/* 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([11, 0, -7, 0, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([29, 29, w^3 - 2*w^2 - 3*w + 7]) primes_array = [ [4, 2, -w^2 + w + 3],\ [5, 5, w + 1],\ [5, 5, -w^3 + w^2 + 4*w - 4],\ [11, 11, w],\ [29, 29, w^3 - 2*w^2 - 3*w + 7],\ [29, 29, -w^3 - 2*w^2 + 3*w + 7],\ [31, 31, -w^3 + w^2 + 4*w - 2],\ [31, 31, -w^3 - w^2 + 4*w + 2],\ [41, 41, w^3 + 2*w^2 - 4*w - 6],\ [41, 41, w^3 - 5*w + 2],\ [49, 7, -w^3 + w^2 + 4*w - 1],\ [49, 7, w^3 + w^2 - 4*w - 1],\ [59, 59, -3*w^2 - w + 10],\ [59, 59, -3*w^2 + w + 10],\ [61, 61, -2*w^3 + 2*w^2 + 7*w - 5],\ [61, 61, 2*w^3 + w^2 - 8*w - 6],\ [71, 71, 2*w^2 + w - 9],\ [71, 71, 2*w^2 - w - 9],\ [81, 3, -3],\ [101, 101, 2*w^2 - w - 10],\ [101, 101, 2*w^2 + w - 10],\ [109, 109, 2*w^3 + w^2 - 8*w - 3],\ [109, 109, -2*w^3 + w^2 + 8*w - 3],\ [121, 11, -w^2 + 7],\ [131, 131, 4*w^2 - 2*w - 15],\ [131, 131, 2*w^2 + 2*w - 3],\ [131, 131, 2*w^2 - 2*w - 3],\ [131, 131, -w^3 - w^2 + 3*w + 7],\ [139, 139, w^3 + 4*w^2 - 5*w - 16],\ [139, 139, w^3 + 3*w^2 - 6*w - 10],\ [139, 139, 2*w^3 - w^2 - 8*w + 1],\ [139, 139, -4*w^2 - w + 12],\ [149, 149, 2*w^3 - w^2 - 8*w + 4],\ [149, 149, 2*w^3 + w^2 - 8*w - 4],\ [151, 151, 3*w^2 - 3*w - 8],\ [151, 151, 2*w^3 - 4*w^2 - 8*w + 17],\ [151, 151, w^2 - 2*w - 7],\ [151, 151, w^3 + 2*w^2 - 6*w - 8],\ [179, 179, w^3 - w^2 - 5*w + 1],\ [179, 179, -w^3 - w^2 + 5*w + 1],\ [191, 191, 2*w^3 - w^2 - 7*w + 2],\ [191, 191, 3*w^3 - 2*w^2 - 10*w + 2],\ [199, 199, w^3 - 4*w^2 - 3*w + 13],\ [199, 199, -w^3 - 4*w^2 + 3*w + 13],\ [211, 211, -w^3 + w^2 + 6*w - 5],\ [211, 211, w^3 - 5*w - 5],\ [211, 211, -w^3 + 5*w - 5],\ [211, 211, w^3 + w^2 - 6*w - 5],\ [229, 229, w^3 - 4*w^2 - 2*w + 14],\ [229, 229, 2*w^3 + 3*w^2 - 7*w - 12],\ [229, 229, 2*w^3 - 3*w^2 - 7*w + 12],\ [229, 229, 2*w^3 + w^2 - 7*w - 8],\ [241, 241, w^3 + w^2 - 6*w - 4],\ [241, 241, -w^3 + w^2 + 6*w - 4],\ [251, 251, 2*w^2 - w - 2],\ [251, 251, 2*w^2 + w - 2],\ [271, 271, 2*w^3 - 2*w^2 - 8*w + 5],\ [271, 271, -4*w^2 - w + 14],\ [271, 271, -4*w^2 + w + 14],\ [271, 271, 2*w^3 + 2*w^2 - 8*w - 5],\ [281, 281, -2*w^3 - w^2 + 9*w + 2],\ [281, 281, 2*w^3 - w^2 - 9*w + 2],\ [311, 311, -w^3 + 3*w - 5],\ [311, 311, w^3 - 3*w - 5],\ [331, 331, -3*w^3 + 3*w^2 + 12*w - 13],\ [331, 331, 3*w^3 + 3*w^2 - 12*w - 13],\ [349, 349, 2*w^3 + 2*w^2 - 7*w - 10],\ [349, 349, -2*w^3 + 2*w^2 + 7*w - 10],\ [361, 19, 4*w^2 - 15],\ [361, 19, -4*w^2 + 13],\ [379, 379, -w^3 + 3*w^2 + 3*w - 13],\ [379, 379, w^3 + 3*w^2 - 3*w - 13],\ [409, 409, 3*w^3 - 2*w^2 - 11*w + 2],\ [409, 409, w^3 + 3*w^2 - 6*w - 9],\ [419, 419, -w^3 + w^2 + 6*w + 2],\ [419, 419, -3*w^3 + 5*w^2 + 13*w - 19],\ [421, 421, -w^3 + 3*w^2 + 7*w - 13],\ [421, 421, 2*w^3 - 2*w^2 - 9*w + 6],\ [421, 421, -2*w^3 - 2*w^2 + 9*w + 6],\ [421, 421, w^3 + 3*w^2 - 7*w - 13],\ [439, 439, 3*w^2 + 2*w - 13],\ [439, 439, 3*w^3 + 2*w^2 - 12*w - 10],\ [439, 439, -3*w^3 + 2*w^2 + 12*w - 10],\ [439, 439, 3*w^2 - 2*w - 13],\ [449, 449, 2*w^3 - 4*w^2 - 9*w + 14],\ [449, 449, -3*w^3 + 3*w^2 + 11*w - 9],\ [449, 449, 3*w^3 + 3*w^2 - 11*w - 9],\ [449, 449, w^3 - 2*w^2 - 5*w + 2],\ [461, 461, -w - 5],\ [461, 461, w - 5],\ [499, 499, w^3 - 5*w^2 - w + 17],\ [499, 499, -w^3 - 5*w^2 + w + 17],\ [509, 509, w^3 + 5*w^2 - 4*w - 16],\ [509, 509, w^2 - 2*w - 10],\ [509, 509, w^2 + 2*w - 10],\ [509, 509, w^3 - w^2 - 5*w + 9],\ [541, 541, 3*w^3 - 10*w - 6],\ [541, 541, -2*w^3 + 5*w^2 + 5*w - 16],\ [569, 569, 3*w^3 + 2*w^2 - 13*w - 6],\ [569, 569, -2*w^3 + 4*w^2 + 7*w - 10],\ [599, 599, w^3 + 2*w^2 - 7*w - 10],\ [599, 599, -w^3 + 2*w^2 + 7*w - 10],\ [601, 601, -3*w^3 - w^2 + 12*w + 6],\ [601, 601, 3*w^3 - w^2 - 12*w + 6],\ [619, 619, 3*w^3 - 12*w + 2],\ [619, 619, -3*w^3 + 12*w + 2],\ [631, 631, -w^3 + w^2 + 2*w - 8],\ [631, 631, w^3 + w^2 - 2*w - 8],\ [641, 641, w^3 + w^2 - 7*w - 1],\ [641, 641, -3*w^3 - 2*w^2 + 12*w + 4],\ [641, 641, 3*w^3 - 2*w^2 - 12*w + 4],\ [641, 641, w^3 - w^2 - 7*w + 1],\ [659, 659, -2*w^3 + w^2 + 10*w - 1],\ [659, 659, -w^3 + 5*w^2 + 4*w - 19],\ [659, 659, w^3 + 5*w^2 - 4*w - 19],\ [659, 659, 2*w^3 + w^2 - 10*w - 1],\ [691, 691, 3*w^2 - 2*w - 14],\ [691, 691, 3*w^2 + 2*w - 14],\ [701, 701, w^3 + 4*w^2 - 7*w - 12],\ [701, 701, 3*w^3 - w^2 - 12*w - 1],\ [719, 719, -w^3 + 7*w - 3],\ [719, 719, w^3 - 7*w - 3],\ [739, 739, 3*w^3 + w^2 - 12*w - 3],\ [739, 739, w^3 + 2*w^2 - 3*w - 1],\ [739, 739, -w^3 + 2*w^2 + 3*w - 1],\ [739, 739, -3*w^3 + w^2 + 12*w - 3],\ [751, 751, -2*w^3 + 6*w^2 + 7*w - 20],\ [751, 751, 2*w^3 + 6*w^2 - 7*w - 20],\ [761, 761, 2*w^3 + 6*w^2 - 8*w - 21],\ [761, 761, -2*w^3 + 6*w^2 + 8*w - 21],\ [769, 769, -2*w^3 + 6*w^2 + 8*w - 23],\ [769, 769, w^2 + 2*w + 3],\ [809, 809, 2*w^3 + 3*w^2 - 7*w - 14],\ [809, 809, -2*w^3 + 3*w^2 + 7*w - 14],\ [821, 821, -3*w^3 - w^2 + 11*w + 1],\ [821, 821, 3*w^3 - w^2 - 11*w + 1],\ [829, 829, 3*w^3 - w^2 - 12*w + 4],\ [829, 829, w^3 - 4*w^2 - 6*w + 12],\ [829, 829, -w^3 - 4*w^2 + 6*w + 12],\ [829, 829, -3*w^3 - w^2 + 12*w + 4],\ [839, 839, -w^3 + 3*w - 6],\ [839, 839, w^3 - 3*w - 6],\ [841, 29, 5*w^2 - 19],\ [859, 859, w^3 + 2*w^2 - 7*w - 9],\ [859, 859, -w^3 + 2*w^2 + 7*w - 9],\ [911, 911, -2*w^3 + 4*w^2 + 9*w - 13],\ [911, 911, 2*w^3 + 4*w^2 - 9*w - 13],\ [919, 919, 2*w^3 + 4*w^2 - 7*w - 17],\ [919, 919, 3*w^3 + 2*w^2 - 11*w - 6],\ [919, 919, 3*w^3 - 2*w^2 - 11*w + 6],\ [919, 919, 2*w^3 - 4*w^2 - 7*w + 17],\ [941, 941, -2*w^3 + 2*w^2 + 7*w - 15],\ [941, 941, 2*w^3 + 2*w^2 - 7*w - 15],\ [961, 31, -2*w^2 + 13],\ [971, 971, w^3 + 2*w^2 - 7*w - 7],\ [971, 971, -w^3 + 2*w^2 + 7*w - 7],\ [991, 991, 3*w^3 + 2*w^2 - 11*w - 7],\ [991, 991, -3*w^3 + 2*w^2 + 11*w - 7]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^6 - 15*x^4 + 32*x^2 - 16 K. = NumberField(heckePol) hecke_eigenvalues_array = [e, -3/8*e^5 + 41/8*e^3 - 9/2*e, 1/4*e^5 - 15/4*e^3 + 7*e, -5/8*e^5 + 71/8*e^3 - 25/2*e, -1, e^4 - 14*e^2 + 18, -3/8*e^5 + 41/8*e^3 - 15/2*e, 3/4*e^5 - 41/4*e^3 + 11*e, 3/4*e^5 - 41/4*e^3 + 11*e, 3/4*e^5 - 41/4*e^3 + 11*e, 1/4*e^5 - 19/4*e^3 + 17*e, 3/2*e^5 - 41/2*e^3 + 24*e, 2*e^4 - 27*e^2 + 28, 1/2*e^4 - 15/2*e^2 + 20, 3/8*e^5 - 41/8*e^3 + 9/2*e, -3/2*e^5 + 41/2*e^3 - 22*e, 3/2*e^4 - 43/2*e^2 + 28, 2*e^2 - 12, e^4 - 15*e^2 + 18, e^4 - 12*e^2 + 6, -3/2*e^4 + 41/2*e^2 - 16, -9/8*e^5 + 123/8*e^3 - 31/2*e, 3/8*e^5 - 41/8*e^3 + 1/2*e, -6, 3/2*e^4 - 45/2*e^2 + 28, -2*e^5 + 29*e^3 - 46*e, -5/8*e^5 + 71/8*e^3 - 37/2*e, 1/2*e^4 - 11/2*e^2 + 8, -7/2*e^4 + 97/2*e^2 - 56, -6*e, 2*e^5 - 27*e^3 + 30*e, 7/2*e^4 - 93/2*e^2 + 48, -25/8*e^5 + 347/8*e^3 - 103/2*e, -15/8*e^5 + 205/8*e^3 - 57/2*e, -17/8*e^5 + 243/8*e^3 - 85/2*e, e^4 - 17*e^2 + 36, -5/2*e^4 + 69/2*e^2 - 48, e^5 - 15*e^3 + 36*e, 11/8*e^5 - 153/8*e^3 + 43/2*e, -17/8*e^5 + 235/8*e^3 - 77/2*e, 2*e^5 - 28*e^3 + 32*e, -3*e^5 + 41*e^3 - 44*e, 2*e^2 - 20, 1/2*e^4 - 15/2*e^2 + 24, -1/8*e^5 + 27/8*e^3 - 41/2*e, e^2, -1/2*e^4 + 7/2*e^2 + 12, -11/8*e^5 + 153/8*e^3 - 55/2*e, -e^4 + 12*e^2 - 6, 1/2*e^4 - 15/2*e^2 + 20, -2*e^4 + 28*e^2 - 26, -3/2*e^4 + 37/2*e^2, 7/2*e^5 - 97/2*e^3 + 56*e, 9/8*e^5 - 131/8*e^3 + 71/2*e, -3/2*e^4 + 41/2*e^2 - 36, e^4 - 14*e^2 + 28, -29/8*e^5 + 399/8*e^3 - 113/2*e, 7/2*e^4 - 95/2*e^2 + 52, -3*e^4 + 41*e^2 - 32, 3/8*e^5 - 41/8*e^3 - 5/2*e, -9/4*e^5 + 123/4*e^3 - 37*e, 11/4*e^5 - 153/4*e^3 + 39*e, -9/2*e^4 + 121/2*e^2 - 68, -5/2*e^4 + 69/2*e^2 - 40, 3/8*e^5 - 33/8*e^3 - 5/2*e, 3/2*e^5 - 39/2*e^3 + 16*e, -1/2*e^4 + 19/2*e^2 - 8, e^4 - 14*e^2 + 6, 3*e^4 - 37*e^2 + 22, -13/2*e^4 + 177/2*e^2 - 96, -4*e^4 + 55*e^2 - 56, -1/2*e^4 + 11/2*e^2 + 12, -27/8*e^5 + 377/8*e^3 - 137/2*e, -3*e^5 + 41*e^3 - 50*e, -5/8*e^5 + 71/8*e^3 - 25/2*e, -1/2*e^5 + 13/2*e^3 - 8*e, -3/2*e^5 + 39/2*e^3 - 10*e, e^5 - 14*e^3 + 20*e, 7/4*e^5 - 105/4*e^3 + 57*e, -23/8*e^5 + 325/8*e^3 - 117/2*e, -7/2*e^4 + 103/2*e^2 - 68, 27/8*e^5 - 369/8*e^3 + 91/2*e, -3/2*e^5 + 45/2*e^3 - 42*e, -4*e^4 + 54*e^2 - 56, 43/8*e^5 - 601/8*e^3 + 177/2*e, 29/8*e^5 - 391/8*e^3 + 103/2*e, 5/4*e^5 - 63/4*e^3 - 3*e, 5/8*e^5 - 71/8*e^3 + 11/2*e, -4*e^4 + 56*e^2 - 50, -3/2*e^4 + 41/2*e^2 - 4, 9/2*e^4 - 119/2*e^2 + 56, 9/2*e^4 - 119/2*e^2 + 56, -4*e^4 + 54*e^2 - 54, 3/2*e^4 - 45/2*e^2 + 40, 1/2*e^4 - 19/2*e^2, -1/2*e^4 + 19/2*e^2, -7/2*e^4 + 97/2*e^2 - 36, -2*e^4 + 24*e^2 - 14, -13/8*e^5 + 175/8*e^3 - 31/2*e, 11/8*e^5 - 137/8*e^3 + 1/2*e, 13/8*e^5 - 175/8*e^3 + 53/2*e, 23/8*e^5 - 333/8*e^3 + 131/2*e, -9/4*e^5 + 123/4*e^3 - 41*e, 11/8*e^5 - 145/8*e^3 + 41/2*e, -33/8*e^5 + 459/8*e^3 - 141/2*e, -31/8*e^5 + 421/8*e^3 - 123/2*e, -11/2*e^4 + 155/2*e^2 - 92, 7/2*e^4 - 89/2*e^2 + 48, 23/8*e^5 - 325/8*e^3 + 121/2*e, -3*e^5 + 41*e^3 - 50*e, 3/8*e^5 - 65/8*e^3 + 85/2*e, 35/8*e^5 - 465/8*e^3 + 105/2*e, 1/8*e^5 + 21/8*e^3 - 79/2*e, e^2 - 12, 2*e^4 - 23*e^2, 15/4*e^5 - 201/4*e^3 + 41*e, 3/2*e^4 - 29/2*e^2 - 20, 6*e^4 - 82*e^2 + 76, -3/2*e^5 + 43/2*e^3 - 34*e, 17/8*e^5 - 243/8*e^3 + 103/2*e, -3/2*e^5 + 45/2*e^3 - 54*e, -21/8*e^5 + 311/8*e^3 - 161/2*e, -3/4*e^5 + 33/4*e^3 + 3*e, 11/2*e^4 - 149/2*e^2 + 80, -3/2*e^4 + 45/2*e^2 - 40, 15/8*e^5 - 229/8*e^3 + 143/2*e, e^4 - 15*e^2 + 8, -6*e^4 + 86*e^2 - 92, 22, e^4 - 10*e^2 - 14, 3*e^4 - 37*e^2 + 14, -9/2*e^4 + 125/2*e^2 - 68, -6*e^4 + 86*e^2 - 94, -7*e^4 + 94*e^2 - 94, 25/8*e^5 - 347/8*e^3 + 111/2*e, 9/8*e^5 - 139/8*e^3 + 55/2*e, 19/8*e^5 - 249/8*e^3 + 21/2*e, 7/8*e^5 - 85/8*e^3 - 19/2*e, -1/4*e^5 + 7/4*e^3 + 9*e, 1/8*e^5 - 19/8*e^3 + 31/2*e, -3/2*e^4 + 35/2*e^2 + 32, -e^4 + 13*e^2 - 16, -4*e^4 + 58*e^2 - 78, -2*e^5 + 31*e^3 - 70*e, -17/4*e^5 + 247/4*e^3 - 107*e, -1/4*e^5 + 27/4*e^3 - 33*e, -15/8*e^5 + 237/8*e^3 - 159/2*e, e^4 - 11*e^2 + 8, 9/2*e^5 - 127/2*e^3 + 78*e, -29/8*e^5 + 423/8*e^3 - 185/2*e, -7/2*e^4 + 81/2*e^2 - 12, -3*e^4 + 41*e^2 - 86, -2*e^4 + 31*e^2 - 70, 7/2*e^4 - 91/2*e^2 + 28, 27/8*e^5 - 377/8*e^3 + 91/2*e, -23/4*e^5 + 329/4*e^3 - 113*e, 3/4*e^5 - 41/4*e^3 + 3*e, -19/8*e^5 + 241/8*e^3 - 19/2*e] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([29, 29, w^3 - 2*w^2 - 3*w + 7])] = 1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]