, w^5 + 2*w^4 - 9*w^3 - 8*w^2 + 16*w + 2],\ [729, 3, -3],\ [743, 743, w^5 - 4*w^4 + 13*w^2 - 8*w - 5],\ [743, 743, -w^3 + w^2 + w - 2],\ [743, 743, w^4 - 4*w^3 - w^2 + 10*w - 2],\ [743, 743, 3*w^3 - 3*w^2 - 11*w + 6],\ [757, 757, -w^5 + 4*w^4 - 11*w^2 + 5*w - 2],\ [757, 757, -3*w^4 + 3*w^3 + 11*w^2 - 5*w - 2],\ [757, 757, w^5 - 3*w^4 - 2*w^3 + 10*w^2 - 4],\ [757, 757, w^5 - 8*w^3 + w^2 + 15*w - 4],\ [769, 769, w^5 - 5*w^3 - 3*w^2 + 4*w + 6],\ [769, 769, -2*w^5 + 3*w^4 + 8*w^3 - 9*w^2 - 4*w + 3],\ [769, 769, w^4 - 3*w^3 - w^2 + 9*w - 3],\ [769, 769, w^5 - 4*w^4 - w^3 + 13*w^2 - 4*w - 5],\ [797, 797, w^5 + w^4 - 8*w^3 - 4*w^2 + 11*w + 2],\ [797, 797, w^5 - 2*w^4 - 6*w^3 + 9*w^2 + 9*w - 6],\ [811, 811, w^5 - 3*w^4 - 4*w^3 + 11*w^2 + 4*w - 4],\ [811, 811, 2*w^4 - 2*w^3 - 6*w^2 + 3*w - 4],\ [811, 811, -2*w^5 + 3*w^4 + 9*w^3 - 8*w^2 - 9*w + 1],\ [811, 811, -w^5 + w^4 + 2*w^3 + w^2 + 4*w - 5],\ [827, 827, w^5 + w^4 - 9*w^3 - 2*w^2 + 14*w - 3],\ [827, 827, w^5 - 3*w^4 + w^3 + 6*w^2 - 11*w + 4],\ [827, 827, w^5 - 3*w^4 + 9*w^2 - 9*w - 4],\ [827, 827, w^5 - w^4 - 4*w^3 + 4*w + 5],\ [827, 827, -w^5 + 4*w^4 + 2*w^3 - 13*w^2 - w + 3],\ [827, 827, -w^5 + w^4 + 7*w^3 - 4*w^2 - 11*w + 3],\ [839, 839, -w^5 + w^4 + 7*w^3 - 4*w^2 - 13*w + 4],\ [839, 839, -w^5 + 3*w^4 + w^3 - 8*w^2 + 3*w - 1],\ [841, 29, -2*w^4 + 5*w^3 + 5*w^2 - 13*w + 1],\ [853, 853, -2*w^4 + 10*w^2 + 3*w - 6],\ [853, 853, 2*w^5 - 4*w^4 - 8*w^3 + 13*w^2 + 8*w - 6],\ [853, 853, -2*w^4 + w^3 + 9*w^2 - 2*w - 4],\ [853, 853, -2*w^5 + 3*w^4 + 8*w^3 - 8*w^2 - 7*w + 1],\ [881, 881, w^5 - 5*w^3 - 2*w^2 + 5*w - 1],\ [881, 881, -w^5 + 3*w^4 + 4*w^3 - 12*w^2 - 5*w + 5],\ [883, 883, 3*w^4 - 4*w^3 - 11*w^2 + 10*w + 5],\ [883, 883, -3*w^4 + 3*w^3 + 12*w^2 - 5*w - 7],\ [883, 883, w^5 - 2*w^4 - 4*w^3 + 7*w^2 + 5*w - 2],\ [883, 883, w^5 - w^4 - 6*w^3 + 4*w^2 + 10*w - 6],\ [911, 911, -w^5 + w^4 + 6*w^3 - 5*w^2 - 7*w + 4],\ [911, 911, 2*w^5 - 4*w^4 - 6*w^3 + 10*w^2 + 2*w + 1],\ [911, 911, w^5 - w^4 - 3*w^3 + 2*w^2 - 2*w - 3],\ [911, 911, -w^5 + 3*w^4 + 4*w^3 - 10*w^2 - 5*w + 2],\ [911, 911, w^5 - 2*w^4 - w^3 + 2*w^2 - 5*w + 6],\ [911, 911, -2*w^5 + 4*w^4 + 8*w^3 - 12*w^2 - 8*w + 3],\ [937, 937, w^4 - 2*w^3 - 5*w^2 + 7*w + 5],\ [937, 937, w^5 + w^4 - 7*w^3 - 5*w^2 + 9*w + 4],\ [937, 937, 3*w^4 - 4*w^3 - 9*w^2 + 7*w - 1],\ [937, 937, -w^5 + 4*w^4 + w^3 - 14*w^2 + 4*w + 8],\ [953, 953, w^5 - 3*w^4 - w^3 + 7*w^2 - 5*w + 4],\ [953, 953, w^5 - 2*w^4 - 3*w^3 + 4*w^2 + 2],\ [967, 967, w^5 - 6*w^4 + 2*w^3 + 20*w^2 - 12*w - 6],\ [967, 967, w^5 - 4*w^4 - 2*w^3 + 14*w^2 + w - 5],\ [967, 967, -w^5 + 8*w^3 - 2*w^2 - 14*w + 10],\ [967, 967, w^5 - 3*w^4 + 7*w^2 - 9*w],\ [967, 967, -w^5 + 6*w^3 + 2*w^2 - 9*w - 2],\ [967, 967, -w^5 + w^4 + 5*w^3 - 2*w^2 - 4*w - 3],\ [1009, 1009, w^5 - 3*w^4 - 2*w^3 + 9*w^2 - 2*w - 4],\ [1009, 1009, -2*w^5 + 4*w^4 + 6*w^3 - 11*w^2 - w + 2],\ [1009, 1009, 2*w^5 - 3*w^4 - 9*w^3 + 9*w^2 + 9*w - 3],\ [1009, 1009, -w^5 + w^4 + 4*w^3 - w^2 - 2*w - 5],\ [1021, 1021, w^5 - w^4 - 5*w^3 + 5*w^2 + 6*w - 10],\ [1021, 1021, w^5 - 8*w^3 + 3*w^2 + 14*w - 6],\ [1021, 1021, 2*w^5 - 4*w^4 - 7*w^3 + 12*w^2 + 4*w - 4],\ [1021, 1021, -2*w^5 + 3*w^4 + 8*w^3 - 8*w^2 - 6*w + 1],\ [1051, 1051, w^5 - w^4 - 3*w^3 + w^2 - w - 2],\ [1051, 1051, 3*w^4 - 6*w^3 - 8*w^2 + 16*w - 3],\ [1091, 1091, -w^5 + 4*w^4 - 13*w^2 + 6*w + 5],\ [1091, 1091, -w^5 - w^4 + 9*w^3 + 3*w^2 - 16*w + 1],\ [1093, 1093, w^2 + w - 5],\ [1093, 1093, w^4 - 3*w^3 - w^2 + 9*w - 2],\ [1093, 1093, 2*w^5 - 5*w^4 - 5*w^3 + 14*w^2 - w - 2],\ [1093, 1093, 2*w^5 - 3*w^4 - 8*w^3 + 7*w^2 + 6*w + 2],\ [1163, 1163, -w^3 + 3*w^2 + 2*w - 5],\ [1163, 1163, 2*w^4 - 2*w^3 - 6*w^2 + 3*w - 3],\ [1217, 1217, w^5 + w^4 - 6*w^3 - 6*w^2 + 3*w + 5],\ [1217, 1217, w^5 - w^4 - 5*w^3 + w^2 + 9*w],\ [1231, 1231, -w^5 + 4*w^4 - w^3 - 10*w^2 + 10*w - 1],\ [1231, 1231, -w^5 + w^4 + 5*w^3 - w^2 - 5*w - 5],\ [1259, 1259, w^5 - w^4 - 5*w^3 + 4*w^2 + 6*w - 7],\ [1259, 1259, w^5 - w^4 - 6*w^3 + 5*w^2 + 9*w - 4],\ [1289, 1289, w^5 + w^4 - 11*w^3 + 23*w - 7],\ [1289, 1289, -w^5 + w^4 + 5*w^3 - 2*w^2 - 8*w - 2],\ [1289, 1289, 2*w^5 - 13*w^3 - w^2 + 19*w - 1],\ [1289, 1289, -2*w^5 + 4*w^4 + 10*w^3 - 15*w^2 - 14*w + 9],\ [1301, 1301, -2*w^5 + 3*w^4 + 8*w^3 - 9*w^2 - 6*w + 4],\ [1301, 1301, -w^5 - w^4 + 11*w^3 + 2*w^2 - 23*w + 3],\ [1301, 1301, -2*w^5 + w^4 + 11*w^3 - 2*w^2 - 12*w + 1],\ [1301, 1301, -2*w^5 + 3*w^4 + 9*w^3 - 10*w^2 - 9*w + 6],\ [1303, 1303, w^5 - 4*w^4 - w^3 + 13*w^2 - 2*w - 2],\ [1303, 1303, w^5 - 4*w^4 - w^3 + 14*w^2 - 3*w - 4],\ [1331, 11, -w^4 + 4*w^3 - 2*w^2 - 10*w + 8],\ [1331, 11, w^5 - 5*w^4 + w^3 + 16*w^2 - 10*w - 4],\ [1399, 1399, -w^5 + 6*w^4 - w^3 - 21*w^2 + 8*w + 8],\ [1399, 1399, 2*w^4 - 4*w^3 - 6*w^2 + 13*w - 4],\ [1399, 1399, 2*w^5 - 4*w^4 - 9*w^3 + 15*w^2 + 9*w - 7],\ [1399, 1399, w^5 - 7*w^3 + w^2 + 10*w],\ [1427, 1427, 2*w^5 - 5*w^4 - 7*w^3 + 17*w^2 + 6*w - 6],\ [1427, 1427, -w^5 + 3*w^4 + 2*w^3 - 12*w^2 + 3*w + 9],\ [1471, 1471, w^5 - 5*w^3 - w^2 + 4*w - 3],\ [1471, 1471, w^5 - w^4 - 6*w^3 + 3*w^2 + 7*w - 1],\ [1471, 1471, -w^5 + 3*w^4 - w^3 - 6*w^2 + 12*w - 4],\ [1471, 1471, w^5 - 2*w^4 - 5*w^3 + 9*w^2 + 6*w - 6],\ [1483, 1483, -2*w^4 + 2*w^3 + 9*w^2 - 6*w - 6],\ [1483, 1483, -w^5 + 4*w^4 - 2*w^3 - 8*w^2 + 12*w - 7],\ [1483, 1483, -w^5 + 5*w^4 + w^3 - 18*w^2 + w + 3],\ [1483, 1483, w^5 - 4*w^4 - 4*w^3 + 17*w^2 + 7*w - 8],\ [1499, 1499, -2*w^5 + 6*w^4 + 6*w^3 - 21*w^2 - 2*w + 8],\ [1499, 1499, 2*w^5 - 5*w^4 - 7*w^3 + 17*w^2 + 5*w - 7],\ [1567, 1567, 2*w^5 - 7*w^4 - w^3 + 19*w^2 - 11*w],\ [1567, 1567, -w^5 + w^4 + 4*w^3 - 2*w^2 - w - 3],\ [1567, 1567, w^5 - 2*w^4 - 3*w^3 + 6*w^2 - w - 5],\ [1567, 1567, -w^5 + 2*w^4 + 5*w^3 - 7*w^2 - 5*w - 2],\ [1583, 1583, 2*w^4 - 6*w^3 - 4*w^2 + 17*w - 3],\ [1583, 1583, -2*w^4 + 5*w^3 + 5*w^2 - 12*w + 1],\ [1609, 1609, -4*w^4 + 5*w^3 + 13*w^2 - 11*w + 2],\ [1609, 1609, 2*w^5 - 2*w^4 - 10*w^3 + 3*w^2 + 12*w + 5],\ [1637, 1637, -w^5 - 2*w^4 + 10*w^3 + 6*w^2 - 18*w],\ [1637, 1637, -w^4 + 6*w^2 - 10],\ [1637, 1637, -2*w^4 + w^3 + 8*w^2 - 7],\ [1637, 1637, -w^5 + 4*w^4 - 3*w^3 - 11*w^2 + 17*w + 2],\ [1667, 1667, 2*w^4 - w^3 - 10*w^2 + 2*w + 7],\ [1667, 1667, 3*w^4 - 2*w^3 - 12*w^2 + 2*w + 4],\ [1667, 1667, w^5 - 7*w^3 + w^2 + 9*w - 2],\ [1667, 1667, w^5 - 2*w^4 - 2*w^3 + 6*w^2 - 4*w - 5],\ [1681, 41, -w^4 + 4*w^3 + w^2 - 11*w + 4],\ [1681, 41, -3*w^4 + 6*w^3 + 9*w^2 - 15*w + 1],\ [1693, 1693, w^5 - 3*w^4 - 2*w^3 + 11*w^2 - w - 7],\ [1693, 1693, -w^4 - 3*w^3 + 7*w^2 + 12*w - 7],\ [1693, 1693, -w^4 + 5*w^2 - w + 1],\ [1693, 1693, -w^5 - w^4 + 9*w^3 + 2*w^2 - 16*w + 4],\ [1709, 1709, -w^5 + 7*w^3 - 12*w + 3],\ [1709, 1709, -w^5 + 3*w^4 + 3*w^3 - 11*w^2 - 3*w + 5],\ [1723, 1723, 2*w^5 - w^4 - 10*w^3 + 11*w + 2],\ [1723, 1723, 2*w^5 - 6*w^4 - 6*w^3 + 21*w^2 + 4*w - 9],\ [1723, 1723, -w^5 + 2*w^4 + 6*w^3 - 11*w^2 - 8*w + 10],\ [1723, 1723, -w^5 - w^4 + 5*w^3 + 5*w^2 - 2*w + 2],\ [1777, 1777, w^5 - 2*w^4 - 5*w^3 + 6*w^2 + 8*w - 2],\ [1777, 1777, w^5 - 5*w^4 + 17*w^2 - 5*w - 4],\ [1777, 1777, -2*w^5 + 6*w^4 + 5*w^3 - 20*w^2 + 9],\ [1777, 1777, -w^5 + 6*w^4 - 3*w^3 - 17*w^2 + 11*w],\ [1847, 1847, 3*w^5 - 4*w^4 - 14*w^3 + 11*w^2 + 14*w - 3],\ [1847, 1847, 2*w^3 - 3*w^2 - 8*w + 3],\ [1849, 43, 2*w^4 - 5*w^3 - 5*w^2 + 13*w - 2],\ [1849, 43, -w^4 - w^3 + 6*w^2 + 4*w - 3],\ [1861, 1861, 3*w^4 - 5*w^3 - 8*w^2 + 13*w - 6],\ [1861, 1861, 3*w^4 - w^3 - 12*w^2 - w + 2],\ [1861, 1861, -w^5 + w^4 + 6*w^3 - 4*w^2 - 11*w + 6],\ [1861, 1861, -w^5 + 4*w^4 + 3*w^3 - 13*w^2 - 4*w + 2],\ [1933, 1933, -2*w^5 + 4*w^4 + 7*w^3 - 11*w^2 - 4*w + 2],\ [1933, 1933, -2*w^5 + 4*w^4 + 6*w^3 - 10*w^2 - w - 2],\ [1933, 1933, 2*w^4 - 3*w^3 - 7*w^2 + 8*w + 6],\ [1933, 1933, -2*w^4 + 2*w^3 + 8*w^2 - 3*w - 8],\ [1973, 1973, w^4 - 4*w^3 + 12*w - 4],\ [1973, 1973, w^3 - 2*w - 3],\ [1987, 1987, -3*w^5 + 6*w^4 + 10*w^3 - 19*w^2 - 3*w + 9],\ [1987, 1987, -w^4 + 4*w^3 + w^2 - 14*w + 1]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^2 + 4*x + 2 K. = NumberField(heckePol) hecke_eigenvalues_array = [e, e + 4, -4, -4, 4, -e, -e - 4, 1, e, e + 4, -1, -8*e - 18, 8*e + 14, -9*e - 20, -9*e - 16, 2, 2, -4, 4, e + 16, e - 12, -6*e, 3*e + 20, -6*e - 24, 4*e - 8, 3*e - 8, -4*e - 24, 8*e + 4, -8*e - 28, 4*e + 8, -4*e - 8, 4, -10, 14*e + 28, 14*e + 28, -10*e - 20, -10*e - 20, -9*e - 28, -9*e - 8, 4*e - 6, -4*e - 22, 7*e + 16, 7*e + 12, 16*e + 30, -16*e - 34, 12*e + 24, 11*e + 28, 8*e + 12, -12*e - 24, -8*e - 20, 11*e + 16, 12*e + 20, -12*e - 28, 12*e + 14, -12*e - 34, -6*e - 4, 12*e + 36, 8*e + 26, -12*e - 12, -8*e - 6, -6*e - 20, -14*e - 16, -14, -14, -14*e - 40, -4*e - 2, 4*e + 14, 12*e + 42, -12*e - 6, 5*e + 16, 5*e + 4, 20, 20, -3*e - 20, -3*e + 8, 4*e + 12, -4*e - 4, -22*e - 36, -22*e - 52, 3*e + 16, 3*e - 4, -e - 12, -7*e + 12, -e + 8, -7*e - 40, 6*e + 8, 6*e + 16, e + 4, e, 6, 12*e + 8, -12*e - 40, 6, -e + 28, -12, -e - 32, -12, -5*e + 4, -5*e - 24, 20*e + 26, -e + 8, -20*e - 54, -e - 12, 8*e + 10, -24*e - 60, 24*e + 36, -8*e - 22, -16*e - 32, -16*e - 32, 14*e + 56, 4*e - 28, 14*e, -4*e - 44, -12*e - 18, -21*e - 44, 12*e + 30, -21*e - 40, -20*e - 52, -20*e - 28, -8*e - 16, 8*e + 2, 8*e + 16, -8*e - 30, 32, -5*e + 8, -6*e + 24, -5*e - 28, -6*e - 48, 12*e + 14, -12*e - 34, -3*e + 12, -3*e - 24, -15*e - 64, 8*e - 2, -8*e - 34, -15*e + 4, 3*e - 8, 3*e + 20, -12*e - 16, 14*e + 32, 14*e + 24, -12*e - 32, 12*e + 24, 28*e + 56, -12*e - 24, -16*e - 42, 16*e + 22, 28*e + 56, -46, -46, 4, -4*e - 26, -8*e + 14, 4*e - 10, 8*e + 46, 12*e + 60, 12*e - 12, -36*e - 68, 36*e + 76, 8*e + 56, 8*e - 24, -8*e + 14, 22*e + 56, 8*e + 46, 9*e + 60, 9*e - 24, 22*e + 32, 24*e + 68, -12, 24*e + 28, 12, 8*e + 44, -8*e + 12, -4*e + 16, 8, -4*e - 32, -8*e + 8, 8, 8*e + 40, -e + 16, -10, -10, -e - 20, 14*e + 12, 14*e + 44, 10, 10, -3*e - 24, -3*e + 12, 8*e + 56, 8*e - 24, 2, 2, 28*e + 68, 28*e + 44, -4*e - 16, -4*e, -e + 20, -e - 24, 20*e + 16, -20*e - 64, -21*e - 40, -21*e - 44, -8*e - 12, -8*e - 20, 30*e + 44, 30*e + 76, -48*e - 96, 24*e + 62, -24*e - 34, 48*e + 96, 30*e + 32, 30*e + 88, 7*e - 16, 7*e + 44, -8*e - 32, 21*e + 28, 8*e, 21*e + 56, 37*e + 80, 37*e + 68, 17*e, -24*e - 48, -24*e - 48, 17*e + 68, 4*e - 24, -4*e - 40, -6*e - 72, -6*e + 48, -14*e - 32, -14*e - 24, 20*e + 24, -24*e - 48, -24*e - 48, -20*e - 56, 12*e + 68, -12*e + 20, 11*e + 36, 11*e + 8, 11*e + 40, 24*e + 86, -24*e - 10, 11*e + 4, 32*e + 44, -32*e - 84, 4*e + 44, -4*e + 28, 14, -4, 3*e + 68, -4*e - 12, -4*e - 4, 3*e - 56, 15*e + 40, 15*e + 20, -24*e - 16, 24*e + 80, 4*e + 40, 4*e - 24, 16*e - 10, -16*e - 74, -17*e - 16, -17*e - 52, 2, 2, 6, -10, -11*e - 20, 4*e + 38, -4*e + 22, -11*e - 24, 19*e + 28, 19*e + 48, 25*e + 92, 25*e + 8, 24*e + 14, -24*e - 82, 4*e - 4, -4*e - 20] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([41, 41, -w^5 + w^4 + 4*w^3 - w^2 - 3*w - 1])] = -1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]