/* 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, 9, -9, -1, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([16, 2, 2]) primes_array = [ [5, 5, -2*w^3 - 2*w^2 + 13*w + 8],\ [11, 11, 2*w^3 + 3*w^2 - 12*w - 11],\ [11, 11, -w^3 - 2*w^2 + 6*w + 9],\ [11, 11, w^3 + w^2 - 7*w - 4],\ [11, 11, w - 1],\ [16, 2, 2],\ [19, 19, -w^2 + 4],\ [19, 19, 2*w^3 + 2*w^2 - 13*w - 6],\ [19, 19, 3*w^3 + 4*w^2 - 18*w - 16],\ [19, 19, -w^3 - w^2 + 5*w + 3],\ [49, 7, 2*w^3 + 3*w^2 - 13*w - 15],\ [59, 59, -3*w^3 - 4*w^2 + 19*w + 14],\ [59, 59, 3*w^3 + 4*w^2 - 18*w - 17],\ [59, 59, 2*w^3 + 3*w^2 - 11*w - 13],\ [59, 59, -w^3 - 2*w^2 + 7*w + 9],\ [71, 71, 3*w^3 + 3*w^2 - 17*w - 10],\ [71, 71, -2*w^3 - w^2 + 14*w + 2],\ [71, 71, 5*w^3 + 7*w^2 - 30*w - 25],\ [71, 71, -3*w^3 - 4*w^2 + 16*w + 16],\ [81, 3, -3],\ [89, 89, -2*w^3 - 2*w^2 + 13*w + 4],\ [89, 89, 3*w^3 + 4*w^2 - 17*w - 16],\ [89, 89, 3*w^3 + 4*w^2 - 18*w - 18],\ [89, 89, -4*w^3 - 5*w^2 + 25*w + 18],\ [139, 139, -w^3 + 6*w + 1],\ [139, 139, -3*w^3 - 3*w^2 + 19*w + 8],\ [139, 139, 4*w^3 + 5*w^2 - 24*w - 21],\ [139, 139, -2*w^3 - 2*w^2 + 11*w + 5],\ [151, 151, -6*w^3 - 7*w^2 + 37*w + 27],\ [151, 151, 3*w^3 + 5*w^2 - 19*w - 17],\ [151, 151, -6*w^3 - 8*w^2 + 37*w + 31],\ [151, 151, 4*w^3 + 6*w^2 - 25*w - 26],\ [191, 191, 5*w^3 + 7*w^2 - 31*w - 26],\ [191, 191, -2*w^3 - 3*w^2 + 10*w + 13],\ [191, 191, -5*w^3 - 7*w^2 + 31*w + 29],\ [191, 191, 2*w^3 + 4*w^2 - 13*w - 17],\ [199, 199, 2*w^3 + 4*w^2 - 11*w - 20],\ [199, 199, -2*w^3 - 2*w^2 + 11*w + 3],\ [199, 199, 3*w^3 + 5*w^2 - 17*w - 17],\ [199, 199, 3*w^3 + 3*w^2 - 19*w - 6],\ [211, 211, 3*w^3 + 5*w^2 - 18*w - 19],\ [211, 211, -3*w^3 - 5*w^2 + 18*w + 21],\ [211, 211, 2*w^3 + 4*w^2 - 12*w - 17],\ [211, 211, -4*w^3 - 6*w^2 + 24*w + 23],\ [229, 229, 2*w^3 + 4*w^2 - 13*w - 19],\ [229, 229, -2*w^3 - 3*w^2 + 10*w + 15],\ [229, 229, -5*w^3 - 7*w^2 + 31*w + 24],\ [229, 229, 2*w^3 + 2*w^2 - 14*w - 3],\ [269, 269, 2*w^3 + w^2 - 11*w - 1],\ [269, 269, -6*w^3 - 7*w^2 + 37*w + 24],\ [269, 269, 5*w^3 + 6*w^2 - 29*w - 24],\ [269, 269, 3*w^3 + 2*w^2 - 19*w - 7],\ [281, 281, -4*w^3 - 6*w^2 + 25*w + 25],\ [281, 281, -5*w^3 - 7*w^2 + 31*w + 27],\ [281, 281, w^2 + 2*w - 7],\ [281, 281, 3*w^3 + 5*w^2 - 19*w - 18],\ [331, 331, -2*w^3 - 3*w^2 + 12*w + 7],\ [331, 331, w^3 + w^2 - 7*w - 8],\ [331, 331, w^3 + 2*w^2 - 6*w - 13],\ [331, 331, w - 5],\ [349, 349, 2*w^3 + 5*w^2 - 16*w - 12],\ [349, 349, 2*w^3 + 4*w^2 - 15*w - 10],\ [349, 349, 8*w^3 + 9*w^2 - 50*w - 32],\ [349, 349, 2*w^3 + 4*w^2 - 15*w - 9],\ [401, 401, 4*w^3 + 3*w^2 - 21*w - 12],\ [401, 401, 3*w^3 + 2*w^2 - 16*w - 9],\ [401, 401, -3*w^3 - 6*w^2 + 20*w + 19],\ [401, 401, -4*w^3 - 3*w^2 + 27*w + 9],\ [409, 409, -5*w^3 - 6*w^2 + 31*w + 20],\ [409, 409, 3*w^3 + 4*w^2 - 20*w - 12],\ [409, 409, -3*w^3 - 5*w^2 + 17*w + 23],\ [409, 409, 4*w^3 + 5*w^2 - 23*w - 21],\ [419, 419, -4*w^3 - 4*w^2 + 21*w + 18],\ [419, 419, -4*w^3 - 4*w^2 + 20*w + 19],\ [419, 419, -3*w^3 - 4*w^2 + 16*w + 18],\ [419, 419, -2*w^3 - w^2 + 8*w + 10],\ [421, 421, -6*w^3 - 7*w^2 + 36*w + 26],\ [421, 421, 5*w^3 + 5*w^2 - 31*w - 17],\ [421, 421, 3*w^3 + 2*w^2 - 18*w - 6],\ [421, 421, 4*w^3 + 4*w^2 - 23*w - 14],\ [431, 431, 6*w^3 + 7*w^2 - 36*w - 25],\ [431, 431, -3*w^3 - 2*w^2 + 18*w + 5],\ [431, 431, 4*w^3 + 4*w^2 - 23*w - 15],\ [431, 431, -5*w^3 - 5*w^2 + 31*w + 18],\ [439, 439, w^3 + w^2 - 4*w - 1],\ [439, 439, w^3 + 3*w^2 - 6*w - 10],\ [439, 439, 3*w^3 + 3*w^2 - 20*w - 7],\ [439, 439, 5*w^3 + 7*w^2 - 30*w - 30],\ [479, 479, -4*w^3 - 5*w^2 + 25*w + 15],\ [479, 479, -5*w^3 - 6*w^2 + 32*w + 21],\ [479, 479, 5*w^3 + 7*w^2 - 29*w - 28],\ [479, 479, 2*w^2 + w - 9],\ [491, 491, 7*w^3 + 8*w^2 - 43*w - 31],\ [491, 491, 6*w^3 + 6*w^2 - 37*w - 24],\ [491, 491, 4*w^3 + 3*w^2 - 24*w - 6],\ [491, 491, 5*w^3 + 5*w^2 - 29*w - 21],\ [509, 509, -4*w^3 - 5*w^2 + 25*w + 14],\ [509, 509, -3*w^3 - 4*w^2 + 17*w + 20],\ [509, 509, -6*w^3 - 7*w^2 + 38*w + 25],\ [509, 509, -6*w^3 - 8*w^2 + 35*w + 31],\ [541, 541, 9*w^3 + 12*w^2 - 55*w - 46],\ [541, 541, w^3 + 4*w^2 - 7*w - 17],\ [541, 541, -4*w^3 - 5*w^2 + 21*w + 19],\ [541, 541, -4*w^3 - 3*w^2 + 27*w + 8],\ [571, 571, -3*w^3 - 6*w^2 + 18*w + 23],\ [571, 571, 6*w^3 + 9*w^2 - 36*w - 37],\ [571, 571, 7*w^3 + 10*w^2 - 42*w - 36],\ [571, 571, 2*w^3 + 5*w^2 - 12*w - 24],\ [619, 619, -9*w^3 - 11*w^2 + 55*w + 39],\ [619, 619, -5*w^3 - 4*w^2 + 32*w + 14],\ [619, 619, 7*w^3 + 9*w^2 - 42*w - 36],\ [619, 619, 4*w^3 + 6*w^2 - 26*w - 27],\ [631, 631, 2*w^3 + 5*w^2 - 12*w - 21],\ [631, 631, -7*w^3 - 10*w^2 + 42*w + 39],\ [631, 631, 4*w^3 + 4*w^2 - 27*w - 13],\ [631, 631, w^3 + w^2 - 3*w - 4],\ [641, 641, -9*w^3 - 10*w^2 + 56*w + 36],\ [641, 641, -5*w^3 - 9*w^2 + 30*w + 32],\ [641, 641, -3*w^3 - 6*w^2 + 19*w + 21],\ [641, 641, 4*w^3 + 3*w^2 - 22*w - 10],\ [701, 701, -2*w^3 - 5*w^2 + 14*w + 20],\ [701, 701, -6*w^3 - 7*w^2 + 36*w + 34],\ [701, 701, 4*w^3 + 6*w^2 - 23*w - 17],\ [701, 701, -9*w^3 - 12*w^2 + 56*w + 46],\ [719, 719, 3*w^3 + 5*w^2 - 17*w - 25],\ [719, 719, -3*w^3 - 2*w^2 + 17*w + 3],\ [719, 719, 7*w^3 + 8*w^2 - 43*w - 26],\ [719, 719, -6*w^3 - 7*w^2 + 35*w + 29],\ [751, 751, 2*w^3 + 2*w^2 - 15*w - 4],\ [751, 751, 5*w^3 + 8*w^2 - 30*w - 34],\ [751, 751, w^3 + w^2 - 9*w - 5],\ [751, 751, -4*w^3 - 7*w^2 + 24*w + 26],\ [769, 769, 6*w^3 + 8*w^2 - 37*w - 27],\ [769, 769, -w^3 - 3*w^2 + 7*w + 16],\ [769, 769, 2*w^3 + w^2 - 14*w - 5],\ [769, 769, -3*w^3 - 4*w^2 + 16*w + 19],\ [821, 821, 6*w^3 + 6*w^2 - 37*w - 23],\ [821, 821, 5*w^3 + 5*w^2 - 29*w - 20],\ [821, 821, -7*w^3 - 8*w^2 + 42*w + 27],\ [821, 821, 4*w^3 + 3*w^2 - 24*w - 7],\ [829, 829, 7*w^3 + 8*w^2 - 44*w - 28],\ [829, 829, 7*w^3 + 9*w^2 - 41*w - 35],\ [829, 829, -2*w^3 - w^2 + 10*w + 2],\ [829, 829, 2*w^3 - 13*w + 2],\ [839, 839, 4*w^3 + 8*w^2 - 25*w - 26],\ [839, 839, -3*w^3 - 3*w^2 + 15*w + 16],\ [839, 839, -w^3 - 4*w^2 + 8*w + 10],\ [839, 839, -3*w^3 - 7*w^2 + 18*w + 26],\ [841, 29, -5*w^3 - 5*w^2 + 30*w + 16],\ [841, 29, -5*w^3 - 5*w^2 + 30*w + 19],\ [859, 859, 5*w^3 + 6*w^2 - 32*w - 20],\ [859, 859, -4*w^3 - 5*w^2 + 23*w + 23],\ [859, 859, -5*w^3 - 6*w^2 + 31*w + 18],\ [859, 859, -2*w^3 - w^2 + 13*w + 6],\ [911, 911, 9*w^3 + 12*w^2 - 56*w - 48],\ [911, 911, 6*w^3 + 5*w^2 - 38*w - 14],\ [911, 911, 7*w^3 + 8*w^2 - 40*w - 28],\ [911, 911, 10*w^3 + 12*w^2 - 61*w - 46],\ [929, 929, -7*w^3 - 10*w^2 + 39*w + 39],\ [929, 929, -10*w^3 - 13*w^2 + 60*w + 45],\ [929, 929, -6*w^3 - 7*w^2 + 32*w + 29],\ [929, 929, -3*w^3 - w^2 + 12*w + 12],\ [961, 31, -5*w^3 - 5*w^2 + 30*w + 17],\ [961, 31, -5*w^3 - 5*w^2 + 30*w + 18],\ [991, 991, 10*w^3 + 12*w^2 - 61*w - 45],\ [991, 991, -6*w^3 - 9*w^2 + 38*w + 38],\ [991, 991, 6*w^3 + 8*w^2 - 37*w - 38],\ [991, 991, -8*w^3 - 11*w^2 + 49*w + 43]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^4 - 20*x^2 + 2 K. = NumberField(heckePol) hecke_eigenvalues_array = [e, -2/7*e^2 + 20/7, -2/7*e^2 + 20/7, -2/7*e^2 + 20/7, -2/7*e^2 + 20/7, -1, -1/7*e^3 + 31/7*e, -1/7*e^3 + 31/7*e, -1/7*e^3 + 31/7*e, -1/7*e^3 + 31/7*e, 12/7*e^3 - 232/7*e, 10/7*e^3 - 205/7*e, 10/7*e^3 - 205/7*e, 10/7*e^3 - 205/7*e, 10/7*e^3 - 205/7*e, -1/7*e^2 - 4/7, -1/7*e^2 - 4/7, -1/7*e^2 - 4/7, -1/7*e^2 - 4/7, -8/7*e^2 + 80/7, -2*e^3 + 36*e, -2*e^3 + 36*e, -2*e^3 + 36*e, -2*e^3 + 36*e, 10/7*e^3 - 191/7*e, 10/7*e^3 - 191/7*e, 10/7*e^3 - 191/7*e, 10/7*e^3 - 191/7*e, 2/7*e^2 - 62/7, 2/7*e^2 - 62/7, 2/7*e^2 - 62/7, 2/7*e^2 - 62/7, 3/7*e^2 + 12/7, 3/7*e^2 + 12/7, 3/7*e^2 + 12/7, 3/7*e^2 + 12/7, -30/7*e^3 + 622/7*e, -30/7*e^3 + 622/7*e, -30/7*e^3 + 622/7*e, -30/7*e^3 + 622/7*e, -4/7*e^2 - 72/7, -4/7*e^2 - 72/7, -4/7*e^2 - 72/7, -4/7*e^2 - 72/7, -8/7*e^3 + 171/7*e, -8/7*e^3 + 171/7*e, -8/7*e^3 + 171/7*e, -8/7*e^3 + 171/7*e, 31/7*e^3 - 597/7*e, 31/7*e^3 - 597/7*e, 31/7*e^3 - 597/7*e, 31/7*e^3 - 597/7*e, 4/7*e^2 - 40/7, 4/7*e^2 - 40/7, 4/7*e^2 - 40/7, 4/7*e^2 - 40/7, -8/7*e^2 + 164/7, -8/7*e^2 + 164/7, -8/7*e^2 + 164/7, -8/7*e^2 + 164/7, -1/7*e^3 + 59/7*e, -1/7*e^3 + 59/7*e, -1/7*e^3 + 59/7*e, -1/7*e^3 + 59/7*e, -3/7*e^2 - 138/7, -3/7*e^2 - 138/7, -3/7*e^2 - 138/7, -3/7*e^2 - 138/7, 26/7*e^3 - 554/7*e, 26/7*e^3 - 554/7*e, 26/7*e^3 - 554/7*e, 26/7*e^3 - 554/7*e, -11/7*e^3 + 257/7*e, -11/7*e^3 + 257/7*e, -11/7*e^3 + 257/7*e, -11/7*e^3 + 257/7*e, -2*e^2 + 18, -2*e^2 + 18, -2*e^2 + 18, -2*e^2 + 18, -4/7*e^2 + 40/7, -4/7*e^2 + 40/7, -4/7*e^2 + 40/7, -4/7*e^2 + 40/7, 2*e^3 - 44*e, 2*e^3 - 44*e, 2*e^3 - 44*e, 2*e^3 - 44*e, 12/7*e^3 - 246/7*e, 12/7*e^3 - 246/7*e, 12/7*e^3 - 246/7*e, 12/7*e^3 - 246/7*e, 2/7*e^2 + 176/7, 2/7*e^2 + 176/7, 2/7*e^2 + 176/7, 2/7*e^2 + 176/7, -16/7*e^3 + 321/7*e, -16/7*e^3 + 321/7*e, -16/7*e^3 + 321/7*e, -16/7*e^3 + 321/7*e, -12/7*e^2 + 78/7, -12/7*e^2 + 78/7, -12/7*e^2 + 78/7, -12/7*e^2 + 78/7, 18/7*e^2 - 180/7, 18/7*e^2 - 180/7, 18/7*e^2 - 180/7, 18/7*e^2 - 180/7, -2/7*e^3 - 15/7*e, -2/7*e^3 - 15/7*e, -2/7*e^3 - 15/7*e, -2/7*e^3 - 15/7*e, -25/7*e^2 + 292/7, -25/7*e^2 + 292/7, -25/7*e^2 + 292/7, -25/7*e^2 + 292/7, 15/7*e^2 - 262/7, 15/7*e^2 - 262/7, 15/7*e^2 - 262/7, 15/7*e^2 - 262/7, 22/7*e^2 - 94/7, 22/7*e^2 - 94/7, 22/7*e^2 - 94/7, 22/7*e^2 - 94/7, -48/7*e^3 + 970/7*e, -48/7*e^3 + 970/7*e, -48/7*e^3 + 970/7*e, -48/7*e^3 + 970/7*e, 6/7*e^2 - 18/7, 6/7*e^2 - 18/7, 6/7*e^2 - 18/7, 6/7*e^2 - 18/7, 18/7*e^3 - 390/7*e, 18/7*e^3 - 390/7*e, 18/7*e^3 - 390/7*e, 18/7*e^3 - 390/7*e, 6/7*e^2 - 242/7, 6/7*e^2 - 242/7, 6/7*e^2 - 242/7, 6/7*e^2 - 242/7, 48/7*e^3 - 907/7*e, 48/7*e^3 - 907/7*e, 48/7*e^3 - 907/7*e, 48/7*e^3 - 907/7*e, 22/7*e^3 - 458/7*e, 22/7*e^3 - 458/7*e, 22/7*e^3 - 458/7*e, 22/7*e^3 - 458/7*e, -8/7*e^2 + 402/7, -8/7*e^2 + 402/7, -7*e^3 + 129*e, -7*e^3 + 129*e, -7*e^3 + 129*e, -7*e^3 + 129*e, 10/7*e^2 - 198/7, 10/7*e^2 - 198/7, 10/7*e^2 - 198/7, 10/7*e^2 - 198/7, 36/7*e^3 - 738/7*e, 36/7*e^3 - 738/7*e, 36/7*e^3 - 738/7*e, 36/7*e^3 - 738/7*e, -4/7*e^2 + 418/7, -4/7*e^2 + 418/7, -1/7*e^2 - 228/7, -1/7*e^2 - 228/7, -1/7*e^2 - 228/7, -1/7*e^2 - 228/7] hecke_eigenvalues = {} for i in range(len(hecke_eigenvalues_array)): hecke_eigenvalues[primes[i]] = hecke_eigenvalues_array[i] AL_eigenvalues = {} AL_eigenvalues[ZF.ideal([16, 2, 2])] = 1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]