/* 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, -9, 7, 9, -7, -1, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([41,41,4*w^5 + 2*w^4 - 25*w^3 - 2*w^2 + 26*w + 5]) primes_array = [ [8, 2, -2*w^5 - w^4 + 13*w^3 + w^2 - 16*w - 1],\ [8, 2, -3*w^5 - w^4 + 19*w^3 - 2*w^2 - 20*w + 1],\ [13, 13, -w^2 + 3],\ [29, 29, -w^5 - w^4 + 5*w^3 + 2*w^2 - 3*w - 1],\ [41, 41, -2*w^5 + 13*w^3 - 4*w^2 - 12*w],\ [41, 41, 4*w^5 + 2*w^4 - 25*w^3 - 2*w^2 + 26*w + 5],\ [43, 43, -w^2 - w + 2],\ [43, 43, 2*w^5 + w^4 - 13*w^3 - 2*w^2 + 14*w + 5],\ [43, 43, -w^5 + 7*w^3 - w^2 - 8*w - 3],\ [43, 43, 2*w^5 + w^4 - 13*w^3 - 2*w^2 + 15*w + 3],\ [49, 7, -w^5 - w^4 + 6*w^3 + 3*w^2 - 7*w],\ [71, 71, -5*w^5 - 2*w^4 + 32*w^3 - w^2 - 35*w - 2],\ [71, 71, 2*w^5 + w^4 - 13*w^3 - 2*w^2 + 14*w + 6],\ [71, 71, -3*w^5 - 2*w^4 + 19*w^3 + 5*w^2 - 22*w - 7],\ [71, 71, 4*w^5 + 2*w^4 - 25*w^3 - 2*w^2 + 26*w + 6],\ [83, 83, -2*w^5 - w^4 + 12*w^3 - 12*w + 1],\ [83, 83, -2*w^5 - w^4 + 13*w^3 - 16*w + 1],\ [83, 83, -4*w^5 - 2*w^4 + 26*w^3 + 3*w^2 - 29*w - 6],\ [83, 83, 2*w^5 - 13*w^3 + 5*w^2 + 13*w - 1],\ [97, 97, -3*w^5 + 20*w^3 - 8*w^2 - 22*w + 5],\ [97, 97, -4*w^5 - w^4 + 27*w^3 - w^2 - 31*w - 4],\ [113, 113, 4*w^5 + w^4 - 26*w^3 + 4*w^2 + 27*w - 1],\ [113, 113, 6*w^5 + 3*w^4 - 39*w^3 - 3*w^2 + 46*w + 4],\ [113, 113, w^5 - 8*w^3 + w^2 + 12*w],\ [113, 113, -6*w^5 - 2*w^4 + 39*w^3 - w^2 - 42*w - 8],\ [139, 139, -3*w^5 - w^4 + 20*w^3 - 22*w - 4],\ [139, 139, -w^5 + 7*w^3 - 3*w^2 - 9*w + 3],\ [167, 167, -3*w^5 - w^4 + 19*w^3 - 2*w^2 - 19*w + 3],\ [167, 167, -2*w^5 - 2*w^4 + 13*w^3 + 6*w^2 - 18*w - 3],\ [167, 167, -3*w^5 - w^4 + 19*w^3 - 2*w^2 - 21*w + 1],\ [167, 167, 4*w^5 + 2*w^4 - 26*w^3 - 3*w^2 + 30*w + 5],\ [169, 13, -3*w^5 - 2*w^4 + 19*w^3 + 4*w^2 - 22*w - 5],\ [169, 13, w^4 + w^3 - 5*w^2 - w + 2],\ [181, 181, 4*w^5 + 2*w^4 - 26*w^3 - 3*w^2 + 29*w + 8],\ [181, 181, 3*w^5 + 2*w^4 - 19*w^3 - 4*w^2 + 20*w + 4],\ [197, 197, w^5 + 2*w^4 - 6*w^3 - 9*w^2 + 8*w + 7],\ [197, 197, -5*w^5 - w^4 + 34*w^3 - 4*w^2 - 39*w - 2],\ [197, 197, -6*w^5 - w^4 + 40*w^3 - 7*w^2 - 44*w - 1],\ [197, 197, 8*w^5 + 3*w^4 - 51*w^3 + w^2 + 53*w + 7],\ [211, 211, 4*w^5 + 2*w^4 - 25*w^3 - w^2 + 26*w + 4],\ [211, 211, -7*w^5 - 3*w^4 + 45*w^3 - 49*w - 3],\ [211, 211, 4*w^5 + 2*w^4 - 27*w^3 - 3*w^2 + 34*w + 4],\ [211, 211, -w^5 - w^4 + 7*w^3 + 4*w^2 - 11*w - 5],\ [223, 223, 2*w^5 + w^4 - 11*w^3 + 7*w + 1],\ [223, 223, -7*w^5 - 2*w^4 + 45*w^3 - 5*w^2 - 46*w - 3],\ [223, 223, -w^5 - 2*w^4 + 5*w^3 + 8*w^2 - 5*w - 4],\ [223, 223, -4*w^5 - w^4 + 27*w^3 - w^2 - 31*w - 5],\ [239, 239, 2*w^5 + w^4 - 13*w^3 - w^2 + 17*w - 1],\ [239, 239, -7*w^5 - 3*w^4 + 44*w^3 - 46*w - 6],\ [239, 239, -w^5 + w^4 + 7*w^3 - 9*w^2 - 7*w + 7],\ [239, 239, 4*w^5 + w^4 - 26*w^3 + 4*w^2 + 29*w - 3],\ [251, 251, -w^3 - 2*w^2 + 4*w + 3],\ [251, 251, 4*w^5 - 26*w^3 + 10*w^2 + 25*w - 4],\ [251, 251, w^3 + 2*w^2 - 4*w - 4],\ [251, 251, 3*w^5 + 2*w^4 - 17*w^3 - 2*w^2 + 15*w + 1],\ [281, 281, 3*w^5 + w^4 - 19*w^3 + 3*w^2 + 19*w - 1],\ [281, 281, 4*w^5 + 2*w^4 - 26*w^3 - 4*w^2 + 29*w + 8],\ [281, 281, 4*w^5 + 2*w^4 - 24*w^3 - w^2 + 23*w + 4],\ [281, 281, -3*w^5 - w^4 + 21*w^3 + w^2 - 27*w - 6],\ [293, 293, -4*w^5 - 2*w^4 + 25*w^3 + 3*w^2 - 25*w - 6],\ [293, 293, 4*w^5 + w^4 - 26*w^3 + 2*w^2 + 26*w + 6],\ [307, 307, -3*w^5 - w^4 + 19*w^3 - 2*w^2 - 18*w + 2],\ [307, 307, 3*w^5 + w^4 - 19*w^3 + w^2 + 17*w + 3],\ [349, 349, -5*w^5 - 2*w^4 + 32*w^3 + w^2 - 34*w - 5],\ [349, 349, -6*w^5 - 3*w^4 + 39*w^3 + 4*w^2 - 46*w - 7],\ [349, 349, -5*w^5 - 2*w^4 + 32*w^3 - 33*w - 6],\ [349, 349, 3*w^5 - 20*w^3 + 6*w^2 + 20*w - 3],\ [379, 379, -3*w^5 - 2*w^4 + 19*w^3 + 5*w^2 - 23*w - 5],\ [379, 379, 7*w^5 + 2*w^4 - 45*w^3 + 4*w^2 + 47*w + 4],\ [379, 379, -w^5 + w^4 + 8*w^3 - 6*w^2 - 10*w],\ [379, 379, -3*w^5 - 2*w^4 + 18*w^3 + 4*w^2 - 18*w - 7],\ [379, 379, -2*w^5 + 14*w^3 - 2*w^2 - 15*w - 4],\ [379, 379, 2*w^5 + 2*w^4 - 12*w^3 - 5*w^2 + 15*w],\ [419, 419, -w^4 + 6*w^2 - 2*w - 1],\ [419, 419, 5*w^5 + w^4 - 32*w^3 + 6*w^2 + 33*w + 1],\ [421, 421, 5*w^5 + 2*w^4 - 33*w^3 - 2*w^2 + 37*w + 9],\ [421, 421, 2*w^2 + w - 4],\ [433, 433, 3*w^5 + 2*w^4 - 19*w^3 - 4*w^2 + 21*w + 7],\ [433, 433, -7*w^5 - 3*w^4 + 45*w^3 + w^2 - 51*w - 6],\ [433, 433, -4*w^5 - 2*w^4 + 26*w^3 + 2*w^2 - 31*w - 6],\ [433, 433, -4*w^5 + 27*w^3 - 8*w^2 - 28*w],\ [449, 449, 5*w^5 + w^4 - 33*w^3 + 5*w^2 + 36*w - 1],\ [449, 449, -2*w^5 - w^4 + 12*w^3 + w^2 - 12*w - 5],\ [461, 461, -8*w^5 - 3*w^4 + 51*w^3 - 2*w^2 - 55*w - 3],\ [461, 461, 7*w^5 + 3*w^4 - 44*w^3 + w^2 + 46*w + 3],\ [463, 463, 3*w^5 - 21*w^3 + 5*w^2 + 23*w + 1],\ [463, 463, -4*w^5 - w^4 + 26*w^3 - 2*w^2 - 26*w - 5],\ [463, 463, -6*w^5 - 3*w^4 + 38*w^3 + 3*w^2 - 40*w - 8],\ [463, 463, 8*w^5 + 4*w^4 - 51*w^3 - 4*w^2 + 56*w + 9],\ [491, 491, -3*w^5 + 19*w^3 - 7*w^2 - 16*w + 3],\ [491, 491, 2*w^5 - 14*w^3 + 3*w^2 + 16*w + 4],\ [491, 491, w^5 + w^4 - 7*w^3 - 5*w^2 + 11*w + 4],\ [491, 491, -4*w^5 - 2*w^4 + 25*w^3 + 3*w^2 - 26*w - 8],\ [503, 503, 2*w^5 + 2*w^4 - 11*w^3 - 5*w^2 + 11*w + 3],\ [503, 503, -3*w^5 + 19*w^3 - 8*w^2 - 16*w + 5],\ [503, 503, 7*w^5 + 2*w^4 - 45*w^3 + 4*w^2 + 46*w + 4],\ [503, 503, -w^5 + 8*w^3 - 13*w - 2],\ [547, 547, -5*w^5 - 3*w^4 + 32*w^3 + 6*w^2 - 37*w - 6],\ [547, 547, -3*w^5 - 2*w^4 + 20*w^3 + 6*w^2 - 25*w - 6],\ [587, 587, -2*w^5 - w^4 + 14*w^3 + w^2 - 20*w - 1],\ [587, 587, -2*w^5 + 13*w^3 - 4*w^2 - 11*w + 2],\ [587, 587, 7*w^5 + 3*w^4 - 45*w^3 - w^2 + 48*w + 7],\ [587, 587, 3*w^5 + 2*w^4 - 18*w^3 - 4*w^2 + 17*w + 8],\ [601, 601, 4*w^5 + 2*w^4 - 25*w^3 - 2*w^2 + 24*w + 5],\ [601, 601, -2*w^5 - w^4 + 14*w^3 + 3*w^2 - 20*w - 5],\ [601, 601, 3*w^5 + w^4 - 20*w^3 - w^2 + 22*w + 7],\ [601, 601, 3*w^5 + w^4 - 20*w^3 - w^2 + 22*w + 5],\ [617, 617, -4*w^5 - w^4 + 26*w^3 - 3*w^2 - 26*w - 4],\ [617, 617, -4*w^5 - w^4 + 26*w^3 - 2*w^2 - 26*w - 4],\ [617, 617, -5*w^5 - 2*w^4 + 33*w^3 + w^2 - 37*w - 7],\ [617, 617, 2*w^5 + w^4 - 13*w^3 - w^2 + 16*w - 2],\ [617, 617, -4*w^5 - 3*w^4 + 24*w^3 + 6*w^2 - 26*w - 4],\ [617, 617, w^5 + w^4 - 6*w^3 - 3*w^2 + 6*w - 1],\ [643, 643, -3*w^5 + 20*w^3 - 7*w^2 - 22*w + 1],\ [643, 643, -3*w^5 - 2*w^4 + 20*w^3 + 5*w^2 - 25*w - 6],\ [643, 643, 2*w^5 + w^4 - 12*w^3 + 13*w - 2],\ [643, 643, -7*w^5 - 3*w^4 + 45*w^3 - 50*w - 4],\ [673, 673, -6*w^5 - 2*w^4 + 39*w^3 - 3*w^2 - 42*w],\ [673, 673, -3*w^5 - w^4 + 21*w^3 - 27*w - 1],\ [701, 701, 2*w^3 + 2*w^2 - 7*w - 2],\ [701, 701, 4*w^5 + 2*w^4 - 27*w^3 - 4*w^2 + 34*w + 6],\ [727, 727, 7*w^5 + 3*w^4 - 46*w^3 - 2*w^2 + 53*w + 7],\ [727, 727, 3*w^5 + w^4 - 19*w^3 + 19*w + 5],\ [727, 727, -2*w^5 - w^4 + 11*w^3 - 2*w^2 - 8*w + 3],\ [727, 727, -4*w^5 - 2*w^4 + 26*w^3 + w^2 - 30*w - 1],\ [729, 3, -3],\ [743, 743, 3*w^5 + 2*w^4 - 18*w^3 - 2*w^2 + 18*w],\ [743, 743, 3*w^5 + 2*w^4 - 20*w^3 - 6*w^2 + 26*w + 7],\ [757, 757, 3*w^5 + w^4 - 19*w^3 + 2*w^2 + 22*w],\ [757, 757, 4*w^5 + 2*w^4 - 25*w^3 - w^2 + 28*w],\ [757, 757, -w^5 + w^4 + 7*w^3 - 8*w^2 - 7*w + 3],\ [757, 757, -3*w^5 + 20*w^3 - 7*w^2 - 23*w + 4],\ [769, 769, 5*w^5 + 2*w^4 - 33*w^3 - 2*w^2 + 37*w + 6],\ [769, 769, 8*w^5 + 3*w^4 - 51*w^3 + 3*w^2 + 54*w + 2],\ [797, 797, 7*w^5 + 3*w^4 - 44*w^3 + 47*w + 3],\ [797, 797, -2*w^5 + 13*w^3 - 5*w^2 - 15*w + 3],\ [797, 797, -4*w^5 - w^4 + 26*w^3 - 4*w^2 - 30*w + 1],\ [797, 797, 4*w^5 - 26*w^3 + 9*w^2 + 26*w - 1],\ [811, 811, 6*w^5 + w^4 - 39*w^3 + 9*w^2 + 41*w - 2],\ [811, 811, 4*w^5 + w^4 - 26*w^3 + 4*w^2 + 26*w],\ [811, 811, -2*w^5 + 15*w^3 - 3*w^2 - 21*w],\ [811, 811, 7*w^5 + 3*w^4 - 44*w^3 + w^2 + 45*w + 3],\ [827, 827, w^5 - 5*w^3 + 4*w^2 - w - 2],\ [827, 827, 3*w^5 + 2*w^4 - 19*w^3 - 4*w^2 + 24*w + 5],\ [827, 827, -4*w^5 - 2*w^4 + 24*w^3 - 21*w - 1],\ [827, 827, -3*w^5 - w^4 + 20*w^3 - 21*w - 4],\ [827, 827, w^5 + 2*w^4 - 5*w^3 - 8*w^2 + 4*w + 7],\ [827, 827, -5*w^5 - 3*w^4 + 31*w^3 + 4*w^2 - 35*w - 5],\ [839, 839, -2*w^5 - 2*w^4 + 13*w^3 + 7*w^2 - 18*w - 8],\ [839, 839, 5*w^5 + 3*w^4 - 30*w^3 - 4*w^2 + 29*w + 5],\ [841, 29, -4*w^5 - 3*w^4 + 25*w^3 + 7*w^2 - 29*w - 7],\ [841, 29, -5*w^5 - 2*w^4 + 33*w^3 - 38*w - 2],\ [853, 853, 3*w^5 - 20*w^3 + 5*w^2 + 20*w + 1],\ [853, 853, -6*w^5 - 3*w^4 + 38*w^3 + 4*w^2 - 41*w - 10],\ [881, 881, -8*w^5 - 4*w^4 + 51*w^3 + 4*w^2 - 57*w - 8],\ [881, 881, -7*w^5 - 3*w^4 + 44*w^3 - 47*w - 6],\ [881, 881, w^5 + w^4 - 7*w^3 - 5*w^2 + 10*w + 3],\ [881, 881, -7*w^5 - 2*w^4 + 45*w^3 - 6*w^2 - 48*w],\ [883, 883, -4*w^5 + 26*w^3 - 9*w^2 - 24*w + 2],\ [883, 883, 5*w^5 + 3*w^4 - 30*w^3 - 4*w^2 + 29*w + 6],\ [911, 911, -3*w^5 - 2*w^4 + 18*w^3 + 3*w^2 - 18*w - 6],\ [911, 911, 4*w^5 + 3*w^4 - 26*w^3 - 8*w^2 + 32*w + 7],\ [911, 911, 7*w^5 + 3*w^4 - 45*w^3 - w^2 + 49*w + 8],\ [911, 911, -8*w^5 - 4*w^4 + 51*w^3 + 3*w^2 - 57*w - 5],\ [937, 937, -2*w^5 + w^4 + 14*w^3 - 9*w^2 - 14*w + 3],\ [937, 937, 5*w^5 + 3*w^4 - 31*w^3 - 5*w^2 + 34*w + 7],\ [953, 953, -8*w^5 - 3*w^4 + 52*w^3 - 2*w^2 - 58*w - 2],\ [953, 953, -4*w^5 - 2*w^4 + 27*w^3 + 3*w^2 - 33*w - 4],\ [967, 967, w^4 - 5*w^2 + 2*w - 1],\ [967, 967, 7*w^5 + 3*w^4 - 45*w^3 - w^2 + 50*w + 3]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x K = QQ e = 1 hecke_eigenvalues_array = [1, -1, 0, -2, -2, 1, 8, 6, -8, -2, -4, -10, -8, 8, -2, -12, -12, -6, -14, 18, -2, 6, 0, -18, -14, -20, 16, 16, -12, -16, -20, 10, -16, -14, 6, -6, -22, 10, -6, 16, -10, 20, 20, -2, -8, -4, 12, 20, -12, -8, 6, -12, 10, 4, -28, 4, 24, 10, -26, -8, 4, 4, 20, 6, -10, 24, -30, 8, -8, -4, -2, -16, 20, -20, -36, 26, 24, -14, 4, 22, -14, 22, -30, 0, -26, -32, 14, 24, 20, -14, 20, -12, 2, -44, -16, -16, 6, 28, 4, -4, 12, 30, 32, -26, -2, -10, -26, 18, 12, 2, 46, 26, 10, -34, -26, 16, 36, 48, -2, -30, 2, -24, -24, 40, 20, 44, 16, -50, -34, -10, 16, 42, 14, 8, -42, -36, 50, 8, 12, 24, 2, 28, 40, -12, -44, -8, -22, -36, 0, -30, -10, -48, 2, 4, 42, -50, 22, 42, 50, 2, -60, 0, 38, 32, 34, 28, 48, -6, 28, -44] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([41,41,4*w^5 + 2*w^4 - 25*w^3 - 2*w^2 + 26*w + 5])] = -1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]