/* 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([44, 22, w^3 + w^2 - 3*w]) 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^4 + x^3 - 9*x^2 + 2*x + 4 K. = NumberField(heckePol) hecke_eigenvalues_array = [-1, e, -1/2*e^3 - 1/2*e^2 + 9/2*e - 1, -1, -1/2*e^3 - 1/2*e^2 + 9/2*e + 1, e + 2, 1/2*e^3 + 5/2*e^2 - 3/2*e - 6, -1/2*e^3 - 5/2*e^2 + 3/2*e + 10, 1/2*e^3 - 1/2*e^2 - 9/2*e + 6, -1/2*e^3 + 1/2*e^2 + 9/2*e - 5, 2*e^3 + 4*e^2 - 12*e - 4, -2*e^3 - 4*e^2 + 12*e + 6, -3/2*e^3 - 5/2*e^2 + 21/2*e + 8, e^3 + 2*e^2 - 5*e + 4, e^3 + 2*e^2 - 3*e - 4, -5/2*e^3 - 7/2*e^2 + 39/2*e - 2, 2*e^3 + 4*e^2 - 10*e - 6, -3*e^3 - 5*e^2 + 21*e + 2, -e^3 - e^2 + 11*e + 2, -2*e^2 - 5*e + 8, 3/2*e^3 + 7/2*e^2 - 23/2*e - 7, -e^2 - 4*e + 4, 3/2*e^3 + 5/2*e^2 - 25/2*e - 2, 3/2*e^3 + 3/2*e^2 - 33/2*e + 4, 1/2*e^3 + 7/2*e^2 + 9/2*e - 12, 3*e^3 + 3*e^2 - 22*e + 4, -5/2*e^3 - 5/2*e^2 + 33/2*e - 7, -3*e^3 - 6*e^2 + 23*e + 8, 3*e^3 + 6*e^2 - 17*e - 6, 1/2*e^3 - 1/2*e^2 - 23/2*e + 2, 3*e^3 + 4*e^2 - 27*e - 2, -7/2*e^3 - 13/2*e^2 + 45/2*e + 8, -e^3 - 2*e^2 + 3*e - 4, 5/2*e^3 + 7/2*e^2 - 39/2*e - 6, 2*e^3 + 2*e^2 - 16*e + 10, -e^3 - 5*e^2 + e + 20, 2*e^3 + 6*e^2 - 12*e - 10, -e^3 - e^2 + 5*e + 4, -3*e^3 - 3*e^2 + 28*e - 8, -1/2*e^3 - 1/2*e^2 + 21/2*e - 3, -1/2*e^3 - 1/2*e^2 + 5/2*e + 5, e^3 + e^2 - 8*e + 8, -1/2*e^3 - 1/2*e^2 + 23/2*e + 9, -7/2*e^3 - 7/2*e^2 + 65/2*e + 3, -2*e^2 - 3*e + 12, -5/2*e^3 - 5/2*e^2 + 45/2*e - 1, 5*e + 4, 1/2*e^3 + 5/2*e^2 - 5/2*e - 5, -e^3 - 3*e^2 + 2*e - 2, 5/2*e^3 + 11/2*e^2 - 39/2*e - 4, -3*e^2 - 8*e + 18, 5/2*e^3 + 9/2*e^2 - 37/2*e - 13, 7/2*e^3 + 7/2*e^2 - 59/2*e + 7, -e^3 - e^2 + 2*e - 2, 3/2*e^3 + 3/2*e^2 - 27/2*e - 1, -3*e - 4, 3/2*e^3 - 5/2*e^2 - 41/2*e + 19, e^3 + 7*e^2 + 2*e - 24, -5/2*e^3 - 17/2*e^2 + 29/2*e + 23, 3/2*e^3 + 11/2*e^2 - 25/2*e - 17, -2*e^2 - 4*e + 4, e^3 + 3*e^2 - 7*e - 12, -e^2 + 2*e + 20, -3/2*e^3 - 1/2*e^2 + 29/2*e + 8, 2*e^3 + e^2 - 16*e + 2, -3/2*e^3 - 1/2*e^2 + 21/2*e - 14, 7/2*e^3 + 11/2*e^2 - 43/2*e - 3, -4*e^3 - 6*e^2 + 27*e, 5/2*e^3 + 5/2*e^2 - 55/2*e + 1, -1/2*e^3 - 1/2*e^2 + 11/2*e - 1, -3*e^3 - 4*e^2 + 15*e, 11/2*e^3 + 13/2*e^2 - 85/2*e + 8, -2*e^3 - 10*e^2 + 6*e + 30, 2*e^3 + 10*e^2 - 6*e - 34, 5*e^3 + 9*e^2 - 36*e - 16, -5/2*e^3 - 13/2*e^2 + 17/2*e + 5, -4*e^3 - 10*e^2 + 25*e + 6, -1/2*e^3 - 1/2*e^2 - 11/2*e - 1, 5*e^3 + 5*e^2 - 44*e + 10, 5/2*e^3 + 17/2*e^2 - 17/2*e - 35, 7/2*e^3 + 5/2*e^2 - 67/2*e + 4, 6*e^3 + 10*e^2 - 42*e + 6, -4*e^3 - 8*e^2 + 20*e + 22, 1/2*e^3 + 3/2*e^2 - 21/2*e - 11, -11/2*e^3 - 11/2*e^2 + 97/2*e + 7, -2*e^3 - 2*e^2 + 20*e + 24, -e^3 - e^2 + 13*e + 26, 1/2*e^3 + 1/2*e^2 + 13/2*e + 19, -13/2*e^3 - 19/2*e^2 + 95/2*e + 8, 4*e^3 + 7*e^2 - 20*e + 2, -1/2*e^3 + 5/2*e^2 + 23/2*e - 20, -2*e^3 - 5*e^2 + 16*e + 4, 4*e^3 + 9*e^2 - 28*e - 2, 3/2*e^3 + 13/2*e^2 + 3/2*e - 20, -5*e^3 - 10*e^2 + 37*e + 12, -3/2*e^3 - 13/2*e^2 + 1/2*e + 32, -1/2*e^3 + 9/2*e^2 + 19/2*e - 28, -5*e^2 - 4*e + 18, -3*e^3 - 7*e^2 + 11*e + 18, 6*e^3 + 10*e^2 - 44*e, 2*e^3 + 2*e^2 - 12*e + 28, -3*e^3 - 3*e^2 + 23*e + 18, -3*e^3 - 11*e^2 + 15*e + 18, 2*e^3 + 10*e^2 - 4*e - 44, 3*e^3 + 5*e^2 - 21*e + 28, -2*e^3 - 4*e^2 + 10*e + 36, 6*e^3 + 6*e^2 - 52*e + 6, -e^3 - e^2 - 3*e - 8, -3*e^3 - 5*e^2 + 13*e - 12, -e^3 - 3*e^2 + 9*e + 18, -e^3 + e^2 + 13*e, 6*e^3 + 8*e^2 - 46*e - 12, -e^3 + 5*e - 30, 7*e^3 + 8*e^2 - 53*e + 2, -9/2*e^3 - 11/2*e^2 + 51/2*e - 12, 5/2*e^3 + 3/2*e^2 - 43/2*e - 14, 5/2*e^3 + 1/2*e^2 - 53/2*e + 1, e^3 + 3*e^2 - 12*e - 20, 9/2*e^3 + 13/2*e^2 - 81/2*e - 3, e^3 - e^2 - 20*e + 8, -2*e^3 - 2*e^2 + 18*e + 14, 4*e + 18, 4*e^3 + 10*e^2 - 29*e - 36, 5/2*e^3 + 23/2*e^2 - 23/2*e - 36, -e^3 - 10*e^2 - 5*e + 38, -1/2*e^3 - 13/2*e^2 - 19/2*e + 9, -e^3 + 5*e^2 + 21*e - 28, -3*e^3 - 9*e^2 + 23*e + 22, 6*e^3 + 12*e^2 - 46*e - 10, -e^3 - 7*e^2 - 9*e + 30, -1/2*e^3 + 7/2*e^2 + 3/2*e - 33, 7/2*e^3 - 1/2*e^2 - 69/2*e + 11, 9/2*e^3 + 15/2*e^2 - 41/2*e - 5, -17/2*e^3 - 23/2*e^2 + 129/2*e - 4, -2*e^3 - 4*e^2 + 20*e + 30, -2*e^3 + 24*e + 12, -3/2*e^3 - 9/2*e^2 + 1/2*e + 14, -2*e^3 - 2*e^2 + 12*e - 30, 3*e^3 + 3*e^2 - 23*e - 20, 5*e^3 + 8*e^2 - 39*e, -9/2*e^3 - 27/2*e^2 + 21/2*e + 43, 21/2*e^3 + 39/2*e^2 - 153/2*e - 8, -7/2*e^3 - 7/2*e^2 + 77/2*e - 16, -3/2*e^3 - 5/2*e^2 + 41/2*e - 2, -4*e^3 - 3*e^2 + 40*e - 16, -17/2*e^3 - 15/2*e^2 + 141/2*e - 15, 7/2*e^3 + 5/2*e^2 - 31/2*e + 18, -13/2*e^3 - 31/2*e^2 + 77/2*e + 16, e^3 + 9*e^2 + 17*e - 42, -9*e^3 - 17*e^2 + 71*e + 10, 11/2*e^3 + 29/2*e^2 - 55/2*e - 41, -5/2*e^3 - 7/2*e^2 + 15/2*e + 12, 7*e^3 + 8*e^2 - 57*e + 22, -4*e^3 - 4*e^2 + 44*e - 4, -5/2*e^3 - 9/2*e^2 + 21/2*e + 5, 5*e^3 + 7*e^2 - 38*e + 2, -e^3 + e^2 + 25*e - 6, -7*e^3 - 9*e^2 + 63*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([4, 2, -w^2 + w + 3])] = 1 AL_eigenvalues[ZF.ideal([11, 11, w])] = 1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]