/* 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([-49, -1, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([4, 2, 2]) primes_array = [ [4, 2, 2],\ [7, 7, w - 7],\ [7, 7, w + 6],\ [9, 3, 3],\ [19, 19, w + 5],\ [19, 19, w - 6],\ [23, 23, w + 8],\ [23, 23, -w + 9],\ [25, 5, 5],\ [29, 29, -w - 4],\ [29, 29, w - 5],\ [37, 37, -w - 3],\ [37, 37, w - 4],\ [41, 41, -w - 9],\ [41, 41, w - 10],\ [43, 43, -w - 2],\ [43, 43, w - 3],\ [47, 47, -w - 1],\ [47, 47, w - 2],\ [53, 53, 2*w - 13],\ [53, 53, -2*w - 11],\ [59, 59, -4*w - 25],\ [59, 59, -4*w + 29],\ [61, 61, -w - 10],\ [61, 61, w - 11],\ [83, 83, -w - 11],\ [83, 83, w - 12],\ [97, 97, 2*w - 11],\ [97, 97, -2*w - 9],\ [101, 101, 3*w - 20],\ [101, 101, -3*w - 17],\ [107, 107, -w - 12],\ [107, 107, w - 13],\ [109, 109, -5*w - 31],\ [109, 109, -5*w + 36],\ [121, 11, -11],\ [127, 127, 2*w - 19],\ [127, 127, -2*w - 17],\ [137, 137, -3*w - 16],\ [137, 137, 3*w - 19],\ [157, 157, -3*w - 23],\ [157, 157, 3*w - 26],\ [163, 163, -4*w + 27],\ [163, 163, 4*w + 23],\ [169, 13, -13],\ [173, 173, -6*w - 37],\ [173, 173, -6*w + 43],\ [181, 181, 2*w - 5],\ [181, 181, -2*w - 3],\ [191, 191, -w - 15],\ [191, 191, w - 16],\ [193, 193, 2*w - 3],\ [193, 193, -2*w - 1],\ [197, 197, 2*w - 1],\ [223, 223, -w - 16],\ [223, 223, w - 17],\ [233, 233, -3*w - 13],\ [233, 233, 3*w - 16],\ [239, 239, -5*w + 34],\ [239, 239, -5*w - 29],\ [251, 251, 5*w - 41],\ [251, 251, 5*w + 36],\ [257, 257, -w - 17],\ [257, 257, w - 18],\ [289, 17, -17],\ [293, 293, -w - 18],\ [293, 293, w - 19],\ [311, 311, -3*w - 10],\ [311, 311, 3*w - 13],\ [313, 313, -3*w - 26],\ [313, 313, 3*w - 29],\ [331, 331, -w - 19],\ [331, 331, w - 20],\ [347, 347, 4*w - 23],\ [347, 347, -4*w - 19],\ [353, 353, 3*w - 11],\ [353, 353, -3*w - 8],\ [379, 379, 2*w - 25],\ [379, 379, -2*w - 23],\ [401, 401, 3*w - 8],\ [401, 401, -3*w - 5],\ [409, 409, 5*w - 43],\ [409, 409, 5*w + 38],\ [419, 419, -5*w - 26],\ [419, 419, 5*w - 31],\ [431, 431, 3*w - 5],\ [431, 431, -3*w - 2],\ [433, 433, -7*w + 48],\ [433, 433, -7*w - 41],\ [443, 443, 3*w - 2],\ [443, 443, 3*w - 1],\ [449, 449, -9*w - 55],\ [449, 449, -9*w + 64],\ [457, 457, -w - 22],\ [457, 457, w - 23],\ [479, 479, 2*w - 27],\ [479, 479, -2*w - 25],\ [487, 487, -3*w - 29],\ [487, 487, 3*w - 32],\ [491, 491, -5*w - 39],\ [491, 491, 5*w - 44],\ [499, 499, 4*w - 19],\ [499, 499, -4*w - 15],\ [503, 503, -w - 23],\ [503, 503, w - 24],\ [521, 521, 11*w - 86],\ [521, 521, -7*w + 47],\ [557, 557, 7*w + 51],\ [557, 557, 7*w - 58],\ [563, 563, -4*w - 13],\ [563, 563, 4*w - 17],\ [569, 569, -10*w - 61],\ [569, 569, -10*w + 71],\ [587, 587, 2*w - 29],\ [587, 587, -2*w - 27],\ [601, 601, -w - 25],\ [601, 601, w - 26],\ [607, 607, -7*w - 39],\ [607, 607, 7*w - 46],\ [613, 613, 3*w - 34],\ [613, 613, -3*w - 31],\ [617, 617, -6*w - 31],\ [617, 617, 6*w - 37],\ [619, 619, 4*w - 15],\ [619, 619, -4*w - 11],\ [631, 631, 5*w - 27],\ [631, 631, -5*w - 22],\ [653, 653, -w - 26],\ [653, 653, w - 27],\ [661, 661, 5*w - 46],\ [661, 661, -5*w - 41],\ [683, 683, -9*w + 62],\ [683, 683, -9*w - 53],\ [691, 691, 7*w - 45],\ [691, 691, -7*w - 38],\ [727, 727, -6*w - 47],\ [727, 727, 6*w - 53],\ [733, 733, 4*w - 41],\ [733, 733, -4*w - 37],\ [739, 739, -4*w - 5],\ [739, 739, 4*w - 9],\ [751, 751, 8*w - 53],\ [751, 751, -8*w - 45],\ [769, 769, 5*w - 24],\ [769, 769, -5*w - 19],\ [773, 773, 7*w - 44],\ [773, 773, -7*w - 37],\ [787, 787, 4*w - 3],\ [787, 787, 4*w - 1],\ [797, 797, -9*w - 52],\ [797, 797, -9*w + 61],\ [811, 811, -5*w - 18],\ [811, 811, 5*w - 23],\ [821, 821, -w - 29],\ [821, 821, w - 30],\ [827, 827, 2*w - 33],\ [827, 827, -2*w - 31],\ [829, 829, -10*w + 69],\ [829, 829, -10*w - 59],\ [839, 839, 5*w - 48],\ [839, 839, -5*w - 43],\ [853, 853, -7*w - 36],\ [853, 853, 7*w - 43],\ [881, 881, -w - 30],\ [881, 881, w - 31],\ [961, 31, -31],\ [991, 991, -5*w - 13],\ [991, 991, 5*w - 18]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^4 + 2*x^3 - 18*x^2 - 34*x + 29 K. = NumberField(heckePol) hecke_eigenvalues_array = [1, 1/5*e^3 - 18/5*e - 8/5, e, -3, -1/2*e^3 + 1/2*e^2 + 15/2*e - 5/2, -1/10*e^3 - 1/2*e^2 + 3/10*e + 53/10, -2/5*e^3 + 1/2*e^2 + 26/5*e - 73/10, -1/5*e^3 - 1/2*e^2 + 13/5*e + 21/10, 1/10*e^3 - 13/10*e - 24/5, 1/2*e^3 - 15/2*e - 2, 3/10*e^3 - 29/10*e - 2/5, 1/10*e^3 - e^2 - 23/10*e + 61/5, -1/2*e^3 + e^2 + 15/2*e - 5, -1/5*e^3 + 23/5*e - 17/5, 1/5*e^3 - 23/5*e - 33/5, -1/10*e^3 + 1/2*e^2 + 3/10*e - 97/10, -1/10*e^3 - 1/2*e^2 + 23/10*e + 13/10, -e - 2, -1/5*e^3 + 18/5*e - 2/5, -1/10*e^3 - 1/2*e^2 - 7/10*e + 43/10, -7/10*e^3 + 1/2*e^2 + 111/10*e - 19/10, -1/2*e^2 + 2*e + 23/2, 1/5*e^3 + 1/2*e^2 - 23/5*e - 11/10, 2/5*e^3 - 3/2*e^2 - 31/5*e + 93/10, -2/5*e^3 + 3/2*e^2 + 31/5*e - 173/10, 2/5*e^3 + 1/2*e^2 - 16/5*e - 87/10, e^3 - 1/2*e^2 - 15*e - 5/2, -1/10*e^3 + e^2 + 13/10*e - 66/5, 3/10*e^3 - e^2 - 39/10*e + 28/5, 3/10*e^3 + 1/2*e^2 - 39/10*e - 19/10, 1/2*e^3 - 1/2*e^2 - 13/2*e + 15/2, 1/10*e^3 - 3/10*e - 49/5, 3/10*e^3 - 49/10*e - 57/5, -1/5*e^3 + 23/5*e - 2/5, 1/5*e^3 - 23/5*e - 18/5, 3/10*e^3 - 39/10*e - 32/5, 3/2*e^2 - e - 15/2, 2/5*e^3 - 3/2*e^2 - 21/5*e + 223/10, e^3 + e^2 - 15*e - 8, e^3 - e^2 - 11*e + 14, -2/5*e^3 + 3/2*e^2 + 31/5*e - 153/10, 2/5*e^3 - 3/2*e^2 - 31/5*e + 113/10, 7/10*e^3 - 1/2*e^2 - 101/10*e + 9/10, 3/10*e^3 + 1/2*e^2 - 29/10*e - 69/10, 8/5*e^3 - 104/5*e - 29/5, -e^2 - 2*e + 19, -4/5*e^3 + e^2 + 62/5*e + 17/5, 3/10*e^3 + 1/2*e^2 - 59/10*e - 79/10, 1/10*e^3 - 1/2*e^2 + 7/10*e + 47/10, -6/5*e^3 + 1/2*e^2 + 78/5*e - 69/10, -e^3 - 1/2*e^2 + 13*e + 5/2, 1/5*e^3 - 23/5*e - 48/5, -1/5*e^3 + 23/5*e - 32/5, -2/5*e^3 + 26/5*e + 96/5, 4/5*e^3 - e^2 - 67/5*e - 17/5, -1/5*e^3 + e^2 + 28/5*e - 87/5, -3/5*e^3 + 2*e^2 + 39/5*e - 91/5, 1/5*e^3 - 2*e^2 - 13/5*e + 97/5, -e^3 + 3/2*e^2 + 12*e - 11/2, -3/5*e^3 - 3/2*e^2 + 44/5*e + 243/10, -2/5*e^3 + 3/2*e^2 + 36/5*e - 113/10, 3/5*e^3 - 3/2*e^2 - 49/5*e + 137/10, -3/5*e^3 + 2*e^2 + 19/5*e - 151/5, -3/5*e^3 - 2*e^2 + 59/5*e + 69/5, -23/10*e^3 + 299/10*e + 12/5, 1/10*e^3 + 5/2*e^2 - 13/10*e - 263/10, 11/10*e^3 - 5/2*e^2 - 143/10*e + 207/10, -2/5*e^3 - 1/2*e^2 + 21/5*e + 17/10, -4/5*e^3 + 1/2*e^2 + 57/5*e - 61/10, -e^3 + 3*e^2 + 15*e - 31, 3/5*e^3 - 3*e^2 - 49/5*e + 111/5, 19/10*e^3 - 5/2*e^2 - 277/10*e + 153/10, 3/10*e^3 + 5/2*e^2 - 9/10*e - 269/10, -1/10*e^3 - 2*e^2 + 43/10*e + 89/5, -3/10*e^3 + 2*e^2 + 9/10*e - 123/5, -1/5*e^3 - 17/5*e + 13/5, -7/5*e^3 + 121/5*e + 61/5, -19/10*e^3 + 2*e^2 + 277/10*e - 9/5, -1/2*e^3 - 2*e^2 + 7/2*e + 31, 7/5*e^3 + 2*e^2 - 111/5*e - 81/5, 7/5*e^3 - 2*e^2 - 71/5*e + 139/5, 8/5*e^3 - 3*e^2 - 104/5*e + 196/5, 2/5*e^3 + 3*e^2 - 26/5*e - 86/5, 17/10*e^3 + 3/2*e^2 - 231/10*e - 191/10, 21/10*e^3 - 3/2*e^2 - 263/10*e + 107/10, -6/5*e^3 + e^2 + 68/5*e - 157/5, -6/5*e^3 - e^2 + 88/5*e - 47/5, 5/2*e^3 - e^2 - 61/2*e + 9, 5/2*e^3 + e^2 - 69/2*e - 13, -9/10*e^3 + 3*e^2 + 87/10*e - 164/5, -3/10*e^3 - 3*e^2 + 69/10*e + 142/5, -6/5*e^3 + 3*e^2 + 98/5*e - 127/5, 4/5*e^3 - 3*e^2 - 72/5*e + 123/5, 1/10*e^3 + 2*e^2 - 53/10*e - 134/5, 1/10*e^3 - 2*e^2 + 27/10*e + 86/5, -3/5*e^3 + 1/2*e^2 + 64/5*e + 63/10, 3/5*e^3 - 1/2*e^2 - 64/5*e + 77/10, 7/5*e^3 - 1/2*e^2 - 126/5*e - 107/10, -1/5*e^3 + 1/2*e^2 + 48/5*e - 89/10, -6*e + 8, -6/5*e^3 + 108/5*e + 88/5, -1/10*e^3 + 3/2*e^2 - 17/10*e - 247/10, -1/10*e^3 - 3/2*e^2 + 43/10*e + 83/10, -8/5*e^3 - e^2 + 124/5*e + 79/5, -6/5*e^3 + e^2 + 58/5*e - 47/5, -2/5*e^3 - 2*e^2 + 56/5*e + 71/5, 2*e^2 - 6*e - 33, 1/5*e^3 - e^2 - 33/5*e - 3/5, -e^3 + e^2 + 17*e - 13, -1/10*e^3 - 1/2*e^2 + 23/10*e + 253/10, -1/10*e^3 + 1/2*e^2 + 3/10*e + 143/10, -9/10*e^3 + 2*e^2 + 77/10*e - 129/5, -9/10*e^3 - 2*e^2 + 157/10*e + 91/5, -e^3 + 9*e + 20, -9/5*e^3 + 137/5*e + 132/5, -4/5*e^3 + 2*e^2 + 72/5*e - 128/5, 4/5*e^3 - 2*e^2 - 72/5*e + 28/5, -11/5*e^3 + e^2 + 133/5*e - 17/5, -11/5*e^3 - e^2 + 153/5*e + 93/5, -9/5*e^3 + 147/5*e + 52/5, -3/5*e^3 + 9/5*e + 4/5, -9/10*e^3 - e^2 + 157/10*e + 76/5, -1/2*e^3 + e^2 + 5/2*e - 10, -3/5*e^3 - 3/2*e^2 + 29/5*e + 213/10, -8/5*e^3 + 3/2*e^2 + 114/5*e - 37/10, 4/5*e^3 + 1/2*e^2 - 57/5*e - 169/10, 4/5*e^3 - 1/2*e^2 - 47/5*e - 59/10, 8/5*e^3 - 3*e^2 - 124/5*e + 181/5, -2/5*e^3 + 3*e^2 + 46/5*e - 69/5, -12/5*e^3 + 156/5*e + 36/5, -12/5*e^3 + 156/5*e + 36/5, -9/5*e^3 + 2*e^2 + 117/5*e - 28/5, -e^3 - 2*e^2 + 13*e + 32, -7/5*e^3 - 2*e^2 + 101/5*e + 66/5, -9/5*e^3 + 2*e^2 + 107/5*e - 138/5, 2/5*e^3 + e^2 - 56/5*e - 151/5, -2/5*e^3 - e^2 + 56/5*e - 9/5, 3/2*e^3 - 5/2*e^2 - 27/2*e + 87/2, 17/10*e^3 + 5/2*e^2 - 281/10*e - 131/10, 7/2*e^2 - 2*e - 73/2, e^3 - 7/2*e^2 - 11*e + 65/2, 1/5*e^3 + 5/2*e^2 - 63/5*e - 381/10, -4/5*e^3 - 5/2*e^2 + 102/5*e + 249/10, -39/10*e^3 + 507/10*e + 96/5, -39/10*e^3 + 507/10*e + 96/5, 3/5*e^3 - 3/2*e^2 - 44/5*e - 103/10, -1/5*e^3 + 3/2*e^2 + 18/5*e - 369/10, 4/5*e^3 + e^2 - 52/5*e - 67/5, 6/5*e^3 - e^2 - 78/5*e + 27/5, -3*e^2 - 4*e + 37, -2*e^3 + 3*e^2 + 30*e - 13, -6/5*e^3 - 4*e^2 + 98/5*e + 188/5, -2*e^3 + 4*e^2 + 22*e - 44, -1/5*e^3 - 2*e^2 + 53/5*e + 108/5, 3/5*e^3 + 2*e^2 - 79/5*e - 144/5, 12/5*e^3 + 3*e^2 - 176/5*e - 111/5, 14/5*e^3 - 3*e^2 - 162/5*e + 203/5, -21/10*e^3 + 11/2*e^2 + 253/10*e - 597/10, -3/10*e^3 - 11/2*e^2 + 59/10*e + 469/10, 1/5*e^3 - 33/5*e + 12/5, -3/5*e^3 + 59/5*e + 44/5, 19/10*e^3 - 3*e^2 - 277/10*e + 159/5, 1/10*e^3 + 3*e^2 + 17/10*e - 99/5, 13/10*e^3 - 5*e^2 - 189/10*e + 238/5, -11/10*e^3 + 5*e^2 + 163/10*e - 216/5, 14/5*e^3 - 182/5*e - 27/5, 2/5*e^3 + e^2 - 41/5*e + 59/5, 1/5*e^3 - e^2 + 2/5*e + 177/5] 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]