/* 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([-27, -1, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([4, 2, 2]) primes_array = [ [3, 3, -w + 6],\ [3, 3, w + 5],\ [4, 2, 2],\ [5, 5, -3*w + 17],\ [5, 5, -3*w - 14],\ [7, 7, w - 5],\ [7, 7, w + 4],\ [29, 29, -w - 7],\ [29, 29, -w + 8],\ [31, 31, -5*w + 28],\ [31, 31, -5*w - 23],\ [43, 43, 6*w + 29],\ [43, 43, -6*w + 35],\ [61, 61, 3*w - 19],\ [61, 61, -3*w - 16],\ [71, 71, -7*w - 34],\ [71, 71, 7*w - 41],\ [73, 73, 2*w - 7],\ [73, 73, -2*w - 5],\ [83, 83, -w - 10],\ [83, 83, w - 11],\ [89, 89, 3*w - 14],\ [89, 89, 3*w + 11],\ [97, 97, 3*w + 17],\ [97, 97, 3*w - 20],\ [109, 109, 2*w - 1],\ [113, 113, -3*w - 10],\ [113, 113, 3*w - 13],\ [121, 11, -11],\ [131, 131, 5*w - 31],\ [131, 131, -5*w - 26],\ [137, 137, -9*w - 41],\ [137, 137, 9*w - 50],\ [157, 157, -14*w + 79],\ [157, 157, 14*w + 65],\ [169, 13, -13],\ [173, 173, -3*w - 7],\ [173, 173, 3*w - 10],\ [191, 191, -10*w - 49],\ [191, 191, 10*w - 59],\ [193, 193, 22*w + 103],\ [193, 193, 22*w - 125],\ [197, 197, 6*w + 25],\ [197, 197, -6*w + 31],\ [211, 211, -4*w - 13],\ [211, 211, 4*w - 17],\ [223, 223, 8*w + 35],\ [223, 223, -8*w + 43],\ [227, 227, 9*w - 49],\ [227, 227, 9*w + 40],\ [233, 233, 3*w - 5],\ [233, 233, -3*w - 2],\ [239, 239, -3*w - 1],\ [239, 239, 3*w - 4],\ [263, 263, 33*w - 188],\ [263, 263, 33*w + 155],\ [281, 281, 8*w - 49],\ [281, 281, -8*w - 41],\ [289, 17, -17],\ [293, 293, 4*w - 29],\ [293, 293, 4*w + 25],\ [307, 307, 3*w - 25],\ [307, 307, -3*w - 22],\ [311, 311, -5*w - 29],\ [311, 311, 5*w - 34],\ [331, 331, 19*w + 88],\ [331, 331, -19*w + 107],\ [347, 347, 23*w - 133],\ [347, 347, 36*w - 205],\ [349, 349, 15*w - 88],\ [349, 349, -15*w - 73],\ [353, 353, -w - 19],\ [353, 353, w - 20],\ [361, 19, -19],\ [373, 373, 23*w - 130],\ [373, 373, 23*w + 107],\ [401, 401, -9*w + 47],\ [401, 401, 9*w + 38],\ [409, 409, 5*w - 19],\ [409, 409, -5*w - 14],\ [421, 421, -10*w + 53],\ [421, 421, 10*w + 43],\ [431, 431, -15*w - 68],\ [431, 431, 15*w - 83],\ [433, 433, 43*w - 245],\ [433, 433, 43*w + 202],\ [439, 439, 35*w - 199],\ [439, 439, 35*w + 164],\ [443, 443, 12*w + 53],\ [443, 443, 12*w - 65],\ [457, 457, 3*w - 28],\ [457, 457, -3*w - 25],\ [461, 461, 21*w - 118],\ [461, 461, 21*w + 97],\ [463, 463, -6*w - 35],\ [463, 463, 6*w - 41],\ [467, 467, 2*w - 25],\ [467, 467, -2*w - 23],\ [479, 479, -w - 22],\ [479, 479, w - 23],\ [499, 499, -5*w - 11],\ [499, 499, 5*w - 16],\ [509, 509, -5*w - 32],\ [509, 509, 5*w - 37],\ [523, 523, -7*w - 25],\ [523, 523, 7*w - 32],\ [529, 23, -23],\ [541, 541, 11*w + 47],\ [541, 541, -11*w + 58],\ [557, 557, 7*w - 47],\ [557, 557, -7*w - 40],\ [571, 571, -5*w - 8],\ [571, 571, 5*w - 13],\ [593, 593, -16*w - 79],\ [593, 593, 16*w - 95],\ [619, 619, 6*w - 43],\ [619, 619, 6*w + 37],\ [647, 647, -9*w + 44],\ [647, 647, -9*w - 35],\ [653, 653, -4*w - 31],\ [653, 653, 4*w - 35],\ [659, 659, 45*w + 211],\ [659, 659, 45*w - 256],\ [661, 661, 5*w - 7],\ [661, 661, -5*w - 2],\ [683, 683, 27*w + 125],\ [683, 683, -27*w + 152],\ [727, 727, -27*w - 130],\ [727, 727, 27*w - 157],\ [743, 743, 14*w - 85],\ [743, 743, -14*w - 71],\ [751, 751, 40*w + 187],\ [751, 751, 40*w - 227],\ [797, 797, -13*w - 67],\ [797, 797, 13*w - 80],\ [809, 809, -30*w + 169],\ [809, 809, 30*w + 139],\ [811, 811, 3*w - 34],\ [811, 811, -3*w - 31],\ [823, 823, -13*w + 68],\ [823, 823, 13*w + 55],\ [827, 827, 7*w - 50],\ [827, 827, 7*w + 43],\ [829, 829, -7*w - 19],\ [829, 829, 7*w - 26],\ [857, 857, -8*w - 47],\ [857, 857, 8*w - 55],\ [863, 863, -11*w + 70],\ [863, 863, -11*w - 59],\ [877, 877, -3*w - 32],\ [877, 877, 3*w - 35],\ [881, 881, -6*w - 7],\ [881, 881, 6*w - 13],\ [887, 887, -21*w - 95],\ [887, 887, 21*w - 116],\ [907, 907, 19*w - 104],\ [907, 907, 19*w + 85],\ [947, 947, 9*w - 40],\ [947, 947, -9*w - 31],\ [953, 953, 29*w + 140],\ [953, 953, -29*w + 169],\ [977, 977, 6*w - 5],\ [977, 977, 6*w - 1],\ [997, 997, -26*w - 119],\ [997, 997, 26*w - 145]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^5 + x^4 - 11*x^3 - 12*x^2 + 16*x + 2 K. = NumberField(heckePol) hecke_eigenvalues_array = [e, e, -1, -e^4 + e^3 + 8*e^2 - 3*e - 3, -e^4 + e^3 + 8*e^2 - 3*e - 3, -e^3 + e^2 + 7*e - 2, -e^3 + e^2 + 7*e - 2, -2*e^4 + 2*e^3 + 15*e^2 - 5*e - 1, -2*e^4 + 2*e^3 + 15*e^2 - 5*e - 1, e^3 - e^2 - 9*e, e^3 - e^2 - 9*e, 2*e^4 - 3*e^3 - 14*e^2 + 13*e - 4, 2*e^4 - 3*e^3 - 14*e^2 + 13*e - 4, -e^4 + e^3 + 10*e^2 - 3*e - 9, -e^4 + e^3 + 10*e^2 - 3*e - 9, 3*e^3 - 3*e^2 - 23*e + 10, 3*e^3 - 3*e^2 - 23*e + 10, e^4 - e^3 - 8*e^2 + 6*e + 1, e^4 - e^3 - 8*e^2 + 6*e + 1, -e^4 - e^3 + 10*e^2 + 10*e - 6, -e^4 - e^3 + 10*e^2 + 10*e - 6, -2*e^4 + 2*e^3 + 17*e^2 - 2*e - 15, -2*e^4 + 2*e^3 + 17*e^2 - 2*e - 15, e^4 - 4*e^3 - 4*e^2 + 22*e - 11, e^4 - 4*e^3 - 4*e^2 + 22*e - 11, -6*e^4 + 6*e^3 + 48*e^2 - 14*e - 10, 4*e^4 - 5*e^3 - 29*e^2 + 20*e + 4, 4*e^4 - 5*e^3 - 29*e^2 + 20*e + 4, 2*e^4 - 3*e^3 - 16*e^2 + 16*e + 19, -5*e^4 + 7*e^3 + 38*e^2 - 32*e - 16, -5*e^4 + 7*e^3 + 38*e^2 - 32*e - 16, 2*e^4 - 3*e^3 - 15*e^2 + 14*e - 3, 2*e^4 - 3*e^3 - 15*e^2 + 14*e - 3, -2*e^3 + e^2 + 15*e - 5, -2*e^3 + e^2 + 15*e - 5, -e^4 + 2*e^3 + 6*e^2 - 10*e + 22, -2*e^4 + 2*e^3 + 16*e^2 - 6*e + 6, -2*e^4 + 2*e^3 + 16*e^2 - 6*e + 6, 2*e^4 - 20*e^2 - 12*e + 18, 2*e^4 - 20*e^2 - 12*e + 18, e^4 - e^3 - 10*e^2 + 4*e + 11, e^4 - e^3 - 10*e^2 + 4*e + 11, -4*e^3 + 3*e^2 + 25*e - 9, -4*e^3 + 3*e^2 + 25*e - 9, 3*e^4 - 5*e^3 - 24*e^2 + 26*e + 14, 3*e^4 - 5*e^3 - 24*e^2 + 26*e + 14, -4*e^4 + 3*e^3 + 31*e^2 - 5*e - 10, -4*e^4 + 3*e^3 + 31*e^2 - 5*e - 10, -e^4 + 4*e^3 + 4*e^2 - 27*e + 22, -e^4 + 4*e^3 + 4*e^2 - 27*e + 22, e^4 + 2*e^3 - 12*e^2 - 20*e + 17, e^4 + 2*e^3 - 12*e^2 - 20*e + 17, 4*e^4 - 6*e^3 - 26*e^2 + 22*e - 22, 4*e^4 - 6*e^3 - 26*e^2 + 22*e - 22, 2*e^4 - 4*e^3 - 10*e^2 + 16*e - 16, 2*e^4 - 4*e^3 - 10*e^2 + 16*e - 16, -6*e^3 + 7*e^2 + 48*e - 13, -6*e^3 + 7*e^2 + 48*e - 13, -e^4 - 3*e^3 + 9*e^2 + 24*e + 8, -6*e^4 + 8*e^3 + 49*e^2 - 29*e - 25, -6*e^4 + 8*e^3 + 49*e^2 - 29*e - 25, -4*e^4 + 4*e^3 + 30*e^2 - 11*e - 10, -4*e^4 + 4*e^3 + 30*e^2 - 11*e - 10, 6*e^4 - 13*e^3 - 41*e^2 + 69*e - 2, 6*e^4 - 13*e^3 - 41*e^2 + 69*e - 2, -3*e^4 + 4*e^3 + 20*e^2 - 12*e + 22, -3*e^4 + 4*e^3 + 20*e^2 - 12*e + 22, 3*e^4 - 10*e^3 - 14*e^2 + 64*e - 22, 3*e^4 - 10*e^3 - 14*e^2 + 64*e - 22, 3*e^4 - e^3 - 26*e^2 - 7*e + 5, 3*e^4 - e^3 - 26*e^2 - 7*e + 5, -2*e^3 + 7*e^2 + 12*e - 33, -2*e^3 + 7*e^2 + 12*e - 33, 4*e^4 + e^3 - 36*e^2 - 30*e + 35, 3*e^2 - e - 15, 3*e^2 - e - 15, -4*e^4 + 9*e^3 + 25*e^2 - 48*e - 3, -4*e^4 + 9*e^3 + 25*e^2 - 48*e - 3, -e^4 + 4*e^3 + 4*e^2 - 26*e + 15, -e^4 + 4*e^3 + 4*e^2 - 26*e + 15, 8*e^4 - 10*e^3 - 59*e^2 + 37*e + 5, 8*e^4 - 10*e^3 - 59*e^2 + 37*e + 5, 2*e^4 + 3*e^3 - 19*e^2 - 35*e + 12, 2*e^4 + 3*e^3 - 19*e^2 - 35*e + 12, 2*e^4 - 19*e^2 - 2*e + 14, 2*e^4 - 19*e^2 - 2*e + 14, 6*e^4 - 10*e^3 - 48*e^2 + 44*e + 26, 6*e^4 - 10*e^3 - 48*e^2 + 44*e + 26, -e^4 + 6*e^3 + 2*e^2 - 37*e + 22, -e^4 + 6*e^3 + 2*e^2 - 37*e + 22, 4*e^4 - 3*e^3 - 35*e^2 + 2*e + 23, 4*e^4 - 3*e^3 - 35*e^2 + 2*e + 23, -5*e^4 + 7*e^3 + 37*e^2 - 26*e + 6, -5*e^4 + 7*e^3 + 37*e^2 - 26*e + 6, -2*e^4 + e^3 + 19*e^2 - e - 28, -2*e^4 + e^3 + 19*e^2 - e - 28, 3*e^4 - 2*e^3 - 20*e^2 - 12, 3*e^4 - 2*e^3 - 20*e^2 - 12, 2*e^4 - e^3 - 13*e^2 - 7*e - 12, 2*e^4 - e^3 - 13*e^2 - 7*e - 12, -e^4 + 4*e^3 + 4*e^2 - 24*e + 2, -e^4 + 4*e^3 + 4*e^2 - 24*e + 2, -2*e^4 - 6*e^3 + 26*e^2 + 60*e - 30, -2*e^4 - 6*e^3 + 26*e^2 + 60*e - 30, -5*e^4 - e^3 + 44*e^2 + 32*e - 24, -5*e^4 - e^3 + 44*e^2 + 32*e - 24, -6*e^4 + 16*e^3 + 39*e^2 - 88*e + 25, 6*e^4 - 10*e^3 - 39*e^2 + 45*e - 15, 6*e^4 - 10*e^3 - 39*e^2 + 45*e - 15, 2*e^4 - 20*e^2 - 18*e + 18, 2*e^4 - 20*e^2 - 18*e + 18, -4*e^2 - 3*e + 8, -4*e^2 - 3*e + 8, -10*e^4 + 6*e^3 + 81*e^2 - 4*e - 30, -10*e^4 + 6*e^3 + 81*e^2 - 4*e - 30, 3*e^4 - 3*e^3 - 20*e^2 + 12*e - 14, 3*e^4 - 3*e^3 - 20*e^2 + 12*e - 14, 8*e^4 - 6*e^3 - 62*e^2 + 6, 8*e^4 - 6*e^3 - 62*e^2 + 6, 7*e^4 - 13*e^3 - 50*e^2 + 63*e - 3, 7*e^4 - 13*e^3 - 50*e^2 + 63*e - 3, -2*e^4 - 8*e^3 + 28*e^2 + 62*e - 32, -2*e^4 - 8*e^3 + 28*e^2 + 62*e - 32, -2*e^4 - 6*e^3 + 26*e^2 + 56*e - 42, -2*e^4 - 6*e^3 + 26*e^2 + 56*e - 42, 4*e^4 - 7*e^3 - 36*e^2 + 43*e + 32, 4*e^4 - 7*e^3 - 36*e^2 + 43*e + 32, 4*e^4 + 2*e^3 - 34*e^2 - 34*e + 6, 4*e^4 + 2*e^3 - 34*e^2 - 34*e + 6, 6*e^4 - 6*e^3 - 44*e^2 + 16*e - 18, 6*e^4 - 6*e^3 - 44*e^2 + 16*e - 18, 4*e^4 - 5*e^3 - 27*e^2 + 27*e - 28, 4*e^4 - 5*e^3 - 27*e^2 + 27*e - 28, -2*e^4 + 22*e^2 + 2*e - 30, -2*e^4 + 22*e^2 + 2*e - 30, -7*e^4 + 11*e^3 + 54*e^2 - 60*e - 21, -7*e^4 + 11*e^3 + 54*e^2 - 60*e - 21, 10*e^4 - 7*e^3 - 78*e^2 + 9*e + 12, 10*e^4 - 7*e^3 - 78*e^2 + 9*e + 12, -2*e^4 + 8*e^3 + 10*e^2 - 50*e + 24, -2*e^4 + 8*e^3 + 10*e^2 - 50*e + 24, 6*e^4 - e^3 - 58*e^2 - 19*e + 50, 6*e^4 - e^3 - 58*e^2 - 19*e + 50, -3*e^4 + 3*e^3 + 20*e^2 + 7*e + 13, -3*e^4 + 3*e^3 + 20*e^2 + 7*e + 13, -4*e^4 + 15*e^3 + 19*e^2 - 96*e + 27, -4*e^4 + 15*e^3 + 19*e^2 - 96*e + 27, 6*e^4 - 9*e^3 - 51*e^2 + 39*e + 24, 6*e^4 - 9*e^3 - 51*e^2 + 39*e + 24, 3*e^4 + 5*e^3 - 31*e^2 - 54*e + 22, 3*e^4 + 5*e^3 - 31*e^2 - 54*e + 22, -5*e^4 + 39*e^2 + 20*e + 13, -5*e^4 + 39*e^2 + 20*e + 13, e^3 + 3*e^2 - 11*e - 10, e^3 + 3*e^2 - 11*e - 10, 8*e^4 - 10*e^3 - 62*e^2 + 35*e + 12, 8*e^4 - 10*e^3 - 62*e^2 + 35*e + 12, 8*e^4 - 2*e^3 - 74*e^2 - 26*e + 56, 8*e^4 - 2*e^3 - 74*e^2 - 26*e + 56, 6*e^4 - e^3 - 54*e^2 - 6*e + 35, 6*e^4 - e^3 - 54*e^2 - 6*e + 35, 4*e^4 - 11*e^3 - 23*e^2 + 54*e - 25, 4*e^4 - 11*e^3 - 23*e^2 + 54*e - 25, 7*e^4 - 9*e^3 - 57*e^2 + 34*e + 58, 7*e^4 - 9*e^3 - 57*e^2 + 34*e + 58] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([4, 2, 2])] = 1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]