/* 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([12,6,2*w + 10]) 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^3 - 3*x^2 - 3*x + 8 K. = NumberField(heckePol) hecke_eigenvalues_array = [e, 1, 1, e - 1, -e^2 + e + 3, -e^2 + 2*e + 4, -e, 2*e^2 - 3*e - 5, e^2 - e - 9, 2*e, 2*e^2 - 12, -3*e + 8, e + 4, 3*e^2 - 7*e - 9, -3*e^2 + 5*e + 3, 12, 4*e^2 - 5*e - 8, -3*e^2 + e + 19, e^2 - e - 13, -4*e^2 + e + 24, -2*e^2 - 2*e + 16, -e^2 - 3*e + 7, e^2 + 2*e - 9, -e^2 + e - 1, -e^2 + 5*e + 7, -3*e^2 + 14, -2*e^2 + 6*e + 2, -5*e^2 + 8*e + 18, e^2 + e - 9, -5*e + 8, 2*e^2 - 4*e + 8, -2*e^2 + e + 7, e^2 + 5*e - 9, e^2 - 8*e - 9, 2*e^2 - 7*e - 5, e^2 - 8*e - 6, 4*e^2 - 9*e - 10, 2*e^2 - 6*e - 2, -6*e^2 + 4*e + 32, -e^2 + 2*e + 8, e^2 + 2*e - 25, 2*e^2 - 3*e - 9, -2*e^2 + 5*e + 3, 5*e^2 - 7*e - 25, 4*e^2 - 8*e, 2*e^2 - 24, -4*e^2 + 6*e + 12, 2*e^2 - 4*e + 16, 2*e^2 - 20, 3*e^2 - 6*e, 5*e^2 - 11*e - 9, -3*e^2 + 14*e + 7, -8*e - 4, 4*e^2 - 7*e - 12, 6*e^2 - 6*e - 24, -4*e + 16, -e^2 - e + 11, -3*e^2 - 6*e + 27, 8*e^2 - 6*e - 30, -e^2 + 7*e + 15, 2*e^2 - 3*e - 5, 2*e^2 + 2*e - 8, -2*e^2 + 4*e - 12, 8*e^2 - 8*e - 36, -6*e^2 + 14*e + 16, 2*e^2 - 4*e + 16, -8*e^2 + 10*e + 24, -5*e^2 + 2*e + 24, 6*e^2 - 24, 5*e^2 - 3*e - 33, -7*e^2 + e + 35, e^2 + 3*e - 25, -4*e^2 - 3*e + 31, -8*e^2 + 13*e + 15, -10*e^2 + 17*e + 31, -4*e^2 + e + 11, 3*e^2 - 3*e - 21, -4*e^2 + e + 27, 3*e^2 - 9*e - 1, -9*e^2 + 9*e + 23, e^2 - 13*e - 1, 7*e^2 - 15*e - 5, -4*e^2 + 9*e + 16, -12*e^2 + 18*e + 36, 9*e^2 - 16*e - 30, e^2 - 4*e + 2, 2*e^2 - 10*e - 8, -6*e^2 + 6*e + 32, -10*e^2 + 8*e + 44, -4*e^2 - 4*e + 32, -3*e^2 - 3*e + 11, 6*e^2 - 11*e - 29, 6*e^2 - 12*e - 18, -8*e^2 + 12*e + 14, -6*e^2 + 18*e + 20, 8*e^2 - 16*e - 8, 8*e^2 - 11*e - 12, -2*e^2 - 4, -10*e + 4, -4*e^2 + 5*e + 32, 13*e^2 - 18*e - 32, -4*e + 24, 2*e^2 + 10*e - 18, 4*e^2 - 9*e - 10, -2*e^2 - 8*e + 24, 6*e^2 - 6*e - 16, e^2 - 3*e - 29, 5*e^2 - 4*e - 41, e^2 + 4*e - 21, -8*e^2 + 20*e + 30, -4*e^2 + 8*e + 14, -5*e^2 + 6*e + 16, -11*e^2 + 10*e + 36, -8*e^2 + 12*e + 2, -3*e - 6, -2*e^2 + 8*e - 4, 2*e^2 - 12, -8*e^2 + 8*e + 48, -12*e^2 + 17*e + 40, -6*e^2 + 13*e + 23, 5*e^2 - e - 13, 3*e^2 + 2*e - 20, 6*e^2 - 36, 4*e^2 + 3*e - 2, 5*e^2 - 8*e - 10, 2*e^2 - 6*e + 4, 13*e^2 - 26*e - 32, -4*e^2 + 23*e + 16, -4*e^2 + 20*e - 8, -8*e^2 + 8*e + 48, 4*e^2 - e - 48, -13*e^2 + 10*e + 68, 16*e - 8, 4*e^2 + e - 26, -2*e^2 + 22*e - 2, 4*e^2 - 7*e - 45, 3*e^2 - e + 19, 2*e^2 + 2*e - 32, -2*e^2 - 8*e + 12, 12*e^2 - 5*e - 68, 4*e^2 - 5*e - 36, 10*e^2 - 22*e - 12, 4*e^2 + 8*e - 48, 5*e^2 - 15*e - 21, -9*e^2 + 17*e + 39, -3*e^2 + 14*e + 19, -5*e^2 + 9*e + 11, 6*e^2 - 36, -11*e^2 + 26*e + 24, 4*e^2 - 11*e - 42, -13*e^2 + 20*e + 46, -e^2 + 3*e - 5, e^2 - 12*e + 11, e^2 + 2*e + 24, -6*e^2 + 4*e + 44, -8*e^2 + 28*e + 24, 4*e^2 - 6*e + 28, 4*e^2 + 4*e - 32, 8*e^2 + 7*e - 48, -3*e^2 - 12*e + 27, -e^2 + 13*e + 3, 9*e^2 - 11*e - 49, e^2 - 15*e - 1, 4*e^2 - 26, -8*e^2 + 70] 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 + 5])] = -1 AL_eigenvalues[ZF.ideal([4,2,2])] = -1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]