/* 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([-49, -1, 1]) F. = NumberField(g) ZF = F.ring_of_integers() NN = ZF.ideal([36, 6, 6]) primes_array = [ [4, 2, 2],\ [7, 7, w - 7],\ [7, 7, w + 6],\ [9, 3, 3],\ [19, 19, w + 5],\ [19, 19, w - 6],\ [23, 23, w + 8],\ [23, 23, -w + 9],\ [25, 5, 5],\ [29, 29, -w - 4],\ [29, 29, w - 5],\ [37, 37, -w - 3],\ [37, 37, w - 4],\ [41, 41, -w - 9],\ [41, 41, w - 10],\ [43, 43, -w - 2],\ [43, 43, w - 3],\ [47, 47, -w - 1],\ [47, 47, w - 2],\ [53, 53, 2*w - 13],\ [53, 53, -2*w - 11],\ [59, 59, -4*w - 25],\ [59, 59, -4*w + 29],\ [61, 61, -w - 10],\ [61, 61, w - 11],\ [83, 83, -w - 11],\ [83, 83, w - 12],\ [97, 97, 2*w - 11],\ [97, 97, -2*w - 9],\ [101, 101, 3*w - 20],\ [101, 101, -3*w - 17],\ [107, 107, -w - 12],\ [107, 107, w - 13],\ [109, 109, -5*w - 31],\ [109, 109, -5*w + 36],\ [121, 11, -11],\ [127, 127, 2*w - 19],\ [127, 127, -2*w - 17],\ [137, 137, -3*w - 16],\ [137, 137, 3*w - 19],\ [157, 157, -3*w - 23],\ [157, 157, 3*w - 26],\ [163, 163, -4*w + 27],\ [163, 163, 4*w + 23],\ [169, 13, -13],\ [173, 173, -6*w - 37],\ [173, 173, -6*w + 43],\ [181, 181, 2*w - 5],\ [181, 181, -2*w - 3],\ [191, 191, -w - 15],\ [191, 191, w - 16],\ [193, 193, 2*w - 3],\ [193, 193, -2*w - 1],\ [197, 197, 2*w - 1],\ [223, 223, -w - 16],\ [223, 223, w - 17],\ [233, 233, -3*w - 13],\ [233, 233, 3*w - 16],\ [239, 239, -5*w + 34],\ [239, 239, -5*w - 29],\ [251, 251, 5*w - 41],\ [251, 251, 5*w + 36],\ [257, 257, -w - 17],\ [257, 257, w - 18],\ [289, 17, -17],\ [293, 293, -w - 18],\ [293, 293, w - 19],\ [311, 311, -3*w - 10],\ [311, 311, 3*w - 13],\ [313, 313, -3*w - 26],\ [313, 313, 3*w - 29],\ [331, 331, -w - 19],\ [331, 331, w - 20],\ [347, 347, 4*w - 23],\ [347, 347, -4*w - 19],\ [353, 353, 3*w - 11],\ [353, 353, -3*w - 8],\ [379, 379, 2*w - 25],\ [379, 379, -2*w - 23],\ [401, 401, 3*w - 8],\ [401, 401, -3*w - 5],\ [409, 409, 5*w - 43],\ [409, 409, 5*w + 38],\ [419, 419, -5*w - 26],\ [419, 419, 5*w - 31],\ [431, 431, 3*w - 5],\ [431, 431, -3*w - 2],\ [433, 433, -7*w + 48],\ [433, 433, -7*w - 41],\ [443, 443, 3*w - 2],\ [443, 443, 3*w - 1],\ [449, 449, -9*w - 55],\ [449, 449, -9*w + 64],\ [457, 457, -w - 22],\ [457, 457, w - 23],\ [479, 479, 2*w - 27],\ [479, 479, -2*w - 25],\ [487, 487, -3*w - 29],\ [487, 487, 3*w - 32],\ [491, 491, -5*w - 39],\ [491, 491, 5*w - 44],\ [499, 499, 4*w - 19],\ [499, 499, -4*w - 15],\ [503, 503, -w - 23],\ [503, 503, w - 24],\ [521, 521, 11*w - 86],\ [521, 521, -7*w + 47],\ [557, 557, 7*w + 51],\ [557, 557, 7*w - 58],\ [563, 563, -4*w - 13],\ [563, 563, 4*w - 17],\ [569, 569, -10*w - 61],\ [569, 569, -10*w + 71],\ [587, 587, 2*w - 29],\ [587, 587, -2*w - 27],\ [601, 601, -w - 25],\ [601, 601, w - 26],\ [607, 607, -7*w - 39],\ [607, 607, 7*w - 46],\ [613, 613, 3*w - 34],\ [613, 613, -3*w - 31],\ [617, 617, -6*w - 31],\ [617, 617, 6*w - 37],\ [619, 619, 4*w - 15],\ [619, 619, -4*w - 11],\ [631, 631, 5*w - 27],\ [631, 631, -5*w - 22],\ [653, 653, -w - 26],\ [653, 653, w - 27],\ [661, 661, 5*w - 46],\ [661, 661, -5*w - 41],\ [683, 683, -9*w + 62],\ [683, 683, -9*w - 53],\ [691, 691, 7*w - 45],\ [691, 691, -7*w - 38],\ [727, 727, -6*w - 47],\ [727, 727, 6*w - 53],\ [733, 733, 4*w - 41],\ [733, 733, -4*w - 37],\ [739, 739, -4*w - 5],\ [739, 739, 4*w - 9],\ [751, 751, 8*w - 53],\ [751, 751, -8*w - 45],\ [769, 769, 5*w - 24],\ [769, 769, -5*w - 19],\ [773, 773, 7*w - 44],\ [773, 773, -7*w - 37],\ [787, 787, 4*w - 3],\ [787, 787, 4*w - 1],\ [797, 797, -9*w - 52],\ [797, 797, -9*w + 61],\ [811, 811, -5*w - 18],\ [811, 811, 5*w - 23],\ [821, 821, -w - 29],\ [821, 821, w - 30],\ [827, 827, 2*w - 33],\ [827, 827, -2*w - 31],\ [829, 829, -10*w + 69],\ [829, 829, -10*w - 59],\ [839, 839, 5*w - 48],\ [839, 839, -5*w - 43],\ [853, 853, -7*w - 36],\ [853, 853, 7*w - 43],\ [881, 881, -w - 30],\ [881, 881, w - 31],\ [961, 31, -31],\ [991, 991, -5*w - 13],\ [991, 991, 5*w - 18]] primes = [ZF.ideal(I) for I in primes_array] heckePol = x^4 + 8*x^3 + 14*x^2 - 8*x - 4 K. = NumberField(heckePol) hecke_eigenvalues_array = [-1, -e - 4, e, 1, e^3 + 11/2*e^2 + 5*e - 3, -e^3 - 13/2*e^2 - 9*e + 1, 1/2*e^3 + 4*e^2 + 7*e - 2, -1/2*e^3 - 2*e^2 + e + 2, 3/2*e^2 + 6*e - 5, -1/2*e^3 - 3*e^2 + 6, 1/2*e^3 + 3*e^2 - 10, 1/2*e^3 + 4*e^2 + 5*e - 6, -1/2*e^3 - 2*e^2 + 3*e + 6, -1/2*e^3 - 7/2*e^2 - 6*e + 4, 1/2*e^3 + 5/2*e^2 + 2*e + 4, -1/2*e^3 - 5/2*e^2 + e + 6, 1/2*e^3 + 7/2*e^2 + 3*e - 6, 1/2*e^3 + e^2 - 6*e, -1/2*e^3 - 5*e^2 - 10*e + 8, -e^2 - 5*e + 2, -e^2 - 3*e + 6, 1/2*e^2 - e - 6, 1/2*e^2 + 5*e + 6, 1/2*e^3 + e^2 - 6*e - 6, -1/2*e^3 - 5*e^2 - 10*e + 2, 1/2*e^3 + 7/2*e^2 + 4*e - 5, -1/2*e^3 - 5/2*e^2 + 3, e^3 + 9/2*e^2 - 4*e - 14, -e^3 - 15/2*e^2 - 8*e + 10, -e^3 - 4*e^2 + 2*e + 8, e^3 + 8*e^2 + 14*e, e^3 + 13/2*e^2 + 9*e + 4, -e^3 - 11/2*e^2 - 5*e + 8, -2*e^2 - 7*e + 2, -2*e^2 - 9*e - 2, -7*e^2 - 28*e + 10, 1/2*e^3 - 8*e + 2, -1/2*e^3 - 6*e^2 - 16*e + 2, 1/2*e^3 + 5/2*e^2 + e + 13, -1/2*e^3 - 7/2*e^2 - 5*e + 17, -3/2*e^3 - 5*e^2 + 7*e + 4, 3/2*e^3 + 13*e^2 + 25*e - 8, 2*e^3 + 27/2*e^2 + 19*e - 11, -2*e^3 - 21/2*e^2 - 7*e + 1, 5/2*e^2 + 10*e, -e^3 - 3*e^2 + 9*e, e^3 + 9*e^2 + 15*e - 20, -1/2*e^3 + e^2 + 13*e - 10, 1/2*e^3 + 7*e^2 + 19*e - 14, e^3 + 10*e^2 + 19*e - 12, -e^3 - 2*e^2 + 13*e + 8, 3*e^3 + 39/2*e^2 + 24*e - 11, -3*e^3 - 33/2*e^2 - 12*e + 13, -10*e^2 - 40*e + 8, -5/2*e^3 - 19*e^2 - 31*e + 6, 5/2*e^3 + 11*e^2 - e - 14, 7/2*e^3 + 35/2*e^2 + 8*e - 14, -7/2*e^3 - 49/2*e^2 - 36*e + 10, -3/2*e^3 - 14*e^2 - 34*e + 6, 3/2*e^3 + 4*e^2 - 6*e + 14, 3/2*e^2 + 5*e - 4, 3/2*e^2 + 7*e, -5/2*e^3 - 29/2*e^2 - 17*e - 5, 5/2*e^3 + 31/2*e^2 + 21*e - 9, 5*e^2 + 20*e - 22, -1/2*e^3 + e^2 + 7*e - 8, 1/2*e^3 + 7*e^2 + 25*e + 12, -e^3 - 10*e^2 - 22*e + 18, e^3 + 2*e^2 - 10*e + 10, 5*e^3 + 65/2*e^2 + 40*e - 23, -5*e^3 - 55/2*e^2 - 20*e + 17, -5/2*e^3 - 29/2*e^2 - 11*e + 6, 5/2*e^3 + 31/2*e^2 + 15*e - 22, 3/2*e^3 + 21/2*e^2 + 14*e - 7, -3/2*e^3 - 15/2*e^2 - 2*e + 9, 1/2*e^3 - 5/2*e^2 - 20*e + 16, -1/2*e^3 - 17/2*e^2 - 24*e + 24, -e^3 + 3/2*e^2 + 21*e - 9, e^3 + 27/2*e^2 + 39*e - 5, 1/2*e^3 + 9/2*e^2 + 13*e - 1, -1/2*e^3 - 3/2*e^2 - e - 13, -3*e^3 - 41/2*e^2 - 30*e + 7, 3*e^3 + 31/2*e^2 + 10*e - 9, e^3 + 18*e^2 + 50*e - 18, -e^3 + 6*e^2 + 46*e + 6, -e^3 + 3*e^2 + 33*e - 4, e^3 + 15*e^2 + 39*e - 24, e^3 + 3/2*e^2 - 10*e - 8, -e^3 - 21/2*e^2 - 26*e - 8, -4*e^3 - 26*e^2 - 30*e + 6, 4*e^3 + 22*e^2 + 14*e - 34, -1/2*e^3 - 9/2*e^2 - 4*e + 36, 1/2*e^3 + 3/2*e^2 - 8*e + 12, 4*e^3 + 61/2*e^2 + 56*e - 14, -4*e^3 - 35/2*e^2 - 4*e - 6, 3/2*e^3 + 7*e^2 + 3*e + 8, -3/2*e^3 - 11*e^2 - 19*e + 12, 3/2*e^3 + 10*e^2 + 6*e - 32, -3/2*e^3 - 8*e^2 + 2*e + 8, -1/2*e^3 - 11/2*e^2 - 8*e + 27, 1/2*e^3 + 1/2*e^2 - 12*e + 3, -7/2*e^3 - 43/2*e^2 - 15*e + 20, 7/2*e^3 + 41/2*e^2 + 11*e - 40, 2*e^3 + 16*e^2 + 37*e + 8, -2*e^3 - 8*e^2 - 5*e - 12, 1/2*e^3 - 3/2*e^2 - 23*e - 29, -1/2*e^3 - 15/2*e^2 - 13*e + 7, 1/2*e^3 - 2*e^2 - 20*e + 4, -1/2*e^3 - 8*e^2 - 20*e + 20, -4*e^3 - 22*e^2 - 26*e - 8, 4*e^3 + 26*e^2 + 42*e, -9/2*e^3 - 69/2*e^2 - 61*e + 23, 9/2*e^3 + 39/2*e^2 + e + 3, -e^3 - 13/2*e^2 - 7*e - 6, e^3 + 11/2*e^2 + 3*e - 18, -4*e^3 - 41/2*e^2 - 6*e + 9, 4*e^3 + 55/2*e^2 + 34*e - 39, e^3 + 11*e^2 + 29*e - 16, -e^3 - e^2 + 11*e - 20, e^3 + 15*e^2 + 41*e - 26, -e^3 + 3*e^2 + 31*e - 14, -11/2*e^3 - 65/2*e^2 - 32*e + 26, 11/2*e^3 + 67/2*e^2 + 36*e - 14, 3*e^3 + 9*e^2 - 14*e + 14, -3*e^3 - 27*e^2 - 58*e + 22, -5/2*e^3 - 21*e^2 - 42*e + 2, 5/2*e^3 + 9*e^2 - 6*e - 6, -3/2*e^3 - 7*e^2 - 5*e - 24, 3/2*e^3 + 11*e^2 + 21*e - 20, 5*e^3 + 32*e^2 + 43*e - 30, -5*e^3 - 28*e^2 - 27*e - 10, e^3 + 10*e^2 + 24*e - 8, -e^3 - 2*e^2 + 8*e - 8, 5/2*e^3 + 13/2*e^2 - 17*e, -5/2*e^3 - 47/2*e^2 - 51*e + 12, -4*e^3 - 21*e^2 - 16*e + 16, 4*e^3 + 27*e^2 + 40*e, -4*e^2 - 21*e - 24, -4*e^2 - 11*e - 4, 1/2*e^3 + 7/2*e^2 - 9*e - 34, -1/2*e^3 - 5/2*e^2 + 13*e + 26, -5/2*e^3 - 17*e^2 - 23*e + 36, 5/2*e^3 + 13*e^2 + 7*e + 16, 2*e^3 + 15/2*e^2 - 10*e - 17, -2*e^3 - 33/2*e^2 - 26*e + 15, -4*e^3 - 31*e^2 - 48*e + 20, 4*e^3 + 17*e^2 - 8*e - 28, -3*e^3 - 31/2*e^2 - 11*e - 1, 3*e^3 + 41/2*e^2 + 31*e - 13, -3*e^3 - 14*e^2 + 3*e + 26, 3*e^3 + 22*e^2 + 29*e - 18, -e^3 - 25/2*e^2 - 39*e - 11, e^3 - 1/2*e^2 - 13*e + 9, 3*e^3 + 19*e^2 + 26*e - 38, -3*e^3 - 17*e^2 - 18*e - 30, -7/2*e^3 - 65/2*e^2 - 60*e + 33, 7/2*e^3 + 19/2*e^2 - 32*e - 23, -7/2*e^3 - 35*e^2 - 76*e + 28, 7/2*e^3 + 7*e^2 - 36*e - 4, -1/2*e^3 - e^2 + 9*e + 8, 1/2*e^3 + 5*e^2 + 7*e - 12, -19/2*e^3 - 55*e^2 - 49*e + 46, 19/2*e^3 + 59*e^2 + 65*e - 30, -5/2*e^3 - 19/2*e^2 + 15*e + 25, 5/2*e^3 + 41/2*e^2 + 29*e - 27, 15/2*e^2 + 30*e + 16, -9/2*e^3 - 28*e^2 - 19*e + 36, 9/2*e^3 + 26*e^2 + 11*e - 48] 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, 2])] = 1 AL_eigenvalues[ZF.ideal([9, 3, 3])] = -1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]