/* 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, 1, 5, -3, -2, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([32, 2, 2]) primes_array = [ [3, 3, w^4 - 2*w^3 - 3*w^2 + 4*w + 1],\ [13, 13, w^3 - 2*w^2 - 2*w + 2],\ [23, 23, 2*w^4 - 3*w^3 - 6*w^2 + 5*w + 1],\ [25, 5, -w^2 + 2*w + 2],\ [29, 29, -w^4 + w^3 + 4*w^2 - 3*w - 1],\ [31, 31, w^4 - w^3 - 5*w^2 + 2*w + 4],\ [32, 2, 2],\ [37, 37, w^4 - 2*w^3 - 2*w^2 + 4*w + 1],\ [47, 47, w^4 - w^3 - 5*w^2 + 2*w + 3],\ [47, 47, 2*w^4 - 3*w^3 - 6*w^2 + 6*w + 2],\ [49, 7, -w^4 + 2*w^3 + 4*w^2 - 6*w - 2],\ [53, 53, -2*w^4 + 3*w^3 + 7*w^2 - 7*w - 2],\ [59, 59, -2*w^4 + 3*w^3 + 6*w^2 - 5*w - 2],\ [67, 67, -w^4 + 3*w^3 + 2*w^2 - 7*w],\ [67, 67, -w^4 + 3*w^3 + w^2 - 7*w + 1],\ [71, 71, w^4 - 2*w^3 - 4*w^2 + 3*w + 3],\ [71, 71, -w^2 + 5],\ [79, 79, 2*w^4 - 3*w^3 - 5*w^2 + 3*w + 1],\ [81, 3, -w^4 + 3*w^3 + 3*w^2 - 8*w - 2],\ [83, 83, -3*w^4 + 3*w^3 + 10*w^2 - 2*w - 2],\ [89, 89, -w^4 + 2*w^3 + 3*w^2 - 6*w - 1],\ [101, 101, w^4 - w^3 - 4*w^2 + 3],\ [103, 103, w^4 - w^3 - 5*w^2 + 3*w + 1],\ [103, 103, 2*w^4 - 3*w^3 - 8*w^2 + 8*w + 3],\ [103, 103, -w^4 + 2*w^3 + 2*w^2 - 5*w + 4],\ [109, 109, -2*w^4 + 4*w^3 + 7*w^2 - 10*w - 4],\ [121, 11, w^4 - 3*w^3 - 2*w^2 + 7*w + 1],\ [125, 5, -2*w^4 + 2*w^3 + 7*w^2 - 3*w + 1],\ [127, 127, -2*w^3 + 2*w^2 + 7*w - 4],\ [149, 149, 2*w^4 - 5*w^3 - 4*w^2 + 11*w + 1],\ [157, 157, -2*w^4 + 2*w^3 + 9*w^2 - 5*w - 6],\ [163, 163, 2*w^4 - 2*w^3 - 6*w^2 + w - 1],\ [163, 163, 3*w^4 - 5*w^3 - 9*w^2 + 10*w + 4],\ [163, 163, -3*w^4 + 5*w^3 + 10*w^2 - 12*w - 3],\ [167, 167, w^4 - 2*w^3 - 2*w^2 + 4*w - 4],\ [173, 173, 2*w^4 - 3*w^3 - 6*w^2 + 7*w],\ [173, 173, 2*w^4 - 5*w^3 - 4*w^2 + 10*w],\ [179, 179, -w^4 + 3*w^3 + 2*w^2 - 6*w - 2],\ [181, 181, -w^4 + w^3 + 2*w^2 + w + 2],\ [181, 181, -2*w^4 + 3*w^3 + 7*w^2 - 4*w - 4],\ [193, 193, 2*w^4 - 4*w^3 - 5*w^2 + 8*w + 3],\ [197, 197, 2*w^4 - 2*w^3 - 9*w^2 + 3*w + 6],\ [199, 199, w^4 - 2*w^3 - w^2 + 3*w - 4],\ [211, 211, -w^4 + 6*w^2 + w - 3],\ [223, 223, 3*w^4 - 5*w^3 - 9*w^2 + 11*w + 1],\ [223, 223, -w^3 + w^2 + w - 3],\ [229, 229, 2*w^2 - w - 4],\ [241, 241, -w^4 + 6*w^2 - 6],\ [241, 241, -w^4 + 3*w^3 + 3*w^2 - 8*w],\ [257, 257, -2*w^3 + 2*w^2 + 5*w],\ [269, 269, 2*w^3 - w^2 - 5*w],\ [269, 269, 3*w^4 - 6*w^3 - 7*w^2 + 12*w],\ [277, 277, -w^4 + 3*w^3 + w^2 - 7*w],\ [277, 277, -w^3 - w^2 + 5*w + 4],\ [283, 283, -3*w^4 + 5*w^3 + 8*w^2 - 8*w - 1],\ [283, 283, 3*w^4 - 4*w^3 - 9*w^2 + 7*w + 1],\ [283, 283, 3*w^4 - 5*w^3 - 9*w^2 + 8*w + 4],\ [293, 293, 2*w^4 - 4*w^3 - 7*w^2 + 11*w + 2],\ [307, 307, -w^3 + 3*w^2 - 5],\ [311, 311, w^2 - 2*w - 4],\ [317, 317, w^4 - w^3 - 2*w^2 + w - 4],\ [317, 317, -w^3 + 6*w],\ [331, 331, -w^4 + w^3 + 2*w^2 - 2*w + 3],\ [343, 7, -2*w^3 + w^2 + 7*w - 1],\ [349, 349, 3*w^4 - 5*w^3 - 8*w^2 + 10*w],\ [349, 349, w^4 + w^3 - 6*w^2 - 4*w + 4],\ [353, 353, w^3 - 3*w - 4],\ [359, 359, 2*w^4 - 2*w^3 - 8*w^2 + w + 4],\ [367, 367, -5*w^4 + 8*w^3 + 17*w^2 - 18*w - 8],\ [367, 367, -w^4 + 4*w^3 - 9*w + 1],\ [373, 373, 3*w^4 - 6*w^3 - 8*w^2 + 12*w + 1],\ [379, 379, 4*w^4 - 7*w^3 - 12*w^2 + 15*w + 6],\ [379, 379, -w^4 - w^3 + 6*w^2 + 5*w],\ [379, 379, -3*w^4 + 4*w^3 + 9*w^2 - 6*w - 1],\ [421, 421, -3*w^4 + 4*w^3 + 11*w^2 - 7*w - 3],\ [433, 433, 2*w^4 - 5*w^3 - 5*w^2 + 10*w],\ [439, 439, -4*w^4 + 7*w^3 + 11*w^2 - 15*w - 1],\ [457, 457, -2*w^4 + 4*w^3 + 6*w^2 - 7*w - 2],\ [461, 461, w^3 - 2*w - 3],\ [463, 463, -3*w^4 + 5*w^3 + 10*w^2 - 11*w - 2],\ [463, 463, -2*w^4 + 2*w^3 + 8*w^2 - 2*w - 7],\ [467, 467, w^4 - 2*w^3 - 4*w^2 + 4*w + 6],\ [487, 487, w^2 - 3*w - 3],\ [487, 487, w^2 + w - 5],\ [491, 491, -3*w^4 + 6*w^3 + 8*w^2 - 12*w],\ [491, 491, -w^4 + 3*w^3 + 3*w^2 - 9*w - 1],\ [503, 503, -w^4 + 7*w^2 - 7],\ [503, 503, -w^4 + 2*w^3 + 3*w^2 - 5*w + 3],\ [509, 509, -2*w^4 + 4*w^3 + 7*w^2 - 7*w - 3],\ [521, 521, -w^4 + w^3 + 6*w^2 - 2*w - 5],\ [541, 541, -w^4 + 2*w^3 + 3*w^2 - 4*w - 5],\ [541, 541, -3*w^4 + 5*w^3 + 10*w^2 - 10*w - 2],\ [541, 541, -w^4 + 3*w^2 + 5*w - 1],\ [547, 547, -3*w^4 + 5*w^3 + 10*w^2 - 9*w - 4],\ [557, 557, 5*w^4 - 5*w^3 - 18*w^2 + 6*w + 3],\ [557, 557, w^4 - 2*w^2 - 2*w - 2],\ [563, 563, -2*w^4 + 4*w^3 + 5*w^2 - 6*w],\ [563, 563, 3*w^4 - 4*w^3 - 10*w^2 + 8*w + 1],\ [587, 587, -w^4 + 5*w^2 + 2*w - 6],\ [599, 599, 3*w^4 - 4*w^3 - 11*w^2 + 9*w + 3],\ [599, 599, 3*w^3 - 3*w^2 - 9*w + 1],\ [601, 601, -w^4 + 3*w^3 + w^2 - 8*w],\ [607, 607, 4*w^4 - 8*w^3 - 12*w^2 + 17*w + 6],\ [607, 607, w^4 + 2*w^3 - 7*w^2 - 6*w + 3],\ [613, 613, -3*w^4 + 4*w^3 + 9*w^2 - 7*w - 2],\ [617, 617, 2*w^3 - 4*w^2 - 5*w + 2],\ [619, 619, -w^4 + w^3 + 2*w^2 + 3*w + 2],\ [641, 641, -2*w^4 + 4*w^3 + 6*w^2 - 11*w - 3],\ [641, 641, -2*w^4 + 5*w^3 + 5*w^2 - 11*w - 2],\ [643, 643, -3*w^4 + 6*w^3 + 7*w^2 - 10*w + 3],\ [653, 653, -4*w^4 + 6*w^3 + 15*w^2 - 15*w - 6],\ [659, 659, 2*w^4 - w^3 - 9*w^2 + w + 5],\ [659, 659, -w^4 + w^3 + w^2 + 2*w + 2],\ [659, 659, -2*w^4 + 4*w^3 + 5*w^2 - 9*w - 3],\ [673, 673, 2*w^4 + w^3 - 10*w^2 - 7*w + 3],\ [677, 677, -3*w^4 + w^3 + 14*w^2 + 3*w - 5],\ [677, 677, -3*w^4 + 4*w^3 + 8*w^2 - 4*w - 1],\ [677, 677, -2*w^4 + 2*w^3 + 6*w^2 + w + 1],\ [683, 683, -4*w^4 + 7*w^3 + 12*w^2 - 15*w],\ [691, 691, -3*w^4 + 4*w^3 + 9*w^2 - 4*w + 1],\ [691, 691, 3*w^4 - 5*w^3 - 8*w^2 + 9*w + 1],\ [709, 709, -w^4 + 4*w^3 - 8*w - 1],\ [733, 733, 4*w^4 - 6*w^3 - 13*w^2 + 10*w + 2],\ [743, 743, 2*w^4 - 5*w^3 - 4*w^2 + 14*w - 3],\ [751, 751, -w^4 - 2*w^3 + 7*w^2 + 6*w - 4],\ [757, 757, 3*w^3 - 2*w^2 - 9*w + 1],\ [761, 761, 4*w^4 - 5*w^3 - 14*w^2 + 9*w + 3],\ [761, 761, 5*w^4 - 8*w^3 - 16*w^2 + 15*w + 8],\ [761, 761, w^4 - 2*w^3 - 2*w^2 + 2*w - 3],\ [769, 769, 3*w^3 - 4*w^2 - 8*w + 7],\ [769, 769, 3*w^4 - 5*w^3 - 11*w^2 + 12*w + 4],\ [769, 769, -2*w^3 + 3*w^2 + 6*w - 1],\ [773, 773, 3*w^4 - 3*w^3 - 12*w^2 + 7*w + 8],\ [773, 773, -2*w^4 + 6*w^3 + 3*w^2 - 14*w + 2],\ [773, 773, -2*w^4 + w^3 + 9*w^2 - w - 6],\ [787, 787, w^4 - 4*w^2 - 2*w + 6],\ [809, 809, -2*w^4 + 2*w^3 + 8*w^2 - w - 6],\ [809, 809, -2*w^4 + 3*w^3 + 9*w^2 - 5*w - 8],\ [821, 821, -2*w^4 + 9*w^2 + 3*w + 1],\ [821, 821, w^4 - 4*w^2 - 3*w + 4],\ [829, 829, -w^4 + 3*w^3 - 4*w + 4],\ [853, 853, -3*w^4 + 5*w^3 + 8*w^2 - 9*w - 2],\ [859, 859, 2*w^4 - 2*w^3 - 6*w^2 - 3],\ [863, 863, 5*w^4 - 9*w^3 - 14*w^2 + 17*w + 3],\ [863, 863, 2*w^4 - 4*w^3 - 7*w^2 + 12*w + 6],\ [863, 863, 2*w^4 - 2*w^3 - 11*w^2 + 6*w + 9],\ [877, 877, -2*w^4 + 3*w^3 + 8*w^2 - 7*w - 2],\ [877, 877, 3*w^4 - 6*w^3 - 6*w^2 + 13*w - 1],\ [877, 877, w^4 + w^3 - 4*w^2 - 8*w],\ [881, 881, -w^4 - w^3 + 7*w^2 + 2*w - 4],\ [883, 883, 4*w^4 - 4*w^3 - 16*w^2 + 7*w + 8],\ [887, 887, -2*w^4 + 3*w^3 + 10*w^2 - 8*w - 9],\ [929, 929, -w^3 + 2*w^2 - 4],\ [929, 929, -3*w^4 + 6*w^3 + 9*w^2 - 14*w],\ [929, 929, -4*w^4 + 7*w^3 + 11*w^2 - 14*w - 4],\ [937, 937, 2*w^4 - 2*w^3 - 5*w^2 - w - 2],\ [941, 941, -3*w^4 + 3*w^3 + 13*w^2 - 8*w - 8],\ [941, 941, w^4 - w^3 - 3*w^2 + 3*w - 3],\ [947, 947, -3*w^4 + 6*w^3 + 10*w^2 - 16*w - 3],\ [961, 31, -w^4 + 5*w^3 - 14*w],\ [961, 31, -w^4 - w^3 + 8*w^2 + 2*w - 4],\ [971, 971, -2*w^4 + 3*w^3 + 4*w^2 - 3*w + 2],\ [983, 983, 4*w^4 - 7*w^3 - 11*w^2 + 16*w],\ [983, 983, -3*w^4 + 2*w^3 + 12*w^2],\ [997, 997, 3*w^4 - 4*w^3 - 8*w^2 + 5*w - 2],\ [997, 997, w^4 - 3*w^3 - w^2 + 3*w - 2],\ [997, 997, -4*w^4 + 5*w^3 + 13*w^2 - 9*w - 2]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^6 - 14*x^4 + 46*x^2 - 8 K. = NumberField(heckePol) hecke_eigenvalues_array = [e, -1/4*e^5 + 5/2*e^3 - 7/2*e, 1/2*e^4 - 5*e^2 + 7, -e^3 + 7*e, -1/2*e^5 + 7*e^3 - 23*e, 1/2*e^5 - 7*e^3 + 23*e, 1, -1/2*e^5 + 7*e^3 - 21*e, 1/4*e^5 - 5/2*e^3 + 7/2*e, 1/2*e^4 - 3*e^2 - 1, -1/4*e^5 + 7/2*e^3 - 21/2*e, -2*e, -2, -1/2*e^4 + 2*e^2 + 9, 1/2*e^5 - 6*e^3 + 16*e, e^4 - 8*e^2 + 6, 3/4*e^5 - 19/2*e^3 + 49/2*e, -e^4 + 10*e^2 - 10, 1/2*e^5 - 7*e^3 + 20*e, 1/2*e^4 - 5*e^2 + 5, -3*e^2 + 14, -1/2*e^4 + 5*e^2 - 5, -2*e^2 + 12, -1/2*e^4 + 5*e^2 - 3, 1/2*e^5 - 5*e^3 + 9*e, e^5 - 14*e^3 + 42*e, -e^4 + 8*e^2 + 2, 2*e^3 - 16*e, -1/4*e^5 + 1/2*e^3 + 21/2*e, 3/4*e^5 - 19/2*e^3 + 57/2*e, 1/2*e^4 - 5*e^2 + 17, 1/2*e^4 - 4*e^2 + 7, e^4 - 11*e^2 + 22, -1/2*e^5 + 6*e^3 - 20*e, 3/2*e^4 - 17*e^2 + 25, -5/4*e^5 + 33/2*e^3 - 111/2*e, 4*e^2 - 14, -e^4 + 9*e^2 - 18, -1/2*e^4 + e^2 + 15, -e^4 + 8*e^2, -3/2*e^5 + 20*e^3 - 58*e, -1/2*e^5 + 5*e^3 - 11*e, 1/2*e^4 - e^2 - 5, 7/4*e^5 - 43/2*e^3 + 115/2*e, -5/4*e^5 + 37/2*e^3 - 119/2*e, -1/2*e^5 + 7*e^3 - 21*e, -e^4 + 12*e^2 - 16, -e^4 + 13*e^2 - 20, -3/4*e^5 + 19/2*e^3 - 47/2*e, e^2 - 6, 2*e^2 - 10, 2*e^2 - 10, 1/2*e^5 - 9*e^3 + 39*e, -e^4 + 6*e^2 + 4, 1/2*e^4 - 4*e^2 + 3, 1/2*e^4 - 4*e^2 + 3, -e^5 + 16*e^3 - 53*e, -1/2*e^5 + 7*e^3 - 19*e, e^5 - 14*e^3 + 45*e, -5/4*e^5 + 29/2*e^3 - 71/2*e, 1/2*e^5 - 7*e^3 + 17*e, e^4 - 8*e^2 + 4, -2*e^5 + 26*e^3 - 69*e, 2*e^5 - 28*e^3 + 84*e, -1/2*e^4 + 5*e^2 - 5, -3/2*e^4 + 7*e^2 + 33, 3/2*e^5 - 21*e^3 + 56*e, -2*e^4 + 18*e^2 - 36, 7/4*e^5 - 47/2*e^3 + 133/2*e, e^4 - 14*e^2 + 26, -e^5 + 18*e^3 - 78*e, e^5 - 12*e^3 + 31*e, -3/4*e^5 + 15/2*e^3 - 7/2*e, e^2 - 8, -1/2*e^4 + e^2 + 7, -5/4*e^5 + 33/2*e^3 - 105/2*e, -1/2*e^4 + 3*e^2 - 7, 5/4*e^5 - 31/2*e^3 + 85/2*e, 1/2*e^5 - 11*e^3 + 55*e, 5/2*e^4 - 27*e^2 + 39, 9/4*e^5 - 57/2*e^3 + 147/2*e, -1/4*e^5 + 7/2*e^3 - 29/2*e, 1/2*e^5 - e^3 - 21*e, -e^5 + 16*e^3 - 64*e, -7*e^3 + 49*e, -5/4*e^5 + 41/2*e^3 - 173/2*e, 11/4*e^5 - 71/2*e^3 + 197/2*e, e^4 - 2*e^2 - 30, -2*e^3 + 20*e, 5/2*e^5 - 32*e^3 + 80*e, 2*e^4 - 24*e^2 + 42, -1/2*e^4 + e^2 + 3, -9/4*e^5 + 57/2*e^3 - 171/2*e, 3/4*e^5 - 19/2*e^3 + 39/2*e, -1/2*e^5 + e^3 + 19*e, -1/2*e^4 + e^2 + 19, -7/4*e^5 + 45/2*e^3 - 131/2*e, e^5 - 7*e^3 - 3*e, -3/2*e^5 + 20*e^3 - 68*e, -3/2*e^5 + 19*e^3 - 49*e, -20, 3/2*e^5 - 16*e^3 + 38*e, 10*e^2 - 48, e^4 - 4*e^2 - 18, e^4 - 14*e^2 + 24, -3/4*e^5 + 15/2*e^3 - 43/2*e, -3/4*e^5 + 17/2*e^3 - 43/2*e, 2*e^4 - 15*e^2 - 2, 3*e^4 - 24*e^2 + 10, -2*e^5 + 22*e^3 - 51*e, 4*e^3 - 40*e, -e^4 + 9*e^2 - 34, 5*e^2 - 40, e^5 - 11*e^3 + 23*e, -3/2*e^4 + 14*e^2 - 3, -1/2*e^5 + 5*e^3 - 15*e, -e^4 + 8*e^2, -5/4*e^5 + 45/2*e^3 - 175/2*e, e^4 - 9*e^2 + 14, 1/4*e^5 - 9/2*e^3 + 21/2*e, 10, -e^4 + 10*e^2, 4*e^3 - 20*e, 0, -2*e^4 + 16*e^2 + 16, -2*e^4 + 16*e^2 + 22, 5/2*e^5 - 33*e^3 + 94*e, -e^4 + 12*e^2 - 42, e^5 - 18*e^3 + 83*e, -3*e^4 + 23*e^2 - 8, 5/2*e^4 - 22*e^2 + 17, 2*e^5 - 27*e^3 + 97*e, -8*e^2 + 34, 3*e^4 - 28*e^2 + 20, -1/2*e^4 + 11*e^2 - 49, 3*e^5 - 38*e^3 + 99*e, 4*e^4 - 34*e^2 + 20, -3/2*e^4 + 16*e^2 - 23, 9/4*e^5 - 53/2*e^3 + 107/2*e, 4*e, -3/4*e^5 + 39/2*e^3 - 197/2*e, -3*e^4 + 30*e^2 - 36, -e^5 + 14*e^3 - 63*e, 4*e^3 - 34*e, -3/2*e^4 + 11*e^2 - 13, -e^5 + 12*e^3 - 30*e, -2*e^4 + 18*e^2 - 14, e^5 - 6*e^3 - 18*e, 18*e, e^4 - 13*e^2 + 28, -3*e^4 + 26*e^2 - 4, 3/2*e^4 - 17*e^2 + 1, -7/2*e^5 + 45*e^3 - 134*e, 5*e^2 - 18, -3*e^4 + 20*e^2 + 14, 3/2*e^4 - 15*e^2 + 19, -2*e^4 + 22*e^2 - 14, 5/2*e^5 - 37*e^3 + 141*e, 3*e^5 - 35*e^3 + 91*e, -5/2*e^4 + 27*e^2 - 39, 3*e^5 - 42*e^3 + 129*e, -7/2*e^4 + 36*e^2 - 41, -3/4*e^5 + 23/2*e^3 - 81/2*e, 2*e^2 - 40, 10*e^2 - 30, -3/4*e^5 + 19/2*e^3 - 41/2*e, 9/4*e^5 - 57/2*e^3 + 135/2*e] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([32, 2, 2])] = -1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]