/* 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([28, 14, 2*w - 14]) 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^2 - 2*x - 12 K. = NumberField(heckePol) hecke_eigenvalues_array = [1, 1, e, -2, 3/2*e - 1, 1/2*e + 1, -6, e - 2, -1/2*e - 3, e - 2, 6, -4, e, -3/2*e + 6, -1/2*e + 4, 3/2*e - 4, 3/2*e + 2, e - 8, -e + 2, e - 8, e - 8, -3/2*e + 6, 3/2*e - 6, e + 6, 14, -3/2*e + 9, 3/2*e + 3, -3/2*e + 2, -1/2*e + 12, -2*e + 4, 2*e - 10, -3/2*e + 6, 1/2*e + 8, 2, -3*e + 8, 2*e - 2, -2*e, -4*e - 2, -9/2*e + 3, -1/2*e - 5, 2*e + 4, -3*e + 8, 5/2*e - 9, -5/2*e - 5, 3/2*e - 4, -4*e - 4, -3*e + 12, -e - 14, 2*e + 4, 3*e - 12, -5*e - 2, -13/2*e + 9, 3/2*e + 5, e - 2, -e + 16, -10, -3/2*e + 24, -3/2*e, -3*e + 12, -3*e + 12, 3/2*e - 12, 5/2*e - 20, -3/2*e - 21, -3/2*e + 9, -4*e + 10, -e + 2, -6, -6*e + 6, 3*e + 6, -5/2*e + 25, 7/2*e - 17, -7/2*e + 6, 5/2*e - 12, -3/2*e + 21, -17/2*e + 5, 13/2*e - 4, -5/2*e + 14, 7/2*e - 5, 5/2*e - 21, -9/2*e + 9, 11/2*e - 23, -5/2*e + 25, 3/2*e + 11, -2*e - 8, -4*e + 20, 3*e - 6, 3*e - 18, 13/2*e - 14, 7/2*e - 8, -12, 2*e + 8, -11/2*e + 14, 17/2*e - 2, -5/2*e + 28, 3/2*e + 20, 2*e - 16, -6*e + 24, -4*e + 10, 2*e + 22, -11/2*e + 5, 15/2*e - 15, 7/2*e + 10, -1/2*e - 24, -30, -4*e - 4, -9/2*e - 3, 3/2*e + 15, 3*e + 24, 9*e - 6, 24, 10*e - 20, -21/2*e + 3, 3/2*e + 15, -15/2*e + 12, -5/2*e - 4, -3/2*e - 19, 9/2*e - 1, 2*e - 38, -2*e + 18, -4*e - 20, -2*e + 18, -3/2*e + 12, -3/2*e + 36, 2*e + 28, 2*e - 44, 4*e - 12, 3*e - 4, 9*e - 12, 3*e + 24, 5*e + 10, -e - 2, 24, 4*e - 32, -15/2*e + 32, 3/2*e + 8, -3*e + 14, 4*e - 42, 8*e + 4, -10*e + 4, -5/2*e + 4, 23/2*e - 6, 5*e - 2, 3*e + 2, 3/2*e + 29, 3/2*e + 11, -8*e + 16, -8*e + 4, 15/2*e - 19, -19/2*e + 27, 11*e - 22, -6*e - 24, -15/2*e + 17, -5/2*e + 31, 3*e - 42, -2*e + 10, 13/2*e - 13, 3/2*e + 3, -2*e, 4*e + 6, 3*e - 18, 3*e, -5*e, -2*e - 30, 13/2*e - 1, 25/2*e - 19, -21/2*e + 2, -e + 46, -10*e + 16] 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([7, 7, w - 7])] = -1 # EXAMPLE: # pp = ZF.ideal(2).factor()[0][0] # hecke_eigenvalues[pp]