/* 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([-5, -7, 0, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([17, 17, -w^2 + w + 3]) primes_array = [ [5, 5, w],\ [8, 2, 2],\ [11, 11, -w^2 + 2*w + 4],\ [11, 11, -w + 2],\ [11, 11, w - 1],\ [13, 13, -w^2 + w + 4],\ [17, 17, -w^2 + w + 8],\ [17, 17, -w^2 + w + 3],\ [19, 19, -w^2 + 6],\ [23, 23, -w^2 + 3],\ [25, 5, w^2 - 7],\ [27, 3, -3],\ [31, 31, w^2 - 2*w - 8],\ [37, 37, w^2 - 2*w - 6],\ [41, 41, -w - 4],\ [41, 41, w^2 - 2*w - 7],\ [47, 47, 2*w^2 - 3*w - 7],\ [53, 53, -w^2 + w + 9],\ [61, 61, 3*w^2 - 2*w - 17],\ [67, 67, 2*w - 1],\ [73, 73, 2*w^2 - 2*w - 9],\ [89, 89, 2*w^2 - 3*w - 11],\ [97, 97, 2*w^2 - w - 9],\ [97, 97, w^2 - 3*w - 6],\ [97, 97, 2*w - 3],\ [101, 101, 3*w^2 - 4*w - 13],\ [107, 107, 2*w^2 - 3*w - 14],\ [113, 113, 2*w^2 - 3*w - 12],\ [127, 127, -w^2 + w - 1],\ [127, 127, 2*w^2 - 2*w - 7],\ [127, 127, w^2 + 3*w - 1],\ [131, 131, w^2 - 3*w - 8],\ [137, 137, 2*w^2 - w - 8],\ [139, 139, 4*w^2 - 5*w - 19],\ [149, 149, w^2 + w - 7],\ [151, 151, w^2 - 4*w - 6],\ [157, 157, w^2 + w - 9],\ [163, 163, w^2 + w - 8],\ [169, 13, 2*w^2 - w - 7],\ [173, 173, w^2 - 5*w - 3],\ [179, 179, -w - 6],\ [191, 191, w^2 + 2*w - 4],\ [197, 197, 3*w - 1],\ [229, 229, 3*w^2 - 4*w - 18],\ [233, 233, -w^2 + w - 2],\ [233, 233, -3*w^2 + 4*w + 11],\ [233, 233, 3*w^2 - w - 14],\ [239, 239, 3*w^2 - 5*w - 14],\ [241, 241, w^2 - 4*w - 11],\ [257, 257, -2*w^2 + 2*w + 17],\ [263, 263, 3*w^2 - 3*w - 23],\ [269, 269, w^2 - 2*w - 12],\ [277, 277, w^2 - 4*w - 9],\ [281, 281, 4*w^2 - 2*w - 21],\ [283, 283, 3*w^2 - 3*w - 13],\ [293, 293, 3*w^2 - 4*w - 9],\ [331, 331, w^2 + 3*w - 3],\ [337, 337, w^2 + 2*w - 6],\ [343, 7, -7],\ [361, 19, w^2 - 5*w - 1],\ [367, 367, w^2 - 5*w - 8],\ [373, 373, w^2 + 2*w - 16],\ [373, 373, 2*w^2 + w - 11],\ [373, 373, 3*w^2 - 2*w - 13],\ [379, 379, -w^2 + w - 3],\ [383, 383, 2*w^2 - 6*w - 9],\ [401, 401, w^2 + 2*w - 7],\ [419, 419, 3*w^2 - 6*w - 13],\ [421, 421, 4*w^2 - 5*w - 26],\ [431, 431, 4*w - 1],\ [439, 439, w^2 - 5*w - 9],\ [439, 439, w^2 - 2*w - 13],\ [439, 439, 4*w^2 - 7*w - 17],\ [443, 443, 3*w^2 - 3*w - 8],\ [443, 443, w^2 - 13],\ [443, 443, 3*w^2 - w - 12],\ [449, 449, 4*w^2 - 6*w - 13],\ [457, 457, w^2 + 2*w - 11],\ [461, 461, -w - 8],\ [461, 461, -w^2 + 5*w - 1],\ [461, 461, 2*w^2 + w - 12],\ [463, 463, 3*w^2 + w - 9],\ [487, 487, 5*w^2 - 3*w - 27],\ [499, 499, 4*w^2 - w - 24],\ [499, 499, w^2 - 5*w - 11],\ [499, 499, 5*w^2 - 4*w - 27],\ [503, 503, 3*w^2 - 5*w - 22],\ [503, 503, 3*w^2 - 5*w - 17],\ [503, 503, 5*w^2 - 8*w - 22],\ [509, 509, -w^2 - 4],\ [529, 23, 4*w^2 - 5*w - 16],\ [541, 541, 2*w^2 + w - 17],\ [547, 547, -w^2 + 2*w - 4],\ [547, 547, 3*w^2 - 6*w - 14],\ [547, 547, 2*w^2 - 5*w - 17],\ [557, 557, 5*w^2 - 2*w - 31],\ [563, 563, 3*w^2 - 5*w - 18],\ [571, 571, -w^2 + w - 4],\ [571, 571, w^2 - 6*w - 2],\ [571, 571, 3*w^2 + w - 8],\ [577, 577, 2*w^2 - w - 19],\ [587, 587, 2*w^2 - 5*w - 13],\ [593, 593, 2*w^2 + w - 16],\ [599, 599, 3*w^2 - 19],\ [599, 599, 3*w^2 - 5*w - 19],\ [599, 599, 4*w^2 - w - 18],\ [607, 607, 4*w^2 - 2*w - 19],\ [617, 617, 4*w^2 - 3*w - 19],\ [619, 619, 4*w^2 - 7*w - 18],\ [643, 643, 3*w^2 + w - 7],\ [643, 643, 2*w^2 - 2*w - 19],\ [643, 643, 5*w^2 - 8*w - 16],\ [653, 653, 3*w^2 - w - 9],\ [653, 653, 2*w^2 - 6*w - 11],\ [653, 653, 3*w^2 - w - 7],\ [659, 659, w^2 - 2*w - 14],\ [661, 661, 5*w^2 - 6*w - 31],\ [661, 661, w^2 - 14],\ [661, 661, w - 9],\ [673, 673, -2*w^2 + 4*w - 1],\ [673, 673, 4*w^2 - w - 26],\ [673, 673, 6*w^2 - 5*w - 33],\ [677, 677, 3*w^2 - 8],\ [719, 719, 6*w^2 - 7*w - 29],\ [719, 719, 4*w^2 - 6*w - 29],\ [719, 719, 4*w^2 - 5*w - 14],\ [727, 727, 4*w^2 - 4*w - 31],\ [743, 743, w^2 - w - 14],\ [757, 757, 5*w^2 - 8*w - 23],\ [761, 761, w^2 - 6*w - 11],\ [761, 761, 4*w - 7],\ [761, 761, 4*w^2 - 3*w - 18],\ [773, 773, -5*w^2 + 10*w + 18],\ [787, 787, 5*w^2 - 7*w - 19],\ [797, 797, 4*w^2 - 5*w - 12],\ [797, 797, 3*w^2 + w - 16],\ [797, 797, 3*w^2 - 6*w - 16],\ [811, 811, w^2 + 3*w - 7],\ [821, 821, 4*w^2 - 6*w - 23],\ [829, 829, 2*w^2 + 2*w - 11],\ [829, 829, 2*w^2 - 7*w - 11],\ [829, 829, 5*w^2 - 5*w - 24],\ [853, 853, 2*w^2 - 6*w - 19],\ [857, 857, w^2 + 4*w - 4],\ [877, 877, -w^2 - w - 6],\ [881, 881, -4*w^2 + 4*w + 9],\ [883, 883, 2*w^2 - 6*w - 13],\ [907, 907, 3*w^2 - 6*w - 17],\ [911, 911, w^2 + 3*w - 8],\ [919, 919, 4*w^2 - 8*w - 17],\ [937, 937, 4*w^2 - 6*w - 27],\ [941, 941, 4*w^2 - 2*w - 17],\ [947, 947, 3*w^2 - 8*w - 13],\ [961, 31, 6*w^2 - 7*w - 39],\ [967, 967, 5*w - 2],\ [971, 971, 5*w^2 - 9*w - 21],\ [977, 977, 3*w^2 + w - 17],\ [983, 983, 2*w - 11],\ [991, 991, 6*w^2 - 7*w - 38],\ [991, 991, w^2 + 3*w - 9],\ [991, 991, w^2 + 3*w - 13],\ [997, 997, 4*w^2 - w - 16],\ [997, 997, 5*w - 12],\ [997, 997, 3*w^2 - 2*w - 26]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^5 + 4*x^4 - 6*x^3 - 34*x^2 - 23*x - 2 K. = NumberField(heckePol) hecke_eigenvalues_array = [e, -1/2*e^3 + 9/2*e, e^3 + e^2 - 9*e - 6, -e - 2, 1/2*e^4 + 3/2*e^3 - 9/2*e^2 - 25/2*e - 1, -e - 4, -1/2*e^4 - 3/2*e^3 + 7/2*e^2 + 23/2*e + 5, -1, e^4 + 3*e^3 - 8*e^2 - 26*e - 8, -1/2*e^4 - 5/2*e^3 + 7/2*e^2 + 45/2*e + 5, -3/2*e^4 - 9/2*e^3 + 23/2*e^2 + 75/2*e + 13, -1/2*e^4 - 3/2*e^3 + 9/2*e^2 + 25/2*e + 1, -e^2 + 4, 1/2*e^4 + 1/2*e^3 - 11/2*e^2 - 5/2*e + 9, -e^4 - 4*e^3 + 6*e^2 + 34*e + 14, e^4 + 4*e^3 - 7*e^2 - 32*e - 10, -e^4 - e^3 + 10*e^2 + 6*e - 12, e^2 - e - 4, 1/2*e^4 + 5/2*e^3 - 5/2*e^2 - 39/2*e - 13, 1/2*e^4 + 1/2*e^3 - 9/2*e^2 - 11/2*e - 7, -1/2*e^4 - 1/2*e^3 + 9/2*e^2 + 7/2*e - 3, e^4 + 2*e^3 - 10*e^2 - 16*e + 2, 2*e^4 + 5*e^3 - 18*e^2 - 43*e - 4, 5/2*e^4 + 15/2*e^3 - 39/2*e^2 - 127/2*e - 25, -2*e^4 - 6*e^3 + 16*e^2 + 51*e + 16, e^4 + e^3 - 7*e^2 - 4*e - 10, e^2 + 5*e - 10, e^4 + 2*e^3 - 9*e^2 - 14*e + 2, -e^4 - 2*e^3 + 8*e^2 + 16*e, 2*e^2 - e - 14, -3*e^4 - 8*e^3 + 25*e^2 + 68*e + 20, -1/2*e^4 - 1/2*e^3 + 5/2*e^2 - 1/2*e + 11, -1/2*e^4 - 5/2*e^3 + 13/2*e^2 + 47/2*e - 3, e^4 + 2*e^3 - 7*e^2 - 14*e - 20, e^4 + 6*e^3 - 7*e^2 - 55*e - 20, 2*e^4 + 6*e^3 - 16*e^2 - 52*e - 16, 1/2*e^4 + 9/2*e^3 + 1/2*e^2 - 77/2*e - 27, e^4 + 4*e^3 - 9*e^2 - 35*e - 2, e^3 + 2*e^2 - 9*e, 2*e^3 + 2*e^2 - 15*e - 16, -3*e^4 - 10*e^3 + 25*e^2 + 86*e + 20, -e^4 + 9*e^2 - 3*e - 6, e^4 + 2*e^3 - 9*e^2 - 16*e - 2, -e^4 - 4*e^3 + 7*e^2 + 36*e + 18, -e^4 - 6*e^3 + 5*e^2 + 54*e + 26, -3/2*e^4 - 13/2*e^3 + 19/2*e^2 + 107/2*e + 21, -3/2*e^4 - 9/2*e^3 + 25/2*e^2 + 85/2*e + 7, e^4 + 6*e^3 - 5*e^2 - 54*e - 32, 2*e^4 + 7*e^3 - 17*e^2 - 57*e - 8, -3*e^3 - 4*e^2 + 21*e + 24, 9/2*e^4 + 27/2*e^3 - 71/2*e^2 - 223/2*e - 35, -3/2*e^4 - 13/2*e^3 + 23/2*e^2 + 115/2*e + 13, 2*e^4 + 3*e^3 - 16*e^2 - 19*e - 4, 4*e^4 + 13*e^3 - 30*e^2 - 109*e - 40, -e^4 - 5*e^3 + 7*e^2 + 50*e + 12, -3/2*e^4 - 13/2*e^3 + 19/2*e^2 + 115/2*e + 17, -5*e^3 - 2*e^2 + 42*e, -2*e^4 - 7*e^3 + 16*e^2 + 65*e + 28, -1/2*e^4 + 7/2*e^3 + 13/2*e^2 - 61/2*e - 5, e^3 + e^2 - 12*e - 14, 3*e^3 + e^2 - 30*e, e^4 + 6*e^3 - 6*e^2 - 56*e - 34, -2*e^4 - 9*e^3 + 14*e^2 + 78*e + 30, -e^4 - 3*e^3 + 7*e^2 + 26*e + 18, -7/2*e^4 - 23/2*e^3 + 57/2*e^2 + 203/2*e + 39, e^4 + 5*e^3 - 8*e^2 - 53*e - 6, e^4 + 3*e^3 - 9*e^2 - 26*e - 18, -5*e^3 - 5*e^2 + 42*e + 28, e^4 + 6*e^3 - 7*e^2 - 48*e + 2, -1/2*e^4 - 3/2*e^3 - 1/2*e^2 + 15/2*e + 27, 2*e^4 + 2*e^3 - 23*e^2 - 19*e + 34, 2*e^4 + 2*e^3 - 17*e^2 - 14*e - 8, e^4 + 3*e^3 - 12*e^2 - 27*e + 14, e^4 + 2*e^3 - 9*e^2 - 21*e - 6, -5*e^4 - 16*e^3 + 38*e^2 + 141*e + 50, -7/2*e^4 - 21/2*e^3 + 55/2*e^2 + 187/2*e + 43, 2*e^4 + e^3 - 20*e^2 - 7*e + 12, -3*e^4 - 14*e^3 + 17*e^2 + 119*e + 44, -e^4 - 7*e^3 + 6*e^2 + 59*e + 16, 5/2*e^4 + 17/2*e^3 - 39/2*e^2 - 129/2*e - 7, 3*e^4 + 6*e^3 - 27*e^2 - 50*e + 2, -7/2*e^4 - 15/2*e^3 + 59/2*e^2 + 117/2*e + 5, -e^4 + e^3 + 11*e^2 - 10*e - 8, -2*e^3 - 4*e^2 + 20*e + 20, -3*e^4 - 5*e^3 + 31*e^2 + 45*e - 18, e^4 + 3*e^3 - 6*e^2 - 34*e - 32, 4*e^4 + 13*e^3 - 31*e^2 - 105*e - 26, -3/2*e^4 - 3/2*e^3 + 31/2*e^2 + 21/2*e - 11, 2*e^4 + 6*e^3 - 13*e^2 - 54*e - 40, -3*e^4 - 6*e^3 + 26*e^2 + 49*e + 12, e^4 - 15*e^2 + e + 32, 5/2*e^4 + 11/2*e^3 - 45/2*e^2 - 97/2*e - 19, 4*e^4 + 7*e^3 - 38*e^2 - 61*e + 6, -2*e^4 - 10*e^3 + 15*e^2 + 96*e + 24, -2*e^4 - 8*e^3 + 15*e^2 + 63*e + 18, 3/2*e^4 + 11/2*e^3 - 19/2*e^2 - 81/2*e - 23, -e^4 - 3*e^3 + 7*e^2 + 19*e + 2, 5/2*e^4 + 19/2*e^3 - 33/2*e^2 - 169/2*e - 41, -e^4 - 4*e^3 + 7*e^2 + 38*e + 28, e^4 - 11*e^2 - 2*e + 20, -3*e^4 - 13*e^3 + 21*e^2 + 117*e + 52, 3*e^4 + 10*e^3 - 22*e^2 - 86*e - 32, -3*e^4 - 5*e^3 + 29*e^2 + 38*e - 26, e^4 + 3*e^3 - 13*e^2 - 26*e + 8, -2*e^3 - 2*e^2 + 8*e, 2*e^4 + 9*e^3 - 12*e^2 - 73*e - 22, 1/2*e^4 + 5/2*e^3 - 17/2*e^2 - 47/2*e + 17, 4*e^4 + 12*e^3 - 32*e^2 - 101*e - 24, -3*e^4 - 13*e^3 + 23*e^2 + 113*e + 30, -3*e^4 - 7*e^3 + 24*e^2 + 62*e + 40, -3*e^4 - 5*e^3 + 29*e^2 + 35*e - 22, 7*e^4 + 22*e^3 - 55*e^2 - 187*e - 58, -1/2*e^4 - 3/2*e^3 + 1/2*e^2 + 17/2*e + 47, -1/2*e^4 - 3/2*e^3 + 7/2*e^2 + 11/2*e + 29, 1/2*e^4 - 5/2*e^3 - 15/2*e^2 + 73/2*e + 31, 3*e^4 + 5*e^3 - 26*e^2 - 41*e - 34, -3*e^4 - 2*e^3 + 31*e^2 + 14*e - 18, 2*e^4 + 15*e^3 - 8*e^2 - 129*e - 56, -7*e^4 - 19*e^3 + 56*e^2 + 155*e + 48, -9/2*e^4 - 21/2*e^3 + 85/2*e^2 + 179/2*e - 11, -5*e^4 - 17*e^3 + 37*e^2 + 137*e + 40, 4*e^4 + 10*e^3 - 30*e^2 - 76*e - 50, 4*e^4 + 12*e^3 - 32*e^2 - 107*e - 24, -e^4 - 5*e^3 + 3*e^2 + 40*e + 44, 2*e^4 + 7*e^3 - 15*e^2 - 63*e - 22, 3/2*e^4 + 9/2*e^3 - 15/2*e^2 - 75/2*e - 23, -2*e^4 - 3*e^3 + 19*e^2 + 25*e - 22, 2*e^4 + 11*e^3 - 16*e^2 - 91*e - 2, 6*e^4 + 15*e^3 - 51*e^2 - 131*e - 24, 3*e^4 + 14*e^3 - 21*e^2 - 128*e - 50, -5/2*e^4 - 25/2*e^3 + 33/2*e^2 + 203/2*e + 25, e^4 - 5*e^3 - 19*e^2 + 40*e + 58, -5*e^4 - 13*e^3 + 44*e^2 + 117*e + 28, -3/2*e^4 - 11/2*e^3 + 13/2*e^2 + 75/2*e + 31, 1/2*e^4 + 15/2*e^3 + 1/2*e^2 - 115/2*e - 21, 1/2*e^4 + 5/2*e^3 - 1/2*e^2 - 51/2*e - 25, 5*e^4 + 21*e^3 - 34*e^2 - 171*e - 68, 2*e^4 + e^3 - 17*e^2 - 8*e - 32, 9/2*e^4 + 31/2*e^3 - 65/2*e^2 - 273/2*e - 55, 4*e^4 + 6*e^3 - 35*e^2 - 49*e - 8, -15/2*e^4 - 45/2*e^3 + 113/2*e^2 + 369/2*e + 51, 3*e^4 + 12*e^3 - 21*e^2 - 89*e - 32, -4*e^4 - 13*e^3 + 33*e^2 + 122*e + 22, 7*e^4 + 18*e^3 - 57*e^2 - 157*e - 52, -7*e^3 - 5*e^2 + 61*e + 24, 3/2*e^4 + 17/2*e^3 - 15/2*e^2 - 147/2*e - 49, -2*e^3 - e^2 + 16*e - 24, -7*e^4 - 18*e^3 + 56*e^2 + 154*e + 72, 6*e^4 + 21*e^3 - 44*e^2 - 189*e - 70, 3*e^4 + 2*e^3 - 31*e^2 - 12*e + 44, -4*e^4 - 14*e^3 + 37*e^2 + 123*e + 24, 3/2*e^4 + 1/2*e^3 - 37/2*e^2 + 7/2*e + 29, e^4 + 3*e^3 - 5*e^2 - 37*e - 46, -3/2*e^4 - 11/2*e^3 + 29/2*e^2 + 95/2*e - 15, -4*e^4 - 10*e^3 + 36*e^2 + 77*e - 10, -11*e^4 - 28*e^3 + 92*e^2 + 241*e + 66, e^3 - 2*e^2 + 4*e + 38, -5*e^4 - 16*e^3 + 36*e^2 + 124*e + 64, 5/2*e^4 + 15/2*e^3 - 43/2*e^2 - 115/2*e + 17, -e^4 + 2*e^3 + 15*e^2 - 20*e - 36, 9*e^3 + 4*e^2 - 86*e - 44, 3/2*e^4 + 15/2*e^3 - 31/2*e^2 - 145/2*e + 9, -e^4 + 10*e^2 - 13*e - 12, e^3 + 4*e^2 - 19*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([17, 17, -w^2 + w + 3])] = 1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]