/* 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([-39, -1, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([9, 9, -w + 6]) primes_array = [ [3, 3, w + 6],\ [3, 3, -w + 7],\ [4, 2, 2],\ [11, 11, -3*w - 17],\ [11, 11, 3*w - 20],\ [13, 13, 2*w - 13],\ [13, 13, 2*w + 11],\ [17, 17, w + 7],\ [17, 17, -w + 8],\ [19, 19, -w - 4],\ [19, 19, -w + 5],\ [25, 5, 5],\ [31, 31, -6*w - 35],\ [31, 31, -6*w + 41],\ [37, 37, -w - 1],\ [37, 37, w - 2],\ [47, 47, 3*w + 16],\ [47, 47, -3*w + 19],\ [49, 7, -7],\ [67, 67, 3*w - 22],\ [67, 67, -3*w - 19],\ [71, 71, -w - 10],\ [71, 71, w - 11],\ [89, 89, -8*w - 47],\ [89, 89, 8*w - 55],\ [101, 101, 4*w - 29],\ [101, 101, -4*w - 25],\ [109, 109, -3*w - 20],\ [109, 109, 3*w - 23],\ [113, 113, 3*w - 17],\ [113, 113, -3*w - 14],\ [127, 127, -9*w - 53],\ [127, 127, 9*w - 62],\ [157, 157, 2*w - 1],\ [167, 167, 2*w - 19],\ [167, 167, -2*w - 17],\ [173, 173, 13*w - 89],\ [173, 173, -13*w - 76],\ [193, 193, 11*w - 73],\ [193, 193, -11*w - 62],\ [197, 197, 3*w - 14],\ [197, 197, -3*w - 11],\ [199, 199, 3*w - 25],\ [199, 199, -3*w - 22],\ [233, 233, -w - 16],\ [233, 233, w - 17],\ [239, 239, 7*w - 50],\ [239, 239, -7*w - 43],\ [257, 257, 6*w + 31],\ [257, 257, -6*w + 37],\ [263, 263, 3*w - 11],\ [263, 263, -3*w - 8],\ [277, 277, -12*w - 71],\ [277, 277, 12*w - 83],\ [281, 281, -3*w - 7],\ [281, 281, 3*w - 10],\ [283, 283, -7*w + 44],\ [283, 283, 7*w + 37],\ [311, 311, 3*w - 8],\ [311, 311, -3*w - 5],\ [313, 313, -13*w - 73],\ [313, 313, 13*w - 86],\ [317, 317, -9*w + 58],\ [317, 317, 9*w + 49],\ [331, 331, -5*w - 23],\ [331, 331, 5*w - 28],\ [347, 347, -3*w - 1],\ [347, 347, 3*w - 4],\ [349, 349, 3*w - 28],\ [349, 349, -3*w - 25],\ [353, 353, 3*w - 2],\ [353, 353, 3*w - 1],\ [389, 389, 6*w - 35],\ [389, 389, 6*w + 29],\ [419, 419, 2*w - 25],\ [419, 419, -2*w - 23],\ [431, 431, 10*w - 71],\ [431, 431, -10*w - 61],\ [457, 457, 27*w + 157],\ [457, 457, 27*w - 184],\ [461, 461, -4*w - 31],\ [461, 461, 4*w - 35],\ [467, 467, -w - 22],\ [467, 467, w - 23],\ [487, 487, 8*w + 41],\ [487, 487, -8*w + 49],\ [523, 523, -6*w - 41],\ [523, 523, 6*w - 47],\ [529, 23, -23],\ [547, 547, 4*w - 11],\ [547, 547, -4*w - 7],\ [557, 557, 28*w - 191],\ [557, 557, -28*w - 163],\ [571, 571, 20*w - 133],\ [571, 571, -20*w - 113],\ [577, 577, -3*w - 29],\ [577, 577, 3*w - 32],\ [593, 593, 19*w - 131],\ [593, 593, -19*w - 112],\ [601, 601, 5*w - 22],\ [601, 601, -5*w - 17],\ [617, 617, -18*w - 101],\ [617, 617, 18*w - 119],\ [619, 619, -4*w - 1],\ [619, 619, 4*w - 5],\ [631, 631, 17*w - 112],\ [631, 631, -17*w - 95],\ [641, 641, 15*w + 83],\ [641, 641, 15*w - 98],\ [647, 647, -17*w - 101],\ [647, 647, 17*w - 118],\ [653, 653, -11*w - 68],\ [653, 653, 11*w - 79],\ [659, 659, 5*w - 43],\ [659, 659, -5*w - 38],\ [661, 661, -25*w - 142],\ [661, 661, 25*w - 167],\ [677, 677, 13*w - 92],\ [677, 677, -13*w - 79],\ [709, 709, 5*w - 19],\ [709, 709, -5*w - 14],\ [727, 727, -9*w - 58],\ [727, 727, 9*w - 67],\ [733, 733, 7*w - 38],\ [733, 733, -7*w - 31],\ [739, 739, -18*w - 107],\ [739, 739, 18*w - 125],\ [743, 743, 2*w - 31],\ [743, 743, -2*w - 29],\ [769, 769, -3*w - 32],\ [769, 769, 3*w - 35],\ [773, 773, -w - 28],\ [773, 773, w - 29],\ [797, 797, 21*w - 139],\ [797, 797, -21*w - 118],\ [821, 821, -15*w + 97],\ [821, 821, 15*w + 82],\ [827, 827, 9*w - 53],\ [827, 827, -9*w - 44],\ [829, 829, -19*w - 106],\ [829, 829, 19*w - 125],\ [841, 29, -29],\ [853, 853, 9*w - 68],\ [853, 853, -9*w - 59],\ [907, 907, 3*w - 37],\ [907, 907, -3*w - 34],\ [911, 911, 5*w - 46],\ [911, 911, -5*w - 41],\ [929, 929, -6*w - 19],\ [929, 929, 6*w - 25],\ [941, 941, -20*w - 119],\ [941, 941, 20*w - 139],\ [953, 953, -w - 31],\ [953, 953, w - 32],\ [967, 967, 11*w + 56],\ [967, 967, -11*w + 67],\ [977, 977, 16*w - 113],\ [977, 977, -16*w - 97],\ [991, 991, -8*w - 35],\ [991, 991, 8*w - 43]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^5 + x^4 - 9*x^3 - x^2 + 12*x - 5 K. = NumberField(heckePol) hecke_eigenvalues_array = [0, e, -5/3*e^4 - 3*e^3 + 13*e^2 + 35/3*e - 41/3, -e^4 - 2*e^3 + 7*e^2 + 8*e - 6, e^4 + 2*e^3 - 7*e^2 - 8*e + 6, e^4 + 2*e^3 - 8*e^2 - 9*e + 9, e^4 + 2*e^3 - 8*e^2 - 9*e + 9, -3*e^4 - 5*e^3 + 23*e^2 + 18*e - 22, 3*e^4 + 5*e^3 - 23*e^2 - 18*e + 22, -1/3*e^4 + 4*e^2 - 5/3*e - 19/3, -1/3*e^4 + 4*e^2 - 5/3*e - 19/3, 8/3*e^4 + 4*e^3 - 21*e^2 - 41/3*e + 41/3, 7/3*e^4 + 3*e^3 - 20*e^2 - 25/3*e + 58/3, 7/3*e^4 + 3*e^3 - 20*e^2 - 25/3*e + 58/3, 1/3*e^4 - 4*e^2 - 4/3*e + 10/3, 1/3*e^4 - 4*e^2 - 4/3*e + 10/3, -4*e^4 - 7*e^3 + 31*e^2 + 24*e - 28, 4*e^4 + 7*e^3 - 31*e^2 - 24*e + 28, -3*e^4 - 5*e^3 + 24*e^2 + 21*e - 18, 3*e^4 + 6*e^3 - 22*e^2 - 26*e + 17, 3*e^4 + 6*e^3 - 22*e^2 - 26*e + 17, -2*e^4 - 4*e^3 + 17*e^2 + 19*e - 24, 2*e^4 + 4*e^3 - 17*e^2 - 19*e + 24, -4*e^4 - 6*e^3 + 33*e^2 + 23*e - 33, 4*e^4 + 6*e^3 - 33*e^2 - 23*e + 33, 4*e^2 + 5*e - 14, -4*e^2 - 5*e + 14, -19/3*e^4 - 10*e^3 + 49*e^2 + 112/3*e - 130/3, -19/3*e^4 - 10*e^3 + 49*e^2 + 112/3*e - 130/3, 7*e^4 + 10*e^3 - 57*e^2 - 34*e + 50, -7*e^4 - 10*e^3 + 57*e^2 + 34*e - 50, -25/3*e^4 - 14*e^3 + 66*e^2 + 163/3*e - 190/3, -25/3*e^4 - 14*e^3 + 66*e^2 + 163/3*e - 190/3, -12*e^4 - 20*e^3 + 92*e^2 + 70*e - 80, 6*e^4 + 9*e^3 - 47*e^2 - 29*e + 42, -6*e^4 - 9*e^3 + 47*e^2 + 29*e - 42, 8*e^4 + 14*e^3 - 64*e^2 - 56*e + 65, -8*e^4 - 14*e^3 + 64*e^2 + 56*e - 65, -e^3 + 3*e^2 + 12*e - 10, -e^3 + 3*e^2 + 12*e - 10, 3*e^4 + 6*e^3 - 21*e^2 - 19*e + 8, -3*e^4 - 6*e^3 + 21*e^2 + 19*e - 8, 9*e^4 + 15*e^3 - 70*e^2 - 50*e + 62, 9*e^4 + 15*e^3 - 70*e^2 - 50*e + 62, -6*e^4 - 9*e^3 + 47*e^2 + 24*e - 50, 6*e^4 + 9*e^3 - 47*e^2 - 24*e + 50, e^4 - e^3 - 10*e^2 + 6*e + 5, -e^4 + e^3 + 10*e^2 - 6*e - 5, 3*e^4 + 7*e^3 - 22*e^2 - 28*e + 22, -3*e^4 - 7*e^3 + 22*e^2 + 28*e - 22, -12*e^4 - 22*e^3 + 89*e^2 + 84*e - 81, 12*e^4 + 22*e^3 - 89*e^2 - 84*e + 81, 26/3*e^4 + 17*e^3 - 67*e^2 - 209/3*e + 221/3, 26/3*e^4 + 17*e^3 - 67*e^2 - 209/3*e + 221/3, 6*e^4 + 13*e^3 - 43*e^2 - 60*e + 41, -6*e^4 - 13*e^3 + 43*e^2 + 60*e - 41, 10/3*e^4 + 5*e^3 - 22*e^2 - 43/3*e - 5/3, 10/3*e^4 + 5*e^3 - 22*e^2 - 43/3*e - 5/3, -7*e^4 - 11*e^3 + 59*e^2 + 49*e - 74, 7*e^4 + 11*e^3 - 59*e^2 - 49*e + 74, -46/3*e^4 - 27*e^3 + 119*e^2 + 304/3*e - 370/3, -46/3*e^4 - 27*e^3 + 119*e^2 + 304/3*e - 370/3, -e^4 + 9*e^2 - 7*e - 15, e^4 - 9*e^2 + 7*e + 15, -9*e^4 - 15*e^3 + 66*e^2 + 48*e - 60, -9*e^4 - 15*e^3 + 66*e^2 + 48*e - 60, 2*e^3 + e^2 - 18*e + 10, -2*e^3 - e^2 + 18*e - 10, 3*e^4 + 6*e^3 - 19*e^2 - 20*e + 12, 3*e^4 + 6*e^3 - 19*e^2 - 20*e + 12, -6*e^4 - 13*e^3 + 47*e^2 + 59*e - 49, 6*e^4 + 13*e^3 - 47*e^2 - 59*e + 49, -4*e^4 - 9*e^3 + 30*e^2 + 33*e - 47, 4*e^4 + 9*e^3 - 30*e^2 - 33*e + 47, -8*e^4 - 14*e^3 + 60*e^2 + 43*e - 47, 8*e^4 + 14*e^3 - 60*e^2 - 43*e + 47, -8*e^4 - 14*e^3 + 59*e^2 + 47*e - 39, 8*e^4 + 14*e^3 - 59*e^2 - 47*e + 39, 13*e^4 + 22*e^3 - 98*e^2 - 72*e + 72, 13*e^4 + 22*e^3 - 98*e^2 - 72*e + 72, -e^4 - 4*e^3 + 3*e^2 + 26*e + 5, e^4 + 4*e^3 - 3*e^2 - 26*e - 5, e^3 + 8*e^2 - 25, -e^3 - 8*e^2 + 25, e^4 + e^3 - 2*e^2 + 12*e - 20, e^4 + e^3 - 2*e^2 + 12*e - 20, -25/3*e^4 - 14*e^3 + 61*e^2 + 151/3*e - 187/3, -25/3*e^4 - 14*e^3 + 61*e^2 + 151/3*e - 187/3, 10/3*e^4 + 4*e^3 - 27*e^2 - 22/3*e - 8/3, -47/3*e^4 - 23*e^3 + 124*e^2 + 242/3*e - 329/3, -47/3*e^4 - 23*e^3 + 124*e^2 + 242/3*e - 329/3, -12*e^4 - 22*e^3 + 91*e^2 + 80*e - 90, 12*e^4 + 22*e^3 - 91*e^2 - 80*e + 90, -12*e^4 - 20*e^3 + 89*e^2 + 67*e - 79, -12*e^4 - 20*e^3 + 89*e^2 + 67*e - 79, 35/3*e^4 + 19*e^3 - 94*e^2 - 215/3*e + 275/3, 35/3*e^4 + 19*e^3 - 94*e^2 - 215/3*e + 275/3, -10*e^4 - 22*e^3 + 69*e^2 + 92*e - 76, 10*e^4 + 22*e^3 - 69*e^2 - 92*e + 76, 4*e^2 + 2*e - 29, 4*e^2 + 2*e - 29, 4*e^4 + 11*e^3 - 27*e^2 - 55*e + 15, -4*e^4 - 11*e^3 + 27*e^2 + 55*e - 15, 11/3*e^4 + 5*e^3 - 30*e^2 - 29/3*e + 134/3, 11/3*e^4 + 5*e^3 - 30*e^2 - 29/3*e + 134/3, -3*e^4 - 10*e^3 + 21*e^2 + 48*e - 36, -3*e^4 - 10*e^3 + 21*e^2 + 48*e - 36, -5*e^4 - 6*e^3 + 40*e^2 + 25*e - 25, 5*e^4 + 6*e^3 - 40*e^2 - 25*e + 25, -4*e^4 - 8*e^3 + 27*e^2 + 26*e - 35, 4*e^4 + 8*e^3 - 27*e^2 - 26*e + 35, e^4 - 15*e^2 + 4*e + 45, -e^4 + 15*e^2 - 4*e - 45, 11*e^4 + 20*e^3 - 91*e^2 - 90*e + 100, -11*e^4 - 20*e^3 + 91*e^2 + 90*e - 100, 35/3*e^4 + 21*e^3 - 87*e^2 - 233/3*e + 278/3, 35/3*e^4 + 21*e^3 - 87*e^2 - 233/3*e + 278/3, 11*e^4 + 23*e^3 - 81*e^2 - 101*e + 88, -11*e^4 - 23*e^3 + 81*e^2 + 101*e - 88, 9*e^4 + 11*e^3 - 77*e^2 - 28*e + 78, 9*e^4 + 11*e^3 - 77*e^2 - 28*e + 78, -14*e^4 - 21*e^3 + 113*e^2 + 65*e - 110, -14*e^4 - 21*e^3 + 113*e^2 + 65*e - 110, -31/3*e^4 - 16*e^3 + 81*e^2 + 124/3*e - 265/3, -31/3*e^4 - 16*e^3 + 81*e^2 + 124/3*e - 265/3, -56/3*e^4 - 34*e^3 + 145*e^2 + 401/3*e - 455/3, -56/3*e^4 - 34*e^3 + 145*e^2 + 401/3*e - 455/3, 5*e^4 + 5*e^3 - 44*e^2 - 6*e + 44, -5*e^4 - 5*e^3 + 44*e^2 + 6*e - 44, -14/3*e^4 - 10*e^3 + 33*e^2 + 155/3*e - 128/3, -14/3*e^4 - 10*e^3 + 33*e^2 + 155/3*e - 128/3, -7*e^4 - 15*e^3 + 47*e^2 + 68*e - 35, 7*e^4 + 15*e^3 - 47*e^2 - 68*e + 35, e^4 - 2*e^3 - 12*e^2 + 17*e + 20, -e^4 + 2*e^3 + 12*e^2 - 17*e - 20, -20*e^4 - 32*e^3 + 161*e^2 + 126*e - 164, 20*e^4 + 32*e^3 - 161*e^2 - 126*e + 164, e^4 - 8*e^2 + 2*e - 18, -e^4 + 8*e^2 - 2*e + 18, -8/3*e^4 - 2*e^3 + 17*e^2 - 13/3*e - 5/3, -8/3*e^4 - 2*e^3 + 17*e^2 - 13/3*e - 5/3, -11/3*e^4 - 4*e^3 + 29*e^2 + 8/3*e - 194/3, 34/3*e^4 + 14*e^3 - 94*e^2 - 130/3*e + 280/3, 34/3*e^4 + 14*e^3 - 94*e^2 - 130/3*e + 280/3, -5*e^4 - 11*e^3 + 30*e^2 + 46*e - 8, -5*e^4 - 11*e^3 + 30*e^2 + 46*e - 8, 23*e^4 + 40*e^3 - 179*e^2 - 137*e + 176, -23*e^4 - 40*e^3 + 179*e^2 + 137*e - 176, 20*e^4 + 32*e^3 - 162*e^2 - 114*e + 153, -20*e^4 - 32*e^3 + 162*e^2 + 114*e - 153, -e^4 - e^3 + 10*e^2 + 7*e - 36, e^4 + e^3 - 10*e^2 - 7*e + 36, -9*e^4 - 13*e^3 + 75*e^2 + 45*e - 85, 9*e^4 + 13*e^3 - 75*e^2 - 45*e + 85, -11*e^4 - 20*e^3 + 80*e^2 + 68*e - 57, -11*e^4 - 20*e^3 + 80*e^2 + 68*e - 57, e^4 + 3*e^3 - 4*e^2 - e - 25, -e^4 - 3*e^3 + 4*e^2 + e + 25, 10*e^4 + 18*e^3 - 71*e^2 - 66*e + 56, 10*e^4 + 18*e^3 - 71*e^2 - 66*e + 56] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([3, 3, w + 6])] = -1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]