/* 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, 2, 11, -5, -2, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([13, 13, 3*w^5 - 2*w^4 - 17*w^3 + 10*w^2 + 16*w - 3]) primes_array = [ [13, 13, 3*w^5 - 2*w^4 - 17*w^3 + 10*w^2 + 16*w - 3],\ [13, 13, w^5 - w^4 - 5*w^3 + 5*w^2 + 3*w - 3],\ [13, 13, 2*w^5 - w^4 - 12*w^3 + 4*w^2 + 13*w],\ [13, 13, -w^2 + 3],\ [41, 41, -2*w^5 + w^4 + 11*w^3 - 4*w^2 - 10*w - 1],\ [41, 41, -4*w^5 + 3*w^4 + 23*w^3 - 14*w^2 - 22*w + 4],\ [41, 41, w^5 - 6*w^3 + w^2 + 7*w - 2],\ [41, 41, -3*w^5 + w^4 + 18*w^3 - 4*w^2 - 20*w - 1],\ [43, 43, 3*w^5 - 2*w^4 - 17*w^3 + 11*w^2 + 16*w - 6],\ [43, 43, -2*w^5 + w^4 + 12*w^3 - 6*w^2 - 14*w + 5],\ [49, 7, 2*w^5 - w^4 - 12*w^3 + 5*w^2 + 13*w - 4],\ [64, 2, -2],\ [71, 71, -2*w^5 + w^4 + 12*w^3 - 4*w^2 - 12*w - 1],\ [71, 71, -3*w^5 + 2*w^4 + 17*w^3 - 9*w^2 - 15*w],\ [83, 83, 4*w^5 - 2*w^4 - 24*w^3 + 9*w^2 + 26*w - 1],\ [83, 83, -4*w^5 + 2*w^4 + 23*w^3 - 9*w^2 - 23*w],\ [97, 97, -w^5 + 7*w^3 + w^2 - 11*w - 3],\ [97, 97, -2*w^5 + w^4 + 11*w^3 - 6*w^2 - 9*w + 3],\ [113, 113, -4*w^5 + 2*w^4 + 24*w^3 - 9*w^2 - 27*w + 2],\ [113, 113, 3*w^5 - 2*w^4 - 18*w^3 + 10*w^2 + 20*w - 6],\ [113, 113, -2*w^5 + w^4 + 12*w^3 - 5*w^2 - 12*w + 3],\ [113, 113, 3*w^5 - w^4 - 18*w^3 + 4*w^2 + 20*w],\ [127, 127, -w^5 + w^4 + 6*w^3 - 5*w^2 - 7*w],\ [127, 127, -2*w^5 + w^4 + 11*w^3 - 4*w^2 - 10*w - 2],\ [127, 127, -4*w^5 + 3*w^4 + 23*w^3 - 14*w^2 - 22*w + 5],\ [127, 127, -8*w^5 + 5*w^4 + 46*w^3 - 23*w^2 - 45*w + 4],\ [139, 139, w^3 + w^2 - 4*w - 1],\ [139, 139, 5*w^5 - 2*w^4 - 29*w^3 + 9*w^2 + 29*w + 1],\ [139, 139, -w^5 + w^4 + 5*w^3 - 6*w^2 - 2*w + 5],\ [139, 139, w^5 - 6*w^3 + 5*w + 2],\ [167, 167, w^4 - 4*w^2 + 1],\ [167, 167, -3*w^5 + w^4 + 18*w^3 - 4*w^2 - 21*w - 1],\ [169, 13, -3*w^5 + w^4 + 18*w^3 - 5*w^2 - 20*w],\ [181, 181, -w^4 + w^3 + 5*w^2 - 3*w - 3],\ [181, 181, -5*w^5 + 3*w^4 + 29*w^3 - 15*w^2 - 30*w + 6],\ [181, 181, -w^5 + w^4 + 5*w^3 - 6*w^2 - 2*w + 4],\ [181, 181, w^3 + w^2 - 4*w - 2],\ [197, 197, w^5 - w^4 - 6*w^3 + 4*w^2 + 6*w + 2],\ [197, 197, -4*w^5 + 2*w^4 + 24*w^3 - 9*w^2 - 26*w + 4],\ [197, 197, -2*w^5 + 2*w^4 + 12*w^3 - 11*w^2 - 12*w + 5],\ [197, 197, 2*w^5 - w^4 - 12*w^3 + 6*w^2 + 13*w - 7],\ [197, 197, w^5 - 6*w^3 - w^2 + 7*w + 1],\ [197, 197, -4*w^5 + 3*w^4 + 24*w^3 - 14*w^2 - 25*w + 2],\ [211, 211, 2*w^5 - w^4 - 13*w^3 + 5*w^2 + 18*w - 1],\ [211, 211, -2*w^5 + w^4 + 12*w^3 - 4*w^2 - 12*w - 4],\ [223, 223, 4*w^5 - w^4 - 23*w^3 + 5*w^2 + 24*w - 1],\ [223, 223, -3*w^5 + w^4 + 17*w^3 - 4*w^2 - 15*w - 4],\ [223, 223, 3*w^5 - 3*w^4 - 16*w^3 + 15*w^2 + 11*w - 5],\ [223, 223, 3*w^5 - 3*w^4 - 16*w^3 + 14*w^2 + 13*w - 5],\ [239, 239, -w^5 + w^4 + 6*w^3 - 6*w^2 - 7*w + 4],\ [239, 239, 2*w^5 - w^4 - 12*w^3 + 4*w^2 + 12*w + 2],\ [251, 251, 3*w^5 - 3*w^4 - 16*w^3 + 14*w^2 + 13*w - 4],\ [251, 251, 5*w^5 - 3*w^4 - 29*w^3 + 14*w^2 + 28*w - 1],\ [251, 251, 6*w^5 - 4*w^4 - 35*w^3 + 19*w^2 + 37*w - 7],\ [251, 251, -8*w^5 + 5*w^4 + 46*w^3 - 24*w^2 - 46*w + 8],\ [281, 281, -6*w^5 + 4*w^4 + 35*w^3 - 18*w^2 - 35*w + 3],\ [281, 281, 3*w^5 - 3*w^4 - 17*w^3 + 15*w^2 + 16*w - 10],\ [281, 281, -3*w^5 + w^4 + 17*w^3 - 5*w^2 - 18*w + 3],\ [281, 281, 5*w^5 - 3*w^4 - 30*w^3 + 14*w^2 + 33*w - 5],\ [293, 293, -w^5 + w^4 + 6*w^3 - 5*w^2 - 8*w + 3],\ [307, 307, 5*w^5 - 3*w^4 - 30*w^3 + 13*w^2 + 32*w],\ [307, 307, -4*w^5 + 3*w^4 + 22*w^3 - 15*w^2 - 20*w + 5],\ [307, 307, 3*w^5 - 2*w^4 - 17*w^3 + 8*w^2 + 15*w],\ [307, 307, -2*w^5 + w^4 + 12*w^3 - 5*w^2 - 11*w - 1],\ [307, 307, 5*w^5 - 4*w^4 - 28*w^3 + 20*w^2 + 26*w - 10],\ [307, 307, -4*w^5 + 2*w^4 + 23*w^3 - 8*w^2 - 22*w - 3],\ [337, 337, -w^5 + w^4 + 5*w^3 - 4*w^2 - 3*w + 3],\ [337, 337, 5*w^5 - 3*w^4 - 29*w^3 + 14*w^2 + 29*w],\ [337, 337, 5*w^5 - 3*w^4 - 28*w^3 + 15*w^2 + 27*w - 5],\ [337, 337, -4*w^5 + 4*w^4 + 22*w^3 - 20*w^2 - 19*w + 10],\ [349, 349, -8*w^5 + 5*w^4 + 46*w^3 - 25*w^2 - 45*w + 9],\ [349, 349, -w^4 + 2*w^3 + 5*w^2 - 7*w - 3],\ [419, 419, 5*w^5 - 3*w^4 - 29*w^3 + 13*w^2 + 28*w - 1],\ [419, 419, -6*w^5 + 4*w^4 + 35*w^3 - 20*w^2 - 35*w + 10],\ [421, 421, -6*w^5 + 3*w^4 + 34*w^3 - 15*w^2 - 32*w + 3],\ [421, 421, -5*w^5 + 3*w^4 + 29*w^3 - 14*w^2 - 28*w],\ [421, 421, w^5 - 5*w^3 - w^2 + 2*w + 3],\ [421, 421, -6*w^5 + 3*w^4 + 35*w^3 - 14*w^2 - 37*w + 2],\ [433, 433, -4*w^5 + 4*w^4 + 22*w^3 - 19*w^2 - 19*w + 6],\ [433, 433, 7*w^5 - 4*w^4 - 40*w^3 + 19*w^2 + 40*w - 6],\ [449, 449, 2*w^4 - w^3 - 10*w^2 + 4*w + 6],\ [449, 449, -4*w^5 + 4*w^4 + 23*w^3 - 20*w^2 - 21*w + 10],\ [461, 461, -6*w^5 + 3*w^4 + 36*w^3 - 15*w^2 - 40*w + 4],\ [461, 461, -7*w^5 + 5*w^4 + 40*w^3 - 23*w^2 - 39*w + 6],\ [461, 461, 4*w^5 - 2*w^4 - 22*w^3 + 9*w^2 + 19*w + 1],\ [461, 461, 5*w^5 - 2*w^4 - 30*w^3 + 10*w^2 + 32*w - 1],\ [463, 463, 4*w^5 - 2*w^4 - 23*w^3 + 10*w^2 + 23*w],\ [463, 463, -6*w^5 + 4*w^4 + 35*w^3 - 20*w^2 - 34*w + 7],\ [491, 491, 7*w^5 - 5*w^4 - 41*w^3 + 24*w^2 + 41*w - 8],\ [491, 491, -2*w^5 + w^4 + 12*w^3 - 4*w^2 - 15*w - 2],\ [491, 491, -w^5 + 6*w^3 + w^2 - 7*w],\ [491, 491, w^5 - w^4 - 6*w^3 + 4*w^2 + 6*w + 3],\ [491, 491, 5*w^5 - 3*w^4 - 29*w^3 + 16*w^2 + 30*w - 9],\ [491, 491, -3*w^5 + 2*w^4 + 18*w^3 - 8*w^2 - 20*w - 2],\ [503, 503, 3*w^5 - 2*w^4 - 18*w^3 + 10*w^2 + 20*w - 7],\ [503, 503, 6*w^5 - 3*w^4 - 35*w^3 + 14*w^2 + 37*w - 3],\ [547, 547, 2*w^5 - 12*w^3 - w^2 + 15*w + 3],\ [547, 547, -w^5 + 2*w^4 + 6*w^3 - 9*w^2 - 6*w + 3],\ [547, 547, 7*w^5 - 4*w^4 - 41*w^3 + 19*w^2 + 41*w - 3],\ [547, 547, 2*w^5 - w^4 - 13*w^3 + 6*w^2 + 17*w - 3],\ [547, 547, -4*w^5 + 3*w^4 + 23*w^3 - 13*w^2 - 21*w],\ [547, 547, -7*w^5 + 4*w^4 + 41*w^3 - 20*w^2 - 43*w + 7],\ [587, 587, 2*w^4 - w^3 - 10*w^2 + 4*w + 7],\ [587, 587, 5*w^5 - 2*w^4 - 29*w^3 + 10*w^2 + 31*w - 2],\ [601, 601, w^3 + 2*w^2 - 4*w - 4],\ [601, 601, w^5 - 7*w^3 - 2*w^2 + 11*w + 5],\ [617, 617, -7*w^5 + 4*w^4 + 41*w^3 - 20*w^2 - 43*w + 9],\ [617, 617, -2*w^5 + 13*w^3 - 16*w],\ [643, 643, -2*w^5 + 2*w^4 + 12*w^3 - 8*w^2 - 13*w],\ [643, 643, -3*w^5 + w^4 + 18*w^3 - 3*w^2 - 21*w - 3],\ [659, 659, 5*w^5 - 2*w^4 - 30*w^3 + 8*w^2 + 33*w + 2],\ [659, 659, w^5 - 6*w^3 + 2*w^2 + 7*w - 4],\ [673, 673, -2*w^5 + 2*w^4 + 10*w^3 - 11*w^2 - 5*w + 6],\ [673, 673, 6*w^5 - 4*w^4 - 34*w^3 + 20*w^2 + 33*w - 9],\ [673, 673, -3*w^5 + 2*w^4 + 17*w^3 - 9*w^2 - 16*w + 5],\ [673, 673, 7*w^5 - 4*w^4 - 40*w^3 + 20*w^2 + 38*w - 6],\ [673, 673, -3*w^5 + 2*w^4 + 16*w^3 - 11*w^2 - 12*w + 6],\ [673, 673, w^5 + w^4 - 6*w^3 - 6*w^2 + 8*w + 5],\ [701, 701, 5*w^5 - 4*w^4 - 28*w^3 + 20*w^2 + 26*w - 9],\ [701, 701, 4*w^5 - 3*w^4 - 22*w^3 + 15*w^2 + 20*w - 6],\ [727, 727, -2*w^5 + 2*w^4 + 11*w^3 - 8*w^2 - 9*w - 2],\ [727, 727, -3*w^5 + 3*w^4 + 18*w^3 - 15*w^2 - 20*w + 9],\ [729, 3, -3],\ [743, 743, -3*w^5 + w^4 + 18*w^3 - 4*w^2 - 22*w],\ [743, 743, -5*w^5 + 2*w^4 + 29*w^3 - 9*w^2 - 31*w - 1],\ [743, 743, -5*w^5 + 3*w^4 + 29*w^3 - 14*w^2 - 31*w + 6],\ [743, 743, -3*w^5 + 3*w^4 + 17*w^3 - 14*w^2 - 17*w + 3],\ [757, 757, -6*w^5 + 3*w^4 + 35*w^3 - 13*w^2 - 37*w + 2],\ [757, 757, 3*w^5 - 3*w^4 - 17*w^3 + 13*w^2 + 16*w - 1],\ [757, 757, -8*w^5 + 5*w^4 + 47*w^3 - 25*w^2 - 48*w + 10],\ [757, 757, 4*w^5 - 2*w^4 - 25*w^3 + 10*w^2 + 29*w - 4],\ [797, 797, -6*w^5 + 4*w^4 + 35*w^3 - 19*w^2 - 34*w + 3],\ [797, 797, -w^5 + w^4 + 7*w^3 - 6*w^2 - 10*w + 3],\ [811, 811, 5*w^5 - 4*w^4 - 28*w^3 + 18*w^2 + 25*w - 4],\ [811, 811, -4*w^5 + 3*w^4 + 22*w^3 - 15*w^2 - 17*w + 7],\ [827, 827, w^5 - w^4 - 5*w^3 + 5*w^2 + w - 4],\ [827, 827, -6*w^5 + 3*w^4 + 36*w^3 - 14*w^2 - 39*w + 3],\ [827, 827, -6*w^5 + 3*w^4 + 36*w^3 - 14*w^2 - 39*w + 2],\ [827, 827, -w^5 - w^4 + 7*w^3 + 6*w^2 - 11*w - 4],\ [839, 839, -2*w^5 + 2*w^4 + 10*w^3 - 10*w^2 - 6*w + 7],\ [839, 839, -6*w^5 + 4*w^4 + 34*w^3 - 20*w^2 - 32*w + 5],\ [841, 29, -4*w^5 + 3*w^4 + 24*w^3 - 15*w^2 - 25*w + 7],\ [841, 29, 5*w^5 - 3*w^4 - 30*w^3 + 15*w^2 + 32*w - 6],\ [841, 29, 5*w^5 - 2*w^4 - 30*w^3 + 10*w^2 + 33*w - 2],\ [853, 853, -5*w^5 + 3*w^4 + 29*w^3 - 16*w^2 - 29*w + 9],\ [853, 853, -5*w^5 + 3*w^4 + 29*w^3 - 16*w^2 - 30*w + 7],\ [881, 881, -3*w^5 + 3*w^4 + 16*w^3 - 14*w^2 - 13*w + 3],\ [881, 881, 4*w^5 - 2*w^4 - 24*w^3 + 9*w^2 + 25*w + 2],\ [881, 881, -8*w^5 + 5*w^4 + 46*w^3 - 24*w^2 - 46*w + 9],\ [881, 881, 5*w^5 - 2*w^4 - 30*w^3 + 8*w^2 + 34*w + 2],\ [883, 883, 3*w^5 - w^4 - 17*w^3 + 6*w^2 + 16*w - 5],\ [883, 883, 7*w^5 - 5*w^4 - 40*w^3 + 24*w^2 + 37*w - 8],\ [911, 911, w^5 - 8*w^3 + 14*w],\ [911, 911, -3*w^5 + 2*w^4 + 17*w^3 - 9*w^2 - 17*w - 1],\ [911, 911, -8*w^5 + 5*w^4 + 46*w^3 - 25*w^2 - 44*w + 9],\ [911, 911, 2*w^5 - 2*w^4 - 12*w^3 + 9*w^2 + 11*w],\ [937, 937, 5*w^5 - 2*w^4 - 30*w^3 + 9*w^2 + 32*w - 1],\ [937, 937, 3*w^5 - 3*w^4 - 17*w^3 + 15*w^2 + 15*w - 4],\ [967, 967, -w^5 + 7*w^3 - 9*w + 2],\ [967, 967, 6*w^5 - 4*w^4 - 35*w^3 + 20*w^2 + 36*w - 11]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x K = QQ e = 1 hecke_eigenvalues_array = [1, -7, 1, -6, -5, 9, -6, -6, -3, 11, -5, -3, 6, -15, -5, -12, 10, -18, 6, -13, -6, -8, 15, -6, -6, -6, 11, -10, 4, 4, -22, 13, -1, 22, 15, 22, -20, -10, -6, -19, 22, 18, -12, 22, 15, 8, -27, 15, 29, -11, -4, -29, 10, -18, 27, -22, -10, -8, 4, 12, -16, -28, -21, -23, 0, 7, 23, -33, 22, 1, -16, -2, -7, 14, -16, -23, -30, 33, -4, 24, 30, 9, 14, -22, -29, -14, -6, -27, -1, -8, -33, -40, -36, -36, 16, -12, 22, 15, -4, -4, -27, 15, -20, -41, 36, 8, -18, -18, -6, 29, -14, 0, 2, 19, 28, -21, -19, -16, -22, -1, 28, -7, -44, 5, 47, 16, -19, 14, -7, 5, -2, -31, -24, -16, -2, 4, -3, 11, 4, 28, -7, 41, -50, 34, -9, 26, -7, -10, 28, -24, 36, -20, 18, -39, -24, -32, -52, -38, -41, 22] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([13, 13, 3*w^5 - 2*w^4 - 17*w^3 + 10*w^2 + 16*w - 3])] = -1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]