/* 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, -2, -6, 0, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([16, 2, 2]) primes_array = [ [2, 2, w + 1],\ [3, 3, -w^3 + w^2 + 5*w - 2],\ [11, 11, -w^3 + 5*w + 1],\ [11, 11, -w + 2],\ [13, 13, -w^3 + w^2 + 4*w + 1],\ [17, 17, -w^3 + w^2 + 6*w - 1],\ [17, 17, -w^2 - w + 3],\ [23, 23, w^3 - 6*w],\ [27, 3, w^3 + w^2 - 5*w - 4],\ [41, 41, -w^3 + w^2 + 5*w],\ [41, 41, 2*w^3 - 11*w - 4],\ [53, 53, 2*w^3 - 2*w^2 - 11*w + 6],\ [59, 59, 2*w^3 - 11*w - 2],\ [67, 67, w^3 - 7*w - 1],\ [71, 71, 3*w^3 - 2*w^2 - 15*w + 1],\ [79, 79, -3*w^3 + w^2 + 16*w + 3],\ [89, 89, -4*w^3 + 2*w^2 + 22*w - 3],\ [101, 101, -2*w^3 + 13*w + 6],\ [101, 101, w^3 + w^2 - 6*w - 3],\ [101, 101, -3*w^3 + 2*w^2 + 17*w - 3],\ [101, 101, w^3 - w^2 - 8*w - 3],\ [107, 107, 2*w^3 - 3*w^2 - 6*w],\ [109, 109, -w^3 + w^2 + 4*w - 3],\ [113, 113, w^3 - 8*w - 4],\ [113, 113, 7*w^3 - 4*w^2 - 39*w + 7],\ [121, 11, w^3 - w^2 - 6*w - 1],\ [127, 127, -2*w^3 + w^2 + 9*w + 3],\ [127, 127, 2*w^3 - 13*w],\ [131, 131, 6*w^3 - 3*w^2 - 34*w + 6],\ [139, 139, w^2 - 2*w - 4],\ [149, 149, 3*w^3 - w^2 - 18*w - 3],\ [151, 151, -2*w^3 + 2*w^2 + 9*w - 4],\ [163, 163, 3*w^3 - 2*w^2 - 16*w + 2],\ [163, 163, 2*w^3 - 10*w + 1],\ [167, 167, 3*w^3 - 18*w - 4],\ [173, 173, -3*w^3 + w^2 + 18*w - 3],\ [193, 193, 4*w^3 - 3*w^2 - 23*w + 9],\ [197, 197, 3*w^3 - w^2 - 17*w + 2],\ [197, 197, -3*w^3 - 2*w^2 + 15*w + 9],\ [199, 199, 2*w^3 - w^2 - 8*w - 6],\ [223, 223, -6*w^3 + 3*w^2 + 35*w - 7],\ [227, 227, 2*w^3 - 12*w - 1],\ [227, 227, -w^3 + w^2 + 7*w - 8],\ [227, 227, 3*w^3 - w^2 - 18*w - 1],\ [227, 227, -4*w^3 - 2*w^2 + 21*w + 14],\ [229, 229, -w^3 + 2*w^2 + 3*w - 5],\ [233, 233, w^3 - 3*w - 3],\ [233, 233, -3*w^3 + 3*w^2 + 15*w - 8],\ [241, 241, 4*w^2 - 7*w - 8],\ [241, 241, w^3 + w^2 - 4*w - 5],\ [251, 251, 2*w^3 - w^2 - 13*w - 1],\ [251, 251, 2*w^3 - 9*w - 6],\ [257, 257, w^3 - 8*w],\ [263, 263, -w^3 + 2*w^2 + 5*w - 7],\ [269, 269, 3*w^3 - 3*w^2 - 12*w - 1],\ [271, 271, 2*w^3 - w^2 - 12*w - 2],\ [271, 271, 9*w^3 - 5*w^2 - 50*w + 9],\ [277, 277, w^3 - 2*w^2 - 6*w + 6],\ [277, 277, w^3 + w^2 - 7*w - 2],\ [281, 281, 4*w^3 - 2*w^2 - 24*w + 5],\ [281, 281, -7*w^3 + 4*w^2 + 40*w - 12],\ [289, 17, -5*w^3 + 5*w^2 + 26*w - 15],\ [293, 293, -w^3 + 2*w^2 + 4*w - 4],\ [293, 293, -9*w^3 + 5*w^2 + 51*w - 12],\ [307, 307, -2*w^3 + 2*w^2 + 12*w - 7],\ [311, 311, -4*w^3 + 24*w + 11],\ [313, 313, 2*w^3 - 9*w - 4],\ [317, 317, 2*w^3 - 4*w^2 - 2*w - 3],\ [317, 317, 2*w^3 - 14*w - 7],\ [331, 331, -3*w^3 + 19*w + 3],\ [349, 349, -w^3 - 2*w^2 + 8*w + 8],\ [349, 349, w^3 + w^2 - 5*w - 8],\ [361, 19, -w^3 + w^2 + 7*w - 6],\ [361, 19, w^3 + w^2 - 3*w - 4],\ [373, 373, 2*w^3 - w^2 - 11*w - 3],\ [379, 379, -4*w^3 + 25*w + 14],\ [383, 383, -3*w^3 + w^2 + 15*w - 2],\ [383, 383, 3*w^3 - 4*w^2 - 10*w],\ [401, 401, 4*w^3 - w^2 - 24*w - 4],\ [409, 409, w^2 + 5*w + 5],\ [419, 419, w^2 - 2*w - 6],\ [419, 419, -6*w^3 + 3*w^2 + 34*w - 8],\ [433, 433, 3*w^3 - 16*w - 6],\ [433, 433, 3*w^3 - w^2 - 16*w - 5],\ [443, 443, -w^3 + 2*w^2 + w + 3],\ [449, 449, 2*w^2 - 2*w - 5],\ [457, 457, -4*w^3 + 4*w^2 + 20*w - 9],\ [457, 457, w^2 + w - 7],\ [463, 463, w^3 - 7*w - 7],\ [463, 463, w^3 + 2*w^2 - 5*w - 9],\ [487, 487, w^3 + 2*w^2 - 5*w - 5],\ [487, 487, -w^3 + w^2 + 6*w - 7],\ [509, 509, -2*w^3 - 2*w^2 + 13*w + 10],\ [509, 509, 3*w^3 - 17*w - 3],\ [509, 509, -5*w^3 + 3*w^2 + 30*w - 11],\ [509, 509, -w^3 - 2*w^2 + 4*w + 6],\ [521, 521, -2*w^3 - w^2 + 9*w + 7],\ [541, 541, -2*w^3 - 2*w^2 + 12*w + 9],\ [541, 541, -2*w^3 + 12*w + 9],\ [547, 547, 2*w^3 + w^2 - 7*w - 3],\ [547, 547, 5*w^3 - 3*w^2 - 30*w + 3],\ [557, 557, w^3 + 2*w^2 - 5*w - 11],\ [557, 557, -w^3 + w^2 + 3*w - 4],\ [563, 563, 3*w^3 - w^2 - 15*w - 4],\ [569, 569, 2*w^2 - w - 8],\ [577, 577, 4*w^3 - 22*w - 7],\ [593, 593, -3*w^3 + w^2 + 17*w + 4],\ [607, 607, -w^3 - 3*w^2 + w + 4],\ [613, 613, 3*w^3 - 2*w^2 - 14*w],\ [613, 613, w^2 - 3*w - 5],\ [619, 619, -w^3 + 2*w^2 + 4*w - 10],\ [625, 5, -5],\ [641, 641, -3*w^3 - 3*w^2 + 12*w + 7],\ [643, 643, 2*w^3 + 2*w^2 - 14*w - 13],\ [643, 643, 3*w^3 - 19*w - 7],\ [647, 647, w^3 - 5*w - 7],\ [653, 653, -3*w^3 + 2*w^2 + 14*w - 2],\ [659, 659, -6*w^3 + 2*w^2 + 34*w - 1],\ [659, 659, -2*w^3 + w^2 + 17*w + 9],\ [683, 683, w^3 + 2*w^2 - 7*w - 7],\ [691, 691, 3*w^3 - 15*w - 7],\ [701, 701, w^3 + 2*w^2 - 6*w - 10],\ [701, 701, -w^3 + 3*w^2 + 6*w - 13],\ [701, 701, 5*w^3 - 2*w^2 - 30*w + 4],\ [701, 701, 4*w^3 - 2*w^2 - 21*w + 2],\ [727, 727, -2*w^3 - 3*w^2 + 9*w + 7],\ [743, 743, 3*w - 4],\ [751, 751, w^3 - 9*w - 3],\ [757, 757, w^3 - 5*w^2 + 4*w + 7],\ [757, 757, 4*w^3 - 5*w^2 - 11*w - 5],\ [761, 761, 4*w^3 - 5*w^2 - 20*w + 16],\ [769, 769, -3*w^3 + 16*w + 12],\ [773, 773, 2*w^3 + 2*w^2 - 11*w - 10],\ [773, 773, w^3 - 3*w - 5],\ [773, 773, -w^3 + 2*w^2 + 9*w + 3],\ [773, 773, 5*w^3 - w^2 - 27*w],\ [787, 787, -4*w^3 + 3*w^2 + 22*w - 4],\ [787, 787, -2*w^3 + w^2 + 14*w - 6],\ [797, 797, 2*w^3 - w^2 - 14*w + 2],\ [797, 797, -3*w^3 + 2*w^2 + 19*w - 7],\ [811, 811, 2*w^3 + w^2 - 14*w - 12],\ [811, 811, 2*w^3 + 2*w^2 - 8*w - 3],\ [829, 829, -2*w^3 + 14*w + 11],\ [857, 857, -3*w^3 + 3*w^2 + 14*w - 7],\ [859, 859, 3*w^3 - w^2 - 15*w],\ [859, 859, -11*w^3 + 7*w^2 + 60*w - 15],\ [863, 863, 2*w^2 - 2*w - 13],\ [863, 863, 2*w^3 + w^2 - 12*w - 4],\ [877, 877, -6*w^3 - 2*w^2 + 32*w + 19],\ [887, 887, 3*w^3 + w^2 - 14*w - 7],\ [887, 887, 2*w^3 - w^2 - 15*w - 5],\ [911, 911, 3*w^3 - 18*w - 2],\ [919, 919, w^3 - w^2 - 6*w - 5],\ [919, 919, -4*w^3 + w^2 + 24*w - 2],\ [941, 941, 3*w^3 - 3*w^2 - 13*w],\ [947, 947, 2*w^3 - 2*w^2 - 10*w - 1],\ [947, 947, -3*w^3 + 2*w^2 + 14*w + 6],\ [953, 953, -4*w^3 + 3*w^2 + 21*w - 3],\ [953, 953, 4*w^3 - w^2 - 23*w + 1],\ [961, 31, -2*w^3 + 3*w^2 + 12*w - 8],\ [961, 31, 3*w^2 - 4*w - 6],\ [967, 967, -5*w^3 - 3*w^2 + 23*w + 14],\ [971, 971, 5*w^3 - w^2 - 27*w - 8],\ [971, 971, -9*w^3 + 6*w^2 + 50*w - 12],\ [983, 983, -w^3 + w^2 + 10*w + 3],\ [997, 997, 2*w^3 + w^2 - 11*w - 3],\ [997, 997, w^3 + 2*w^2 - 6*w - 8],\ [997, 997, -5*w^3 + 4*w^2 + 29*w - 15]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^6 - 13*x^4 + 43*x^2 - 16 K. = NumberField(heckePol) hecke_eigenvalues_array = [0, e, e^3 - 7*e, e^2 - 4, -1/2*e^5 + 9/2*e^3 - 15/2*e, -1/2*e^5 + 11/2*e^3 - 25/2*e, e^4 - 8*e^2 + 10, -2*e^2 + 8, e, 1/2*e^5 - 13/2*e^3 + 39/2*e, -1/2*e^5 + 13/2*e^3 - 33/2*e, -1/2*e^5 + 13/2*e^3 - 39/2*e, e^4 - 6*e^2 - 4, -e^4 + 8*e^2 - 12, 0, -4*e^3 + 26*e, -1/2*e^5 + 7/2*e^3 + 3/2*e, 6, 2*e^2 - 2, -1/2*e^5 + 13/2*e^3 - 51/2*e, -1/2*e^5 + 9/2*e^3 - 11/2*e, -2*e^3 + 17*e, -1/2*e^5 + 13/2*e^3 - 35/2*e, e^4 - 10*e^2 + 18, -1/2*e^5 + 7/2*e^3 - 9/2*e, 3/2*e^5 - 27/2*e^3 + 43/2*e, -2*e, -2*e^4 + 16*e^2 - 16, -e^2 + 4, e^5 - 11*e^3 + 29*e, -1/2*e^5 + 17/2*e^3 - 67/2*e, -2*e, e^3 - 11*e, 4*e^2 - 12, 2*e^2 - 8, 4*e^2 - 10, 2*e^4 - 17*e^2 + 26, 2*e^4 - 12*e^2 - 2, 2*e^4 - 16*e^2 + 14, -4*e^2 + 24, -2*e^4 + 12*e^2 + 16, -2*e^4 + 13*e^2 + 4, -e^4 + 6*e^2 + 4, -2*e^4 + 17*e^2 - 12, 2*e^4 - 11*e^2 - 12, -1/2*e^5 + 5/2*e^3 + 21/2*e, e^2 + 2, 3/2*e^5 - 33/2*e^3 + 87/2*e, 1/2*e^5 - 5/2*e^3 - 1/2*e, -3*e^4 + 22*e^2 - 6, -e^4 + 28, e^4 - 2*e^2 - 20, -e^4 + 8*e^2 + 2, 2*e^5 - 22*e^3 + 56*e, -1/2*e^5 + 5/2*e^3 + 5/2*e, 4*e^3 - 30*e, 2*e, -1/2*e^5 + 5/2*e^3 + 9/2*e, -2*e^4 + 14*e^2 - 2, e^4 - 4*e^2 - 30, 2*e^4 - 21*e^2 + 34, -1/2*e^5 + 17/2*e^3 - 53/2*e, -1/2*e^5 + 13/2*e^3 - 27/2*e, -2*e^2 + 14, -e^4 + 8*e^2 + 4, -12*e, -1/2*e^5 + 17/2*e^3 - 53/2*e, -2*e^4 + 18*e^2 - 10, 2*e^4 - 18*e^2 + 22, -2*e^4 + 17*e^2 - 20, -1/2*e^5 - 3/2*e^3 + 69/2*e, -2*e^4 + 22*e^2 - 34, -2*e^4 + 13*e^2 + 2, -2*e^4 + 23*e^2 - 38, -1/2*e^5 + 13/2*e^3 - 35/2*e, e^3 - 9*e, 2*e^5 - 24*e^3 + 58*e, -6*e, 2*e^4 - 21*e^2 + 34, -1/2*e^5 + 1/2*e^3 + 31/2*e, -5*e^3 + 29*e, 4*e^4 - 35*e^2 + 52, -1/2*e^5 + 7/2*e^3 + 19/2*e, 3/2*e^5 - 27/2*e^3 + 55/2*e, 3*e^2 - 36, 3/2*e^5 - 39/2*e^3 + 111/2*e, 3/2*e^5 - 37/2*e^3 + 119/2*e, -3*e^2 + 26, 2*e^4 - 12*e^2, -8*e, 2*e^5 - 24*e^3 + 66*e, -4*e^4 + 24*e^2 + 8, 3/2*e^5 - 31/2*e^3 + 85/2*e, 2*e^4 - 14*e^2 + 6, 4*e^4 - 28*e^2 + 6, -1/2*e^5 + 5/2*e^3 + 29/2*e, -4*e^4 + 33*e^2 - 38, 3/2*e^5 - 47/2*e^3 + 157/2*e, -4*e^2 + 30, -2*e^5 + 15*e^3 - 11*e, 4*e^4 - 36*e^2 + 52, -2*e^2 + 14, -1/2*e^5 + 17/2*e^3 - 67/2*e, -2*e^5 + 24*e^3 - 67*e, 3/2*e^5 - 37/2*e^3 + 103/2*e, 1/2*e^5 - 5/2*e^3 - 33/2*e, 7/2*e^5 - 71/2*e^3 + 163/2*e, -4*e^3 + 36*e, -1/2*e^5 + 9/2*e^3 - 31/2*e, 3/2*e^5 - 19/2*e^3 - 15/2*e, -e^5 + 15*e^3 - 45*e, 3*e^4 - 18*e^2 - 14, -2*e^4 + 21*e^2 - 22, -2*e^5 + 21*e^3 - 45*e, -3*e^4 + 26*e^2 - 12, 4*e^4 - 30*e^2 + 8, -5/2*e^5 + 45/2*e^3 - 67/2*e, 3*e^4 - 20*e^2 + 20, -3*e^4 + 30*e^2 - 60, -2*e^5 + 20*e^3 - 51*e, 2*e^5 - 15*e^3 + 5*e, -5/2*e^5 + 57/2*e^3 - 175/2*e, -6*e^2 + 30, 2*e^4 - 14*e^2 + 30, -1/2*e^5 + 5/2*e^3 - 7/2*e, 2*e^3 - 12*e, -6*e^4 + 54*e^2 - 72, -4*e^4 + 22*e^2 + 32, -1/2*e^5 - 7/2*e^3 + 97/2*e, -2*e^4 + 14*e^2 - 10, -1/2*e^5 + 5/2*e^3 - 13/2*e, -3*e^4 + 22*e^2 - 30, 3/2*e^5 - 43/2*e^3 + 157/2*e, 4*e^4 - 26*e^2 - 2, 3/2*e^5 - 35/2*e^3 + 89/2*e, 2*e^4 - 6*e^2 - 26, -4*e^5 + 43*e^3 - 97*e, -3*e^2 + 4, -2*e^4 + 24*e^2 - 34, -2*e^2 - 10, 2*e^5 - 18*e^3 + 29*e, -3*e^4 + 22*e^2 - 36, -2*e^4 + 12*e^2 + 30, -1/2*e^5 + 17/2*e^3 - 85/2*e, 3*e^5 - 33*e^3 + 67*e, 4*e^5 - 35*e^3 + 53*e, 4*e^5 - 38*e^3 + 70*e, -2*e^4 + 12*e^2 - 16, 3/2*e^5 - 43/2*e^3 + 129/2*e, 2*e^5 - 18*e^3 + 34*e, 4*e^4 - 38*e^2 + 40, -2*e^4 + 14*e^2 - 48, 2*e^4 - 20*e^2 + 56, -4*e^4 + 30*e^2 - 40, 7/2*e^5 - 83/2*e^3 + 205/2*e, -2*e^5 + 24*e^3 - 73*e, 3*e^5 - 37*e^3 + 91*e, -1/2*e^5 + 13/2*e^3 - 57/2*e, 6*e^4 - 51*e^2 + 66, -9/2*e^5 + 91/2*e^3 - 213/2*e, 7/2*e^5 - 81/2*e^3 + 207/2*e, -2*e^5 + 28*e^3 - 84*e, 3*e^5 - 37*e^3 + 91*e, -2*e^5 + 23*e^3 - 69*e, -2*e^2 + 8, -4*e^4 + 32*e^2 - 18, -1/2*e^5 + 1/2*e^3 + 37/2*e, -6*e^4 + 40*e^2 + 6] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([2, 2, w + 1])] = 1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]