/* 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, -6, -1, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([16, 4, w^2 - 3]) primes_array = [ [2, 2, -w + 1],\ [4, 2, -w^2 - w + 3],\ [5, 5, w^3 - 5*w + 1],\ [11, 11, -w^3 + 6*w - 2],\ [13, 13, -w^2 + 4],\ [17, 17, 2*w^3 + w^2 - 10*w - 2],\ [37, 37, w^3 + w^2 - 6*w - 3],\ [37, 37, -w^3 + 6*w - 4],\ [41, 41, -w^3 + 5*w - 5],\ [43, 43, w^3 - 4*w + 4],\ [43, 43, -2*w^3 + 11*w - 2],\ [47, 47, -2*w^3 - w^2 + 12*w],\ [47, 47, 2*w - 1],\ [61, 61, -w^3 + w^2 + 6*w - 5],\ [71, 71, w^3 + w^2 - 4*w - 3],\ [71, 71, 2*w^3 + 2*w^2 - 9*w - 2],\ [79, 79, w^3 + w^2 - 6*w - 5],\ [81, 3, -3],\ [83, 83, -3*w^3 - w^2 + 16*w + 3],\ [97, 97, w^3 + w^2 - 6*w + 1],\ [103, 103, -2*w^3 - 2*w^2 + 7*w - 2],\ [103, 103, -3*w^3 + 14*w - 12],\ [107, 107, -w^3 + 3*w + 3],\ [125, 5, -2*w^3 - w^2 + 8*w - 2],\ [127, 127, w^3 - w^2 - 4*w + 1],\ [131, 131, 2*w^3 - 10*w + 3],\ [137, 137, -5*w^3 - 2*w^2 + 26*w + 2],\ [137, 137, w^3 - 7*w + 1],\ [139, 139, -2*w^3 - w^2 + 8*w],\ [149, 149, w^3 - 2*w^2 - 7*w + 9],\ [149, 149, -2*w^3 + 11*w],\ [151, 151, 2*w^2 + 2*w - 9],\ [151, 151, -5*w^3 - 2*w^2 + 27*w + 1],\ [151, 151, w^3 - 7*w + 3],\ [151, 151, 2*w^3 + 2*w^2 - 11*w - 8],\ [157, 157, w^3 - 7*w - 3],\ [169, 13, 6*w^3 + 4*w^2 - 32*w - 11],\ [173, 173, w^3 + w^2 - 4*w - 7],\ [191, 191, -3*w^3 + 14*w - 10],\ [193, 193, 2*w^3 - 12*w + 3],\ [197, 197, w + 4],\ [197, 197, w^3 + 2*w^2 - 6*w - 6],\ [197, 197, 2*w^3 + w^2 - 10*w + 2],\ [197, 197, 2*w^2 - 3],\ [199, 199, w^3 + 2*w^2 - 4*w - 4],\ [223, 223, -w^3 - 2*w^2 + 5*w + 7],\ [229, 229, 5*w^3 + 3*w^2 - 26*w - 5],\ [233, 233, 2*w + 3],\ [241, 241, 2*w^3 + 2*w^2 - 10*w - 3],\ [251, 251, w^3 - 2*w^2 - 5*w + 7],\ [263, 263, 2*w^3 + w^2 - 12*w + 2],\ [269, 269, 2*w^3 + 3*w^2 - 14*w - 4],\ [271, 271, 3*w^3 + 2*w^2 - 15*w - 7],\ [281, 281, 2*w^2 + w - 4],\ [281, 281, w^2 - 8],\ [283, 283, -3*w^3 + 18*w - 2],\ [283, 283, -w^3 - w^2 + 4*w - 3],\ [283, 283, -w^3 + w^2 + 6*w - 3],\ [283, 283, w^3 + 2*w^2 - 5*w - 3],\ [293, 293, -4*w^3 + 19*w - 16],\ [307, 307, -w^3 + 2*w^2 + 6*w - 8],\ [311, 311, 3*w^3 + 2*w^2 - 17*w - 9],\ [311, 311, w^3 + 2*w^2 - 6*w - 8],\ [313, 313, -4*w^3 - 2*w^2 + 17*w - 6],\ [317, 317, 2*w^3 - 9*w + 2],\ [317, 317, 3*w^2 + 2*w - 10],\ [331, 331, 5*w^3 + 4*w^2 - 26*w - 12],\ [337, 337, 2*w^3 + w^2 - 8*w + 6],\ [347, 347, 2*w^3 - 9*w + 4],\ [353, 353, w^3 - 5*w - 3],\ [359, 359, -w^3 + w + 3],\ [367, 367, -2*w^3 + 12*w - 7],\ [367, 367, 2*w^3 + 4*w^2 - 6*w - 7],\ [379, 379, -w^3 - 2*w^2 + w + 3],\ [379, 379, -3*w^2 - 2*w + 12],\ [397, 397, -2*w^3 - w^2 + 6*w + 2],\ [397, 397, -w^3 + 3*w - 7],\ [397, 397, -2*w^3 - 2*w^2 + 9*w + 6],\ [397, 397, w^3 + 2*w^2 - 5*w - 5],\ [401, 401, 3*w^3 + 3*w^2 - 16*w - 13],\ [401, 401, w^3 + 2*w^2 - 6*w - 4],\ [431, 431, 3*w^3 + 2*w^2 - 14*w - 4],\ [431, 431, 2*w^3 + w^2 - 10*w + 4],\ [433, 433, -2*w^3 + 2*w^2 + 13*w - 12],\ [433, 433, -w^3 + 4*w^2 + 8*w - 18],\ [439, 439, 2*w^3 + 2*w^2 - 8*w + 3],\ [439, 439, -w^3 + 3*w^2 + 6*w - 11],\ [443, 443, -7*w^3 - 2*w^2 + 39*w + 1],\ [457, 457, w^3 + 2*w^2 - 3*w - 7],\ [461, 461, w^2 - 2*w - 4],\ [461, 461, 4*w^3 + 2*w^2 - 20*w - 5],\ [463, 463, 3*w^3 + 2*w^2 - 13*w - 3],\ [467, 467, -3*w^3 + 15*w - 11],\ [467, 467, -4*w^3 + 18*w + 1],\ [479, 479, 2*w^2 + 2*w - 11],\ [487, 487, -3*w^3 - 2*w^2 + 19*w + 1],\ [487, 487, 4*w^3 + 2*w^2 - 19*w],\ [491, 491, 3*w^3 + 2*w^2 - 18*w],\ [499, 499, -w^3 + 3*w - 5],\ [499, 499, -6*w^3 - 2*w^2 + 31*w - 2],\ [523, 523, 2*w^2 + 3*w - 10],\ [541, 541, 2*w^3 + 3*w^2 - 8*w - 6],\ [547, 547, -3*w^3 - w^2 + 16*w - 3],\ [563, 563, -w^3 + 5*w - 7],\ [563, 563, 2*w^2 + w - 2],\ [571, 571, 3*w^3 + 2*w^2 - 12*w + 6],\ [577, 577, -2*w^3 + w^2 + 8*w - 6],\ [577, 577, -2*w^3 - 2*w^2 + 12*w + 11],\ [587, 587, -5*w^3 - w^2 + 26*w - 5],\ [593, 593, -w^3 + 8*w - 4],\ [599, 599, -w^3 + w^2 + 4*w - 9],\ [601, 601, 3*w^3 + w^2 - 14*w + 1],\ [607, 607, 2*w^2 + 2*w - 3],\ [607, 607, 4*w^3 - 24*w + 1],\ [613, 613, -3*w^3 - 3*w^2 + 16*w + 9],\ [617, 617, -2*w^3 + 13*w - 6],\ [619, 619, -4*w^3 + 20*w - 1],\ [631, 631, -5*w^3 - 2*w^2 + 29*w + 5],\ [643, 643, -w^3 - 2*w^2 + 6*w + 10],\ [647, 647, -5*w^3 - 4*w^2 + 20*w - 4],\ [647, 647, -3*w^3 - 3*w^2 + 18*w + 7],\ [653, 653, 4*w^3 - 20*w + 3],\ [653, 653, w^3 - 8*w + 2],\ [659, 659, -9*w^3 - 4*w^2 + 48*w + 4],\ [659, 659, -5*w^3 - 2*w^2 + 25*w - 1],\ [661, 661, 4*w^3 + 2*w^2 - 21*w],\ [673, 673, -3*w^3 + w^2 + 18*w - 9],\ [683, 683, 2*w^3 + 2*w^2 - 7*w - 4],\ [691, 691, -5*w^3 - 3*w^2 + 28*w + 7],\ [691, 691, 3*w^3 - 15*w + 5],\ [701, 701, -2*w^3 + 2*w^2 + 11*w - 10],\ [701, 701, -w^3 + 8*w - 10],\ [709, 709, -4*w^3 - w^2 + 20*w - 4],\ [709, 709, w^3 + 4*w^2 - w - 13],\ [719, 719, -2*w^3 + 12*w - 9],\ [739, 739, -6*w^3 - 4*w^2 + 29*w + 6],\ [739, 739, -w^3 + w^2 + 8*w - 7],\ [769, 769, -w^3 + w^2 + 8*w - 5],\ [769, 769, 3*w^3 - 2*w^2 - 15*w + 19],\ [773, 773, -4*w^3 + 22*w - 9],\ [797, 797, -2*w^3 + 13*w - 2],\ [827, 827, -w^3 + 6*w - 8],\ [827, 827, -4*w^2 - 3*w + 14],\ [839, 839, w^2 + 4*w - 6],\ [857, 857, 7*w^3 + 2*w^2 - 37*w + 1],\ [863, 863, 3*w^3 - 14*w + 2],\ [881, 881, -3*w^3 + 2*w^2 + 15*w - 13],\ [883, 883, 2*w^3 - w^2 - 10*w + 14],\ [887, 887, -5*w^3 - 4*w^2 + 25*w + 11],\ [887, 887, 3*w^3 - 14*w + 8],\ [907, 907, 4*w^3 - 20*w + 13],\ [907, 907, 2*w^2 + 4*w - 5],\ [907, 907, -3*w^3 + 18*w - 4],\ [907, 907, w - 6],\ [947, 947, 3*w^3 + 3*w^2 - 12*w - 5],\ [947, 947, 2*w^3 + 3*w^2 - 10*w - 10],\ [961, 31, 2*w^3 - w^2 - 12*w + 4],\ [961, 31, -w^3 + 2*w^2 + 5*w - 5],\ [971, 971, -4*w^3 - w^2 + 20*w - 2],\ [991, 991, -2*w^3 + 3*w^2 + 14*w - 18],\ [997, 997, 7*w^3 + 2*w^2 - 36*w + 6]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^3 + x^2 - 7*x + 1 K. = NumberField(heckePol) hecke_eigenvalues_array = [0, e, -1/2*e^2 - e + 3/2, -e^2 + 5, 1/2*e^2 + 3/2, 1/2*e^2 - e - 7/2, -1/2*e^2 - 2*e + 17/2, -5/2*e^2 - 3*e + 19/2, 3/2*e^2 + 4*e - 15/2, -2*e^2 - 4*e + 10, -e^2 - 2*e + 3, -2*e^2 - 4*e + 14, 4*e + 4, -e^2 - 5, -e^2 + 1, -e^2 - 2*e + 7, 2*e^2 + 4*e - 14, -1/2*e^2 + 2*e + 17/2, -e^2 + 2*e + 15, 5/2*e^2 + 3*e - 35/2, 2*e^2 + 4*e - 6, e^2 - 2*e - 11, e^2 + 2*e + 5, 3/2*e^2 + 7*e - 1/2, -e^2 + 2*e + 3, -4*e^2 - 8*e + 24, 4*e^2 + 8*e - 18, -1/2*e^2 + 3*e + 35/2, 3*e^2 + 2*e - 13, -3/2*e^2 - 2*e + 35/2, 7/2*e^2 + 7*e - 21/2, 4*e^2 + 8*e - 20, -e^2 + 4*e + 5, 2*e^2 - 10, -e^2 - 4*e - 3, 1/2*e^2 + 2*e - 9/2, -5/2*e^2 + 2*e + 37/2, -1/2*e^2 - 5*e + 11/2, -2*e^2 - 8*e + 10, -e^2 - 4*e + 3, -e^2 - 4*e - 1, 3*e^2 - 17, 5/2*e^2 - 49/2, 5/2*e^2 + 2*e - 29/2, 3*e^2 + 2*e - 9, e^2 + 7, -9/2*e^2 - 3*e + 55/2, 7/2*e^2 + 9*e - 41/2, -9/2*e^2 - 4*e + 29/2, e^2 + 3, 3*e^2 - 3, 11/2*e^2 + 8*e - 39/2, -5*e^2 - 6*e + 15, 7/2*e^2 + e - 25/2, -4*e^2 - 8*e + 22, -2*e^2 - 4*e - 6, 3*e^2 + 2*e + 3, -2*e^2 - 4*e + 2, 2*e^2 + 4*e - 2, 11/2*e^2 + 7*e - 25/2, -3*e^2 - 14*e + 17, -2*e^2 - 4*e + 22, -e^2 + 4*e + 5, 3/2*e^2 + 8*e - 7/2, 11/2*e^2 + 2*e - 67/2, -2, 4*e^2 + 12*e - 20, 11/2*e^2 + 12*e - 55/2, -5*e^2 + 25, 1/2*e^2 + 43/2, e^2 + 2*e + 9, e^2 + 8*e - 1, 7*e^2 + 12*e - 35, 4, -3*e^2 + 4*e + 27, 5/2*e^2 + 3*e - 59/2, e^2 - 8*e - 15, -1/2*e^2 + 4*e + 29/2, -2*e^2 - 8*e + 24, 15/2*e^2 + 11*e - 45/2, 3/2*e^2 - 5*e - 33/2, 5*e^2 + 12*e - 17, 4*e^2 + 12*e - 24, 6*e^2 + 8*e - 28, -3/2*e^2 + 5*e + 25/2, -3*e^2 - 12*e + 7, -2*e^2 + 26, 3*e^2 + 2*e + 3, -3/2*e^2 + 3*e + 21/2, 3*e^2 - 8*e - 33, 3*e^2 - 25, e^2 - 2*e + 13, -e^2 - 8*e + 5, 4*e^2 + 8*e - 16, -e^2 + 2*e + 3, -7*e^2 - 6*e + 41, -11*e^2 - 16*e + 43, 3*e^2 + 10*e - 5, -3*e^2 - 14*e + 17, -4*e^2 + 24, -4*e - 8, 1/2*e^2 - 2*e - 49/2, 4*e^2 + 8*e - 32, 2*e^2 + 4*e + 22, 8*e^2 + 16*e - 44, -8*e^2 - 12*e + 16, 9/2*e^2 + 9*e - 11/2, 3*e^2 + 4*e + 7, -4*e^2 - 16*e + 32, 9/2*e^2 + 10*e - 25/2, 4*e + 12, 2*e^2 - 8*e - 32, -e^2 - 4*e - 27, 4*e^2 - 4*e - 24, -9/2*e^2 - 11*e + 39/2, -13/2*e^2 - 8*e + 9/2, 3*e^2 + 8*e + 9, -2*e^2 + 4*e + 14, -6*e^2 - 24*e + 34, 2*e^2 + 12*e - 30, 4*e^2 + 12*e, -3/2*e^2 - 9*e + 21/2, -1/2*e^2 - 8*e + 37/2, 4*e^2 + 4*e - 4, -4*e^2 + 16, -11/2*e^2 - 7*e + 33/2, 1/2*e^2 + 4*e + 19/2, 4*e^2 + 20*e - 20, 7*e^2 + 4*e - 39, -12*e^2 - 16*e + 40, 9/2*e^2 + 5*e - 43/2, 1/2*e^2 + 7*e + 49/2, -11/2*e^2 - 9*e + 45/2, -5*e^2 - 12*e - 5, -e^2 - 8*e + 1, 7*e^2 + 12*e - 15, 3*e^2 + 4*e - 19, -4*e^2 - 8*e + 30, -3/2*e^2 + 3*e - 27/2, 19/2*e^2 + 6*e - 83/2, -21/2*e^2 - 4*e + 97/2, -4*e^2 - 12*e + 44, -11*e^2 - 6*e + 57, -10*e^2 - 24*e + 58, 3*e^2 + 12*e - 9, 7*e^2 + 20*e - 43, 9*e^2 + 20*e - 47, 4*e + 8, 8*e^2 + 8*e - 24, -4*e^2 - 12*e, -11*e^2 - 12*e + 35, -5*e^2 - 18*e + 23, e^2 + 4*e + 7, -2*e^2 - 4*e + 18, -e^2 + 12*e + 9, e^2 - 37, 4*e^2 + 8*e + 6, 9/2*e^2 + 16*e - 61/2, 6*e^2 + 8*e - 26, -5*e^2 - 18*e + 3, 7/2*e^2 - 7*e - 1/2] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([2, 2, -w + 1])] = 1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]