/* 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([11, 7, -6, -2, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([16, 2, 2]) primes_array = [ [5, 5, -w^2 + 2*w + 3],\ [9, 3, w^3 - 3*w^2 - 2*w + 9],\ [9, 3, -w^3 + 5*w + 5],\ [11, 11, w],\ [11, 11, w - 1],\ [16, 2, 2],\ [19, 19, -w^3 + 2*w^2 + 3*w - 2],\ [19, 19, w^3 - w^2 - 4*w + 2],\ [29, 29, w^3 - 4*w^2 - w + 10],\ [29, 29, -w^3 + 3*w^2 + w - 7],\ [41, 41, 3*w^2 - 2*w - 10],\ [49, 7, -2*w^2 + 3*w + 8],\ [49, 7, w^3 - 2*w^2 - 2*w + 5],\ [71, 71, -w - 3],\ [71, 71, w - 4],\ [79, 79, -w^3 + w^2 + 3*w + 3],\ [79, 79, -w^3 + 2*w^2 + 2*w - 6],\ [89, 89, w^3 - 3*w^2 - 3*w + 7],\ [89, 89, w^3 - 6*w - 2],\ [101, 101, 2*w^3 - 5*w^2 - 3*w + 9],\ [101, 101, -w^3 + 3*w^2 + 3*w - 12],\ [109, 109, -w^3 + 4*w^2 + w - 9],\ [109, 109, -w^3 + 2*w^2 + 4*w - 2],\ [121, 11, -3*w^2 + 3*w + 10],\ [131, 131, -2*w^3 + 5*w^2 + 4*w - 9],\ [131, 131, -w^3 + w^2 + 3*w + 5],\ [131, 131, -w^3 + 2*w^2 + 2*w - 8],\ [131, 131, -w^3 + 6*w^2 - 18],\ [149, 149, -w^3 - 2*w^2 + 7*w + 8],\ [149, 149, -2*w^3 + 6*w^2 + 5*w - 16],\ [151, 151, w^2 + w - 5],\ [151, 151, w^2 - 3*w - 3],\ [169, 13, w^3 - 4*w^2 - w + 6],\ [169, 13, 2*w^3 - 5*w^2 - 4*w + 6],\ [179, 179, w^3 - 4*w - 6],\ [179, 179, -w^3 + 3*w^2 + w - 9],\ [181, 181, w^3 + 3*w^2 - 7*w - 15],\ [181, 181, w^3 - 6*w^2 + 2*w + 18],\ [191, 191, 5*w^2 - 7*w - 15],\ [191, 191, -4*w^2 + 5*w + 12],\ [199, 199, -w^3 + 3*w^2 + 3*w - 6],\ [199, 199, w^3 - 6*w - 1],\ [211, 211, w^3 - 7*w - 4],\ [211, 211, -w^3 + 3*w^2 + 4*w - 10],\ [229, 229, -w^3 - w^2 + 5*w + 9],\ [229, 229, -w^3 + 4*w^2 - 12],\ [239, 239, 4*w^2 - 5*w - 17],\ [239, 239, 4*w^2 - 3*w - 18],\ [241, 241, -w^3 + 3*w^2 + w - 2],\ [241, 241, -w^3 + 3*w^2 + 4*w - 9],\ [241, 241, w^3 - 7*w - 3],\ [241, 241, -w^3 + 4*w - 1],\ [271, 271, -w - 4],\ [271, 271, w^3 - 3*w^2 - 2*w + 2],\ [271, 271, -2*w^3 + 6*w^2 + 2*w - 7],\ [271, 271, w - 5],\ [281, 281, w^3 - 2*w^2 - 3*w - 1],\ [281, 281, -w^3 + w^2 + 4*w - 5],\ [289, 17, -2*w^3 + w^2 + 9*w + 1],\ [289, 17, -2*w^3 + 5*w^2 + 5*w - 9],\ [311, 311, w^3 - w^2 - 6*w + 1],\ [311, 311, -w^3 + 2*w^2 + 5*w - 5],\ [331, 331, -3*w^3 + 5*w^2 + 10*w - 5],\ [331, 331, 3*w^3 - 2*w^2 - 12*w - 6],\ [349, 349, -2*w^3 + 2*w^2 + 10*w + 3],\ [349, 349, -2*w^3 - w^2 + 12*w + 10],\ [349, 349, 2*w^3 + w^2 - 10*w - 12],\ [349, 349, -2*w^3 + 4*w^2 + 8*w - 13],\ [361, 19, -4*w^2 + 4*w + 13],\ [379, 379, w^3 + w^2 - 5*w - 10],\ [379, 379, -3*w^3 + 4*w^2 + 12*w - 7],\ [379, 379, w^3 + 4*w^2 - 8*w - 21],\ [379, 379, w^3 - 3*w - 6],\ [389, 389, 2*w^3 - 6*w^2 - 4*w + 13],\ [389, 389, -5*w^2 + 7*w + 13],\ [389, 389, 2*w^3 - 7*w^2 - 5*w + 20],\ [389, 389, 2*w^3 - 10*w - 5],\ [419, 419, -w^3 + 5*w^2 + w - 13],\ [419, 419, 4*w^2 - 7*w - 14],\ [419, 419, 2*w^3 - 3*w^2 - 5*w - 1],\ [419, 419, -w^3 - 2*w^2 + 8*w + 8],\ [421, 421, -2*w^2 + 5*w + 7],\ [421, 421, w^3 - 6*w^2 + 5*w + 13],\ [439, 439, 2*w^3 - 2*w^2 - 9*w + 3],\ [439, 439, w^3 + w^2 - 7*w - 4],\ [449, 449, 3*w^3 - 6*w^2 - 9*w + 8],\ [449, 449, 5*w^2 - 8*w - 17],\ [449, 449, 2*w^3 - w^2 - 8*w - 7],\ [449, 449, -3*w^3 + 3*w^2 + 12*w - 4],\ [479, 479, -2*w^3 + 3*w^2 + 5*w + 3],\ [479, 479, 3*w^3 - 7*w^2 - 9*w + 15],\ [499, 499, -w^3 + 4*w^2 + 4*w - 9],\ [499, 499, w^3 + w^2 - 9*w - 2],\ [509, 509, 2*w^3 + 2*w^2 - 11*w - 16],\ [509, 509, -2*w^3 + 8*w^2 + w - 23],\ [521, 521, -3*w^3 + w^2 + 13*w + 6],\ [521, 521, w^3 + 4*w^2 - 10*w - 16],\ [541, 541, 3*w^3 - 2*w^2 - 13*w - 3],\ [541, 541, -w^3 + 7*w^2 - w - 25],\ [541, 541, w^3 + 4*w^2 - 10*w - 20],\ [541, 541, -3*w^3 + 7*w^2 + 8*w - 15],\ [571, 571, w^3 + 4*w^2 - 9*w - 15],\ [571, 571, w^3 - 7*w^2 + 2*w + 19],\ [601, 601, -3*w^3 + 2*w^2 + 13*w + 5],\ [601, 601, 3*w^3 - 7*w^2 - 8*w + 17],\ [619, 619, w^3 - 3*w - 7],\ [619, 619, w^3 - 5*w^2 + 3*w + 13],\ [619, 619, 3*w^3 - 4*w^2 - 9*w - 2],\ [619, 619, -w^3 + 3*w^2 - 9],\ [641, 641, 6*w^2 - 4*w - 23],\ [641, 641, -3*w^3 + w^2 + 12*w + 5],\ [659, 659, -2*w^3 + 6*w^2 + 3*w - 17],\ [659, 659, -2*w^3 + 9*w + 10],\ [661, 661, 3*w^3 - 6*w^2 - 9*w + 14],\ [661, 661, 5*w^2 - 6*w - 18],\ [661, 661, -5*w^2 + 4*w + 19],\ [661, 661, 3*w^3 - 3*w^2 - 12*w - 2],\ [691, 691, -3*w^3 + 5*w^2 + 10*w - 6],\ [691, 691, -3*w^3 + 4*w^2 + 11*w - 6],\ [701, 701, -w^3 - 4*w^2 + 9*w + 20],\ [701, 701, -w^3 + 4*w^2 + 3*w - 19],\ [701, 701, -w^3 + 4*w^2 + 4*w - 15],\ [701, 701, w^3 - 7*w^2 + 2*w + 24],\ [709, 709, 3*w^2 - 5*w - 13],\ [709, 709, 3*w^2 - w - 15],\ [719, 719, -w^3 + 3*w^2 + 4*w - 4],\ [719, 719, -w^3 + 7*w - 2],\ [739, 739, -w^3 + 4*w^2 + 3*w - 9],\ [739, 739, -2*w^3 - w^2 + 11*w + 8],\ [739, 739, 2*w^3 - 7*w^2 - 3*w + 16],\ [739, 739, -w^3 - w^2 + 8*w + 3],\ [751, 751, 3*w^3 - 5*w^2 - 10*w + 10],\ [751, 751, 2*w^2 - 5*w - 5],\ [761, 761, 3*w^3 - 3*w^2 - 11*w + 3],\ [761, 761, w^3 - 2*w^2 - 3*w - 2],\ [761, 761, -w^3 + w^2 + 4*w - 6],\ [761, 761, -3*w^3 + 6*w^2 + 8*w - 8],\ [769, 769, -w^3 - w^2 + 7*w + 1],\ [769, 769, -2*w^3 + 6*w^2 + 6*w - 15],\ [769, 769, -2*w^3 + 5*w^2 + 5*w - 7],\ [769, 769, 2*w^3 + w^2 - 12*w - 9],\ [809, 809, -2*w^3 + 8*w^2 + 4*w - 23],\ [809, 809, -3*w^3 + 11*w^2 + 2*w - 28],\ [829, 829, -3*w^3 + 7*w^2 + 7*w - 9],\ [829, 829, w^3 + 2*w^2 - 6*w - 15],\ [829, 829, -w^3 + 5*w^2 - w - 18],\ [829, 829, -3*w^3 + 2*w^2 + 12*w - 2],\ [839, 839, 2*w^3 - 9*w^2 + 20],\ [839, 839, -3*w^3 + 10*w^2 + 5*w - 30],\ [841, 29, 5*w^2 - 5*w - 16],\ [881, 881, -w^3 - 4*w^2 + 8*w + 18],\ [881, 881, 2*w^3 - 7*w^2 - w + 9],\ [881, 881, 2*w^3 - 6*w^2 - 3*w + 6],\ [881, 881, -w^3 + 7*w^2 - 3*w - 21],\ [911, 911, -w^3 - 4*w^2 + 8*w + 16],\ [911, 911, -w^3 - w^2 + 9*w + 5],\ [911, 911, -w^3 + 4*w^2 + 4*w - 12],\ [911, 911, w^3 - 7*w^2 + 3*w + 19],\ [919, 919, -4*w^3 + 7*w^2 + 14*w - 9],\ [919, 919, 4*w^3 - 5*w^2 - 16*w + 8],\ [929, 929, 3*w^2 - w - 16],\ [929, 929, 3*w^2 - 5*w - 14],\ [941, 941, -w^3 - w^2 + 9*w + 7],\ [941, 941, -3*w^3 + 7*w^2 + 7*w - 15],\ [941, 941, 3*w^3 - 2*w^2 - 12*w - 4],\ [941, 941, w^3 + 4*w^2 - 13*w - 16],\ [961, 31, -2*w^2 + 2*w + 13],\ [961, 31, 5*w^2 - 5*w - 18],\ [971, 971, w^3 + 4*w^2 - 9*w - 19],\ [971, 971, w^3 - 7*w^2 + 2*w + 23],\ [991, 991, 3*w^3 - 7*w^2 - 7*w + 14],\ [991, 991, -3*w^3 + 2*w^2 + 12*w + 3]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^6 - 34*x^4 + 296*x^2 - 148 K. = NumberField(heckePol) hecke_eigenvalues_array = [-5/29*e^4 + 84/29*e^2 - 12/29, e, e, -1/29*e^4 + 11/29*e^2 + 102/29, -1/29*e^4 + 11/29*e^2 + 102/29, -1, 1/58*e^5 + 9/29*e^3 - 283/29*e, 1/58*e^5 + 9/29*e^3 - 283/29*e, 7/58*e^5 - 82/29*e^3 + 368/29*e, 7/58*e^5 - 82/29*e^3 + 368/29*e, 14/29*e^4 - 270/29*e^2 + 486/29, -2/29*e^5 + 51/29*e^3 - 289/29*e, -2/29*e^5 + 51/29*e^3 - 289/29*e, -4/29*e^4 + 44/29*e^2 + 118/29, -4/29*e^4 + 44/29*e^2 + 118/29, 1/29*e^5 - 40/29*e^3 + 420/29*e, 1/29*e^5 - 40/29*e^3 + 420/29*e, -1/29*e^5 + 11/29*e^3 + 15/29*e, -1/29*e^5 + 11/29*e^3 + 15/29*e, -1/29*e^4 + 40/29*e^2 - 188/29, -1/29*e^4 + 40/29*e^2 - 188/29, 9/58*e^5 - 122/29*e^3 + 788/29*e, 9/58*e^5 - 122/29*e^3 + 788/29*e, 1/29*e^4 + 18/29*e^2 - 102/29, -11/29*e^4 + 237/29*e^2 - 618/29, 1/29*e^4 - 40/29*e^2 + 72/29, 1/29*e^4 - 40/29*e^2 + 72/29, -11/29*e^4 + 237/29*e^2 - 618/29, -1/58*e^5 - 38/29*e^3 + 718/29*e, -1/58*e^5 - 38/29*e^3 + 718/29*e, -10/29*e^4 + 226/29*e^2 - 778/29, -10/29*e^4 + 226/29*e^2 - 778/29, -2/29*e^5 + 80/29*e^3 - 869/29*e, -2/29*e^5 + 80/29*e^3 - 869/29*e, 1/58*e^5 - 20/29*e^3 + 123/29*e, 1/58*e^5 - 20/29*e^3 + 123/29*e, -3/29*e^4 + 62/29*e^2 - 332/29, -3/29*e^4 + 62/29*e^2 - 332/29, -16/29*e^4 + 292/29*e^2 - 108/29, -16/29*e^4 + 292/29*e^2 - 108/29, 4/29*e^5 - 44/29*e^3 - 408/29*e, 4/29*e^5 - 44/29*e^3 - 408/29*e, 25/29*e^4 - 420/29*e^2 + 176/29, 25/29*e^4 - 420/29*e^2 + 176/29, -21/58*e^5 + 246/29*e^3 - 1162/29*e, -21/58*e^5 + 246/29*e^3 - 1162/29*e, -2/29*e^5 + 22/29*e^3 + 204/29*e, -2/29*e^5 + 22/29*e^3 + 204/29*e, 12/29*e^4 - 277/29*e^2 + 806/29, -13/29*e^4 + 259/29*e^2 - 878/29, -13/29*e^4 + 259/29*e^2 - 878/29, 12/29*e^4 - 277/29*e^2 + 806/29, -2*e^2 + 30, -18/29*e^4 + 314/29*e^2 - 252/29, -18/29*e^4 + 314/29*e^2 - 252/29, -2*e^2 + 30, -17/29*e^4 + 303/29*e^2 - 702/29, -17/29*e^4 + 303/29*e^2 - 702/29, -1/29*e^5 + 40/29*e^3 - 333/29*e, -1/29*e^5 + 40/29*e^3 - 333/29*e, 26/29*e^4 - 460/29*e^2 + 306/29, 26/29*e^4 - 460/29*e^2 + 306/29, -15/29*e^4 + 252/29*e^2 + 196/29, -15/29*e^4 + 252/29*e^2 + 196/29, 5/58*e^5 - 42/29*e^3 + 122/29*e, -19/58*e^5 + 148/29*e^3 + 244/29*e, -19/58*e^5 + 148/29*e^3 + 244/29*e, 5/58*e^5 - 42/29*e^3 + 122/29*e, -24/29*e^4 + 380/29*e^2 + 882/29, 7/58*e^5 - 24/29*e^3 - 473/29*e, -1/58*e^5 - 9/29*e^3 + 457/29*e, -1/58*e^5 - 9/29*e^3 + 457/29*e, 7/58*e^5 - 24/29*e^3 - 473/29*e, -17/58*e^5 + 224/29*e^3 - 1192/29*e, -17/58*e^5 + 166/29*e^3 - 322/29*e, -17/58*e^5 + 166/29*e^3 - 322/29*e, -17/58*e^5 + 224/29*e^3 - 1192/29*e, -13/58*e^5 + 202/29*e^3 - 1599/29*e, 11/58*e^5 - 104/29*e^3 + 19/29*e, 11/58*e^5 - 104/29*e^3 + 19/29*e, -13/58*e^5 + 202/29*e^3 - 1599/29*e, -3/29*e^4 + 4/29*e^2 + 596/29, -3/29*e^4 + 4/29*e^2 + 596/29, 7/29*e^5 - 164/29*e^3 + 852/29*e, 7/29*e^5 - 164/29*e^3 + 852/29*e, 5/29*e^5 - 113/29*e^3 + 447/29*e, -2/29*e^5 - 65/29*e^3 + 1625/29*e, -2/29*e^5 - 65/29*e^3 + 1625/29*e, 5/29*e^5 - 113/29*e^3 + 447/29*e, 2/29*e^5 - 80/29*e^3 + 782/29*e, 2/29*e^5 - 80/29*e^3 + 782/29*e, 23/58*e^5 - 286/29*e^3 + 1495/29*e, 23/58*e^5 - 286/29*e^3 + 1495/29*e, -3/58*e^5 + 2/29*e^3 + 588/29*e, -3/58*e^5 + 2/29*e^3 + 588/29*e, 13/29*e^4 - 201/29*e^2 - 514/29, 13/29*e^4 - 201/29*e^2 - 514/29, 23/29*e^4 - 398/29*e^2 + 380/29, -33/29*e^4 + 624/29*e^2 - 752/29, -33/29*e^4 + 624/29*e^2 - 752/29, 23/29*e^4 - 398/29*e^2 + 380/29, -7/29*e^4 + 193/29*e^2 - 678/29, -7/29*e^4 + 193/29*e^2 - 678/29, 47/29*e^4 - 836/29*e^2 + 774/29, 47/29*e^4 - 836/29*e^2 + 774/29, -15/58*e^5 + 97/29*e^3 + 475/29*e, -21/58*e^5 + 188/29*e^3 - 147/29*e, -21/58*e^5 + 188/29*e^3 - 147/29*e, -15/58*e^5 + 97/29*e^3 + 475/29*e, 19/29*e^4 - 412/29*e^2 + 1194/29, 19/29*e^4 - 412/29*e^2 + 1194/29, 11/58*e^5 - 75/29*e^3 - 213/29*e, 11/58*e^5 - 75/29*e^3 - 213/29*e, -20/29*e^4 + 278/29*e^2 + 590/29, 13/29*e^4 - 288/29*e^2 + 1400/29, 13/29*e^4 - 288/29*e^2 + 1400/29, -20/29*e^4 + 278/29*e^2 + 590/29, -33/29*e^4 + 537/29*e^2 + 466/29, -33/29*e^4 + 537/29*e^2 + 466/29, 46/29*e^4 - 854/29*e^2 + 1398/29, 36/29*e^4 - 570/29*e^2 - 714/29, 36/29*e^4 - 570/29*e^2 - 714/29, 46/29*e^4 - 854/29*e^2 + 1398/29, 7/58*e^5 - 82/29*e^3 + 368/29*e, 7/58*e^5 - 82/29*e^3 + 368/29*e, 1/29*e^5 + 76/29*e^3 - 1436/29*e, 1/29*e^5 + 76/29*e^3 - 1436/29*e, -17/58*e^5 + 224/29*e^3 - 1163/29*e, -1/58*e^5 + 78/29*e^3 - 1283/29*e, -1/58*e^5 + 78/29*e^3 - 1283/29*e, -17/58*e^5 + 224/29*e^3 - 1163/29*e, -32/29*e^4 + 526/29*e^2 + 480/29, -32/29*e^4 + 526/29*e^2 + 480/29, e^4 - 17*e^2 + 10, -30/29*e^4 + 562/29*e^2 - 594/29, -30/29*e^4 + 562/29*e^2 - 594/29, e^4 - 17*e^2 + 10, -4/29*e^5 + 160/29*e^3 - 1477/29*e, 5/29*e^5 - 55/29*e^3 - 481/29*e, 5/29*e^5 - 55/29*e^3 - 481/29*e, -4/29*e^5 + 160/29*e^3 - 1477/29*e, 7/29*e^5 - 222/29*e^3 + 1809/29*e, 7/29*e^5 - 222/29*e^3 + 1809/29*e, -3/58*e^5 + 2/29*e^3 + 530/29*e, -23/58*e^5 + 286/29*e^3 - 1582/29*e, -23/58*e^5 + 286/29*e^3 - 1582/29*e, -3/58*e^5 + 2/29*e^3 + 530/29*e, -1/29*e^5 + 40/29*e^3 - 710/29*e, -1/29*e^5 + 40/29*e^3 - 710/29*e, -5/29*e^4 + 84/29*e^2 + 1554/29, 22/29*e^4 - 445/29*e^2 + 18/29, -e^4 + 14*e^2 + 30, -e^4 + 14*e^2 + 30, 22/29*e^4 - 445/29*e^2 + 18/29, -16/29*e^4 + 350/29*e^2 - 804/29, 72/29*e^4 - 1256/29*e^2 + 834/29, 72/29*e^4 - 1256/29*e^2 + 834/29, -16/29*e^4 + 350/29*e^2 - 804/29, -13/29*e^5 + 288/29*e^3 - 1226/29*e, -13/29*e^5 + 288/29*e^3 - 1226/29*e, 8/29*e^5 - 117/29*e^3 - 323/29*e, 8/29*e^5 - 117/29*e^3 - 323/29*e, 27/29*e^4 - 442/29*e^2 - 840/29, -14/29*e^4 + 328/29*e^2 - 1414/29, -14/29*e^4 + 328/29*e^2 - 1414/29, 27/29*e^4 - 442/29*e^2 - 840/29, -4/29*e^4 + 131/29*e^2 + 466/29, -32/29*e^4 + 555/29*e^2 - 274/29, -42/29*e^4 + 752/29*e^2 - 1284/29, -42/29*e^4 + 752/29*e^2 - 1284/29, 8/29*e^4 - 204/29*e^2 - 62/29, 8/29*e^4 - 204/29*e^2 - 62/29] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([16, 2, 2])] = 1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]