/* 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([1, 7, 11, -2, -7, 0, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([25, 5, -w^5 + w^4 + 6*w^3 - 4*w^2 - 8*w]) primes_array = [ [5, 5, -2*w^5 + w^4 + 13*w^3 - 2*w^2 - 19*w - 5],\ [9, 3, 2*w^5 - w^4 - 14*w^3 + 2*w^2 + 23*w + 6],\ [11, 11, w - 1],\ [25, 5, w^3 + w^2 - 4*w - 3],\ [29, 29, w^5 - w^4 - 7*w^3 + 4*w^2 + 11*w],\ [41, 41, w^4 - w^3 - 5*w^2 + 3*w + 3],\ [49, 7, w^5 - w^4 - 7*w^3 + 4*w^2 + 11*w + 1],\ [59, 59, 2*w^5 - w^4 - 14*w^3 + 2*w^2 + 24*w + 7],\ [59, 59, -w^5 + 8*w^3 + 2*w^2 - 15*w - 8],\ [61, 61, w^5 - 7*w^3 - 2*w^2 + 12*w + 4],\ [61, 61, -w^5 + 7*w^3 - 11*w - 1],\ [64, 2, 2],\ [71, 71, w^3 + w^2 - 5*w - 3],\ [71, 71, -3*w^5 + w^4 + 21*w^3 - w^2 - 33*w - 9],\ [79, 79, -2*w^5 + w^4 + 13*w^3 - 3*w^2 - 19*w - 5],\ [81, 3, 2*w^5 - w^4 - 13*w^3 + w^2 + 19*w + 8],\ [89, 89, 2*w^5 - w^4 - 13*w^3 + 2*w^2 + 20*w + 7],\ [89, 89, -w^5 + 8*w^3 + w^2 - 16*w - 5],\ [89, 89, -3*w^5 + w^4 + 20*w^3 - 30*w - 11],\ [89, 89, -w^5 + 7*w^3 + 2*w^2 - 11*w - 4],\ [89, 89, -2*w^5 + 14*w^3 + 3*w^2 - 22*w - 8],\ [89, 89, 2*w^5 - w^4 - 14*w^3 + 3*w^2 + 22*w + 6],\ [101, 101, 3*w^5 - 2*w^4 - 18*w^3 + 6*w^2 + 22*w + 6],\ [101, 101, w^2 - w - 4],\ [101, 101, -3*w^5 + w^4 + 20*w^3 - 2*w^2 - 29*w - 7],\ [109, 109, w^4 - w^3 - 5*w^2 + 4*w + 5],\ [121, 11, w^5 - 7*w^3 - 2*w^2 + 10*w + 6],\ [131, 131, w^5 - 2*w^4 - 5*w^3 + 9*w^2 + 4*w - 3],\ [131, 131, -2*w^5 + w^4 + 13*w^3 - 3*w^2 - 20*w - 6],\ [131, 131, 3*w^5 - w^4 - 21*w^3 + w^2 + 35*w + 11],\ [131, 131, -w^5 + w^4 + 6*w^3 - 4*w^2 - 7*w + 2],\ [139, 139, -w^5 + 6*w^3 + w^2 - 8*w - 1],\ [151, 151, -w^3 + w^2 + 4*w - 2],\ [179, 179, w^4 - 6*w^2 + 6],\ [181, 181, w^3 + w^2 - 3*w],\ [191, 191, 3*w^5 - w^4 - 20*w^3 + 2*w^2 + 28*w + 5],\ [199, 199, -2*w^5 + w^4 + 12*w^3 - 3*w^2 - 15*w - 2],\ [199, 199, 5*w^5 - w^4 - 34*w^3 - w^2 + 52*w + 15],\ [211, 211, 3*w^5 - w^4 - 19*w^3 + w^2 + 26*w + 9],\ [229, 229, -w^5 + w^4 + 7*w^3 - 3*w^2 - 11*w],\ [229, 229, -w^4 + 2*w^3 + 6*w^2 - 7*w - 6],\ [239, 239, -w^5 + w^4 + 7*w^3 - 3*w^2 - 13*w - 6],\ [239, 239, 3*w^5 - w^4 - 21*w^3 + w^2 + 35*w + 10],\ [239, 239, w^3 - w^2 - 4*w + 1],\ [239, 239, 5*w^5 - 2*w^4 - 34*w^3 + 2*w^2 + 53*w + 17],\ [241, 241, -w^5 + 7*w^3 + w^2 - 13*w - 4],\ [241, 241, -3*w^5 + w^4 + 21*w^3 - 35*w - 10],\ [241, 241, -w^5 + 7*w^3 - 11*w],\ [251, 251, 3*w^5 - w^4 - 21*w^3 + 35*w + 11],\ [251, 251, w^5 - 2*w^4 - 6*w^3 + 9*w^2 + 8*w - 3],\ [269, 269, -w^5 + 8*w^3 + 3*w^2 - 16*w - 8],\ [269, 269, -2*w^5 + w^4 + 13*w^3 - w^2 - 21*w - 9],\ [269, 269, 3*w^5 - 21*w^3 - 3*w^2 + 32*w + 12],\ [271, 271, -5*w^5 + 3*w^4 + 34*w^3 - 8*w^2 - 53*w - 14],\ [281, 281, -w^5 + 8*w^3 + 3*w^2 - 16*w - 11],\ [281, 281, 4*w^5 - w^4 - 29*w^3 + 49*w + 13],\ [281, 281, w^5 - w^4 - 7*w^3 + 3*w^2 + 11*w + 5],\ [281, 281, -5*w^5 + 3*w^4 + 33*w^3 - 8*w^2 - 50*w - 12],\ [311, 311, 2*w^5 - w^4 - 14*w^3 + w^2 + 21*w + 9],\ [311, 311, -3*w^5 + 20*w^3 + 3*w^2 - 30*w - 10],\ [311, 311, 5*w^5 - 2*w^4 - 35*w^3 + 2*w^2 + 57*w + 19],\ [331, 331, -w^5 - w^4 + 7*w^3 + 6*w^2 - 11*w - 7],\ [349, 349, -6*w^5 + 3*w^4 + 41*w^3 - 7*w^2 - 64*w - 17],\ [349, 349, 4*w^5 - w^4 - 29*w^3 - w^2 + 50*w + 17],\ [349, 349, 5*w^5 - w^4 - 34*w^3 - w^2 + 52*w + 17],\ [359, 359, w^5 - 6*w^3 - w^2 + 5*w + 3],\ [379, 379, -w^5 + w^4 + 8*w^3 - 3*w^2 - 17*w - 3],\ [379, 379, 2*w^5 - 14*w^3 - 4*w^2 + 22*w + 11],\ [379, 379, -5*w^5 + 3*w^4 + 34*w^3 - 8*w^2 - 53*w - 13],\ [379, 379, 4*w^5 - 2*w^4 - 27*w^3 + 3*w^2 + 43*w + 14],\ [389, 389, 4*w^5 - 2*w^4 - 28*w^3 + 4*w^2 + 44*w + 13],\ [401, 401, -2*w^5 + 2*w^4 + 13*w^3 - 6*w^2 - 20*w - 6],\ [409, 409, -w^5 + 2*w^4 + 6*w^3 - 8*w^2 - 9*w],\ [419, 419, -5*w^5 + w^4 + 34*w^3 + 2*w^2 - 53*w - 19],\ [419, 419, 2*w^5 - w^4 - 14*w^3 + 4*w^2 + 23*w + 3],\ [421, 421, -w^5 - w^4 + 8*w^3 + 6*w^2 - 13*w - 7],\ [421, 421, 2*w^5 - 2*w^4 - 13*w^3 + 8*w^2 + 18*w + 2],\ [421, 421, 4*w^5 - w^4 - 26*w^3 + 37*w + 11],\ [431, 431, 2*w^5 - w^4 - 13*w^3 + 2*w^2 + 20*w + 9],\ [431, 431, w^5 - 8*w^3 - w^2 + 14*w + 2],\ [449, 449, 4*w^5 - 2*w^4 - 26*w^3 + 4*w^2 + 38*w + 9],\ [449, 449, -3*w^5 + w^4 + 20*w^3 - 31*w - 14],\ [461, 461, 3*w^5 - 2*w^4 - 18*w^3 + 6*w^2 + 23*w + 4],\ [461, 461, -4*w^5 + 2*w^4 + 26*w^3 - 5*w^2 - 36*w - 9],\ [461, 461, -4*w^5 + 2*w^4 + 27*w^3 - 3*w^2 - 42*w - 15],\ [491, 491, -3*w^5 + w^4 + 20*w^3 - 29*w - 10],\ [491, 491, -5*w^5 + w^4 + 35*w^3 + 2*w^2 - 56*w - 18],\ [491, 491, 2*w^5 - 14*w^3 - 4*w^2 + 24*w + 13],\ [491, 491, -3*w^5 + w^4 + 21*w^3 - 2*w^2 - 33*w - 10],\ [499, 499, 6*w^5 - 3*w^4 - 40*w^3 + 6*w^2 + 61*w + 18],\ [499, 499, -4*w^5 + w^4 + 29*w^3 + w^2 - 49*w - 17],\ [499, 499, -w^5 - w^4 + 8*w^3 + 5*w^2 - 14*w - 4],\ [499, 499, -3*w^5 + w^4 + 19*w^3 - 2*w^2 - 26*w - 8],\ [509, 509, -4*w^5 + 2*w^4 + 27*w^3 - 5*w^2 - 43*w - 11],\ [509, 509, -2*w^5 + 16*w^3 + 2*w^2 - 30*w - 9],\ [521, 521, 3*w^5 - 21*w^3 - 3*w^2 + 34*w + 10],\ [521, 521, -3*w^5 + w^4 + 19*w^3 - 26*w - 9],\ [521, 521, 4*w^5 - w^4 - 27*w^3 + 40*w + 15],\ [521, 521, -3*w^5 + 22*w^3 + 3*w^2 - 37*w - 11],\ [541, 541, -4*w^5 + w^4 + 29*w^3 + w^2 - 49*w - 18],\ [569, 569, -6*w^5 + 2*w^4 + 40*w^3 - 2*w^2 - 60*w - 19],\ [569, 569, -w^5 + w^4 + 7*w^3 - 2*w^2 - 12*w - 6],\ [571, 571, 4*w^5 - w^4 - 28*w^3 + 43*w + 14],\ [571, 571, -4*w^5 + 2*w^4 + 28*w^3 - 4*w^2 - 45*w - 14],\ [599, 599, -w^5 + w^4 + 7*w^3 - 2*w^2 - 13*w - 7],\ [601, 601, 4*w^5 - w^4 - 26*w^3 + 38*w + 13],\ [601, 601, 4*w^5 - 2*w^4 - 28*w^3 + 5*w^2 + 44*w + 11],\ [601, 601, 3*w^5 - w^4 - 21*w^3 + 2*w^2 + 35*w + 9],\ [619, 619, 4*w^5 - 2*w^4 - 27*w^3 + 4*w^2 + 40*w + 13],\ [619, 619, -4*w^5 + w^4 + 26*w^3 - 36*w - 12],\ [619, 619, w^4 - w^3 - 6*w^2 + 4*w + 4],\ [631, 631, -3*w^5 + w^4 + 20*w^3 - 30*w - 9],\ [631, 631, -5*w^5 + 2*w^4 + 35*w^3 - 3*w^2 - 57*w - 15],\ [641, 641, 5*w^5 - w^4 - 34*w^3 - w^2 + 52*w + 16],\ [641, 641, -5*w^5 + 2*w^4 + 34*w^3 - 2*w^2 - 52*w - 18],\ [659, 659, 3*w^5 - w^4 - 20*w^3 + w^2 + 32*w + 9],\ [659, 659, -4*w^5 + w^4 + 27*w^3 - w^2 - 40*w - 11],\ [661, 661, 4*w^5 - w^4 - 26*w^3 + 38*w + 12],\ [661, 661, -w^5 + 6*w^3 - 8*w + 1],\ [691, 691, 3*w^5 - 20*w^3 - 3*w^2 + 29*w + 9],\ [691, 691, w^5 + w^4 - 8*w^3 - 7*w^2 + 14*w + 9],\ [691, 691, -w^5 + w^4 + 6*w^3 - 3*w^2 - 7*w + 2],\ [701, 701, -2*w^5 + w^4 + 13*w^3 - 2*w^2 - 19*w - 9],\ [701, 701, 5*w^5 - 2*w^4 - 33*w^3 + 3*w^2 + 48*w + 14],\ [701, 701, w^4 + 3*w^3 - 3*w^2 - 10*w - 1],\ [701, 701, -2*w^5 + w^4 + 14*w^3 - 3*w^2 - 24*w - 3],\ [709, 709, -2*w^5 + 2*w^4 + 14*w^3 - 7*w^2 - 22*w - 5],\ [709, 709, -3*w^5 + 21*w^3 + 4*w^2 - 34*w - 11],\ [719, 719, 5*w^5 - 2*w^4 - 35*w^3 + 3*w^2 + 58*w + 17],\ [719, 719, 4*w^5 - 2*w^4 - 26*w^3 + 4*w^2 + 37*w + 11],\ [719, 719, 4*w^5 - 2*w^4 - 27*w^3 + 4*w^2 + 40*w + 12],\ [719, 719, 2*w^5 - 16*w^3 - 3*w^2 + 30*w + 10],\ [739, 739, -5*w^5 + w^4 + 34*w^3 + w^2 - 51*w - 17],\ [751, 751, -w^5 - w^4 + 9*w^3 + 7*w^2 - 18*w - 10],\ [761, 761, 4*w^5 - 2*w^4 - 28*w^3 + 3*w^2 + 46*w + 16],\ [761, 761, -4*w^5 + 2*w^4 + 27*w^3 - 5*w^2 - 43*w - 12],\ [769, 769, -4*w^5 + w^4 + 28*w^3 - 44*w - 12],\ [769, 769, -w^4 + w^3 + 3*w^2 - 3*w + 1],\ [809, 809, 2*w^5 - 2*w^4 - 13*w^3 + 7*w^2 + 20*w + 1],\ [811, 811, -2*w^5 + 15*w^3 + 2*w^2 - 24*w - 8],\ [811, 811, -w^5 + 9*w^3 + w^2 - 18*w - 6],\ [821, 821, -w^5 + 8*w^3 + 3*w^2 - 17*w - 9],\ [821, 821, w^5 + w^4 - 7*w^3 - 6*w^2 + 11*w + 8],\ [821, 821, 4*w^5 - 2*w^4 - 28*w^3 + 4*w^2 + 46*w + 11],\ [821, 821, 5*w^5 - 2*w^4 - 34*w^3 + 4*w^2 + 51*w + 13],\ [829, 829, -2*w^5 + w^4 + 15*w^3 - w^2 - 28*w - 10],\ [829, 829, -6*w^5 + 3*w^4 + 42*w^3 - 7*w^2 - 68*w - 21],\ [829, 829, 4*w^5 - 2*w^4 - 26*w^3 + 5*w^2 + 38*w + 12],\ [829, 829, w^5 - 7*w^3 - 3*w^2 + 11*w + 6],\ [839, 839, 3*w^5 - 2*w^4 - 20*w^3 + 5*w^2 + 32*w + 11],\ [839, 839, -w^4 + w^3 + 6*w^2 - 3*w - 4],\ [841, 29, -3*w^5 + w^4 + 22*w^3 - 38*w - 11],\ [911, 911, 3*w^5 - w^4 - 21*w^3 + 36*w + 12],\ [919, 919, -w^5 - w^4 + 7*w^3 + 6*w^2 - 12*w - 5],\ [941, 941, -w^5 + w^4 + 7*w^3 - 5*w^2 - 11*w],\ [941, 941, 4*w^5 - w^4 - 29*w^3 + 48*w + 15],\ [971, 971, -4*w^5 + w^4 + 27*w^3 + w^2 - 41*w - 13],\ [991, 991, w^5 - w^4 - 8*w^3 + 3*w^2 + 16*w + 2],\ [991, 991, -4*w^5 + 3*w^4 + 27*w^3 - 9*w^2 - 42*w - 10]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x K = QQ e = 1 hecke_eigenvalues_array = [0, 5, -3, -4, 0, 12, 10, 15, 0, 2, -8, -5, -3, 12, -10, -8, 15, 0, 0, 15, 15, -15, 12, 12, -3, -10, 22, -18, 12, 12, -18, 5, -8, -15, 7, -3, 10, -5, 7, -5, -10, -15, 0, 15, -15, 2, -8, -23, 12, 12, 0, 0, 0, 7, -18, -18, -18, -3, 12, 27, 12, -8, 5, -10, 10, -15, 10, -20, -10, -25, -30, -18, -35, -15, -15, 17, -23, 37, -3, -33, 30, 30, 12, 12, 27, -3, -18, -33, 12, 5, -40, 40, 25, 0, 30, -3, 42, -33, -18, 2, -15, -30, 22, 2, 30, -8, 7, -28, 25, -10, 20, 7, 7, 12, -33, -15, -30, -13, -38, 17, 47, -28, -3, -3, -18, -18, 35, -5, -15, 0, 0, 15, 50, 2, 42, 27, -20, -10, -30, 37, -38, 42, 42, -18, 42, -20, -35, -25, 35, -15, 45, -58, -18, 20, -18, 42, -48, 32, 2] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([5, 5, -2*w^5 + w^4 + 13*w^3 - 2*w^2 - 19*w - 5])] = 1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]