
# q-expansion of newform 2275.2.a.m, downloaded from the LMFDB on 19 April 2024.

# We generate the q-expansion using the Hecke eigenvalues a_p at the primes.
# Each a_p is given as a linear combination
# of the following basis for the coefficient ring.

def make_data():

    from sage.all import prod, floor, prime_powers, gcd, QQ, primes_first_n, next_prime, RR

    def discrete_log(elts, gens, mod):
        # algorithm 2.2, page 16 of https://arxiv.org/abs/0903.2785
        def table_gens(gens, mod):
            T = [1]
            n = len(gens)
            r = [None]*n
            s = [None]*n
            for i in range(n):
                beta = gens[i]
                r[i] = 1
                N = len(T)
                while beta not in T:
                    for Tj in T[:N]:
                        T.append((beta*Tj) % mod)
                    beta = (beta*gens[i]) % mod
                    r[i] += 1
                s[i] = T.index(beta)
            return T, r, s
        T, r, s = table_gens(gens, mod)
        n = len(gens)
        N = [ prod(r[:j]) for j in range(n) ]
        Z = lambda s: [ (floor(s/N[j]) % r[j]) for j in range(n)]
        return [Z(T.index(elt % mod)) for elt in elts]
    def extend_multiplicatively(an):
        for pp in prime_powers(len(an)-1):
            for k in range(1, (len(an) - 1)//pp + 1):
                if gcd(k, pp) == 1:
                    an[pp*k] = an[pp]*an[k]
    from sage.all import PolynomialRing, NumberField, ZZ
    R = PolynomialRing(QQ, "x")
    f = R(poly_data)
    K = NumberField(f, "a")
    betas = [K([c/ZZ(den) for c in num]) for num, den in basis_data]
    convert_elt_to_field = lambda elt: sum(c*beta for c, beta in zip(elt, betas))
    # convert aps to K elements
    primes = primes_first_n(len(aps_data))
    good_primes = [p for p in primes if not p.divides(level)]
    aps = map(convert_elt_to_field, aps_data)
    if not hecke_ring_character_values:
        # trivial character
        char_values = dict(zip(good_primes, [1]*len(good_primes)))
    else:
        gens = [elt[0] for elt in hecke_ring_character_values]
        gens_values = [convert_elt_to_field(elt[1]) for elt in hecke_ring_character_values]
        char_values = dict([(
            p,prod(g**k for g, k in zip(gens_values, elt)))
            for p, elt in zip(good_primes, discrete_log(good_primes, gens, level))
            ])
    an_list_bound = next_prime(primes[-1])
    an = [0]*an_list_bound
    an[1] = 1
    
    from sage.all import PowerSeriesRing
    PS = PowerSeriesRing(K, "q")
    for p, ap in zip(primes, aps):
        if p.divides(level):
            euler_factor = [1, -ap]
        else:
            euler_factor = [1, -ap, p**(weight - 1) * char_values[p]]
        k = RR(an_list_bound).log(p).floor() + 1
        foo = (1/PS(euler_factor)).padded_list(k)
        for i in range(1, k):
            an[p**i] = foo[i]
    extend_multiplicatively(an)
    return PS(an)
level = 2275
weight = 2
poly_data = [2, -4, -1, 1]

# The entries in the following list give a basis for the
# coefficient ring in terms of a root of the defining polynomial above.
# Each line consists of the coefficients of the numerator, and a denominator.
basis_data  = [[[1, 0, 0], 1], [[0, 1, 0], 1], [[-3, 0, 1], 1]]

hecke_ring_character_values = None
aps_data = [[0, -1, 0], [1, -1, 1], [0, 0, 0], [1, 0, 0], [1, -1, 1], [-1, 0, 0], [-1, -1, -1], [-1, -1, 0], [-4, 2, 1], [8, 0, 1], [-1, -1, 2], [1, -3, -1], [0, 2, -2], [-4, 2, 3], [3, -1, 4], [-2, -2, 3], [-2, 2, 4], [-2, 0, 0], [2, 6, -4], [-3, 3, -1], [3, 1, 4], [-6, 4, -1], [1, 9, -4], [-1, 5, -2], [3, 1, 0], [11, -1, 1], [4, 4, -4], [0, 0, -2], [1, -3, -3], [-4, 0, -5], [2, -6, 0], [6, -2, -2], [3, -1, 5], [-7, 3, 3], [7, -1, -1], [1, 7, -1], [1, -7, -3], [2, -6, 2], [5, 1, -4], [-5, 7, -9], [8, 10, -3], [7, 3, -3], [0, -4, 2], [8, -2, 2], [2, -8, 8], [5, -9, 1], [-2, 8, -3], [7, -1, 6], [10, 2, 2], [-6, 8, -2], [-10, -2, 1], [-10, 2, -2], [3, -3, 0], [-11, 7, -1], [1, 9, -1], [4, -6, 1], [4, -10, 4], [20, 4, 0], [0, 4, -3], [3, -9, -1], [-2, 6, 6], [7, -3, 6], [-19, -3, 0], [5, 3, 3], [13, -3, -1], [-10, 0, 2], [0, 8, -10], [0, 12, -1], [-12, 8, 4], [-1, 1, -10], [22, -4, -2], [3, -3, 9], [-9, 9, -7], [-6, 4, -10], [3, -11, 11], [18, -2, -2], [0, 10, -12], [15, -7, 0], [-2, 0, -2], [3, -7, -2], [-5, 1, -11], [-7, 9, -9], [24, -4, -2], [-2, -4, -6], [18, -6, 12], [-14, 0, -5], [-12, 2, 4], [23, -5, 9], [24, 6, -2], [-7, 3, 1], [21, -9, -1], [-3, -7, 8], [-22, -6, 0], [6, 2, -8], [-5, -7, 9], [16, 4, 2], [27, -7, 0], [18, -4, -10], [-4, -12, 10], [-3, -3, 11], [12, 2, 7], [10, 12, -10], [32, 4, 0], [-4, 8, -7], [-20, 10, -13], [-10, -16, 4], [3, -9, -2], [9, 3, -2], [-14, -12, -1], [18, 0, 12], [-13, -19, 5], [10, -4, 10], [10, -4, -12], [-4, 8, -4], [-24, -4, 2], [-10, 10, -3], [6, -18, 2], [-1, -7, 1], [-14, 16, -8], [-2, -4, 3], [-7, 11, -2], [-12, 8, -5], [-7, -11, -1], [18, -6, -2], [11, -1, -2], [20, -16, 7], [9, -19, -3], [-15, 15, -1], [20, 8, -6], [-17, 13, -18], [-15, -1, -3], [-12, -4, 0], [10, -12, 15], [8, 4, 1], [7, -11, 10], [5, -9, 8], [8, 10, -10], [-27, 9, -10], [-18, 20, 4], [-24, 16, -5], [32, -4, -4], [8, 6, 2], [-12, 8, 10], [10, -22, -6], [-15, -15, 3], [-10, 2, -14], [-3, -5, -2], [8, -2, -2], [0, -24, 6], [12, 4, 4], [15, 7, 9], [16, -14, 10], [30, -14, 12], [30, 10, -6], [-22, 8, -5], [24, 10, -1], [12, -4, -12], [-23, 11, -8], [1, 29, -7], [31, 5, -4], [-18, -14, 12], [-12, -20, 5], [-15, 11, 5], [-10, -6, 4], [-1, 7, -5], [-25, 15, 10], [14, 6, 8], [32, -10, 12], [-25, 3, 7], [-11, 13, 1], [-7, 7, 1], [-16, -10, 10], [-6, -12, 11], [-9, -5, -7], [6, 18, -10], [22, 8, 2], [-18, 14, 6], [20, -14, 2], [11, -3, -3], [-15, 1, 3], [22, -6, 10], [39, -11, -7], [-34, 8, 6], [11, -19, 2], [1, 3, -11], [5, -3, 3], [-4, 0, -7], [25, 13, -2], [32, -6, 2], [53, -3, 2], [-9, -25, 15], [10, -10, 12], [18, 4, 1], [-3, 3, 8], [-8, -2, -9], [-9, -1, 7], [23, 11, -5], [14, 6, -5], [-15, 15, -12], [-2, -26, 4], [-13, -1, -15], [-33, 13, -5], [11, 13, -16], [-38, 4, 10], [3, -9, -14], [28, -14, 16], [-1, -1, -14], [-24, -2, -17], [9, -3, -21], [-8, 0, -2], [18, -14, 1], [9, -3, -3], [-16, 8, -16], [-10, -10, 2], [21, -3, -2], [-5, -23, 10], [-2, -20, 5], [-16, -2, 4], [30, 10, 2], [29, -15, 7], [-34, 18, -13], [-22, 26, 0], [20, -22, 0], [4, -8, -18], [2, -2, -16], [-16, 12, 19], [-2, 20, -14], [-24, 10, -1], [13, 3, 7], [13, 11, -7], [0, 14, 4], [3, 5, -1], [8, 0, 12], [8, -10, 24], [-6, 22, -20], [-17, -5, 4], [48, -14, -6], [-26, 16, -18], [62, -4, -1], [2, -22, 4], [-28, 16, -20], [22, -6, 12], [-44, 12, 0], [-8, 14, -6], [-3, -5, 20], [-7, 3, 5], [-45, 15, -4], [3, 23, -10], [-15, 3, -9], [-24, -22, 7], [-14, -4, 10], [-12, -2, 22], [-22, 2, -4], [-32, 18, -26], [-18, -4, 8], [34, -14, -6], [-2, -22, -1], [-24, 12, 12], [-12, -6, 12], [-17, 25, 14], [8, 14, -19], [12, -10, -3], [-9, 33, -18], [7, -1, 1], [30, -8, 20], [-28, 28, 2], [36, -14, 22], [-19, -9, 0], [-9, 1, 3], [-18, 16, 10], [15, -21, 1], [-29, 1, -19], [10, 12, -12], [-28, -14, 1], [-7, 5, 5], [-16, 16, 4], [-13, 1, -7], [7, 27, -19], [-10, -8, 2], [-12, -8, -12], [6, -2, -24], [-44, 8, 2], [0, 0, -4], [21, -13, 8], [14, 10, -20], [-8, -18, 13], [18, -6, 29], [30, 16, -6], [-22, 6, -6], [57, -7, 9], [18, 2, -7], [-22, 10, -18], [34, -12, -4], [-27, 25, 14], [-6, 22, -7], [-30, 8, 22], [2, -30, -4], [10, -28, 14], [-37, 21, -1], [47, -17, -4], [19, -25, -7], [-31, 5, 15], [10, -24, 5], [-48, 18, -15], [18, -32, -1], [51, -5, -5], [-20, -4, 14], [-23, -3, -15], [0, 28, -18], [-20, 0, 21], [11, 9, 11], [23, 23, -5], [4, -14, 4], [-6, 40, -1], [-7, 7, -5], [-6, -20, 18], [-19, 5, 24], [-25, -3, 7], [-23, 3, 6], [-32, -8, 15], [-34, 6, -2], [-52, 14, -16], [-9, -5, -11], [26, 6, -18], [-38, 8, -10], [15, 3, 11], [-48, 12, 0], [13, 5, 0], [10, 10, -4], [32, 6, 4], [5, -23, 1], [-28, 8, -23], [-25, -23, 9], [-7, 5, 18], [-51, 3, -3], [19, -9, 14], [3, 31, 1], [-40, -10, 14], [-14, 4, 2], [-50, 14, 12], [43, -13, -3], [22, 4, 6], [42, -8, 14], [-6, 18, -20], [3, -21, 1], [58, -2, 0], [-15, 33, -25], [-25, 29, -27], [-15, -7, -14], [-21, -13, 13], [-65, 15, 0], [-27, -23, 13], [1, 17, 3], [16, 4, 14], [10, 0, 18], [23, 11, -23], [-7, 17, -30], [-1, 15, -4], [22, -12, 26], [-43, 7, -23], [-20, 22, -22], [40, -14, -22], [-38, 8, -3], [-17, -3, 21], [17, 19, -1], [-54, 18, -7], [-7, -33, 14], [21, -9, 9], [-24, 24, -1], [4, -12, -10], [31, 5, -7], [-21, 5, 21], [-38, 6, -9], [39, -3, 19], [70, -6, 13], [10, -14, -18], [0, -4, -30], [14, 4, -26], [-8, 14, 0], [38, 0, -16], [51, -11, -2], [48, -10, 30], [-3, 27, 3], [51, 15, -15], [-21, -1, 6], [-43, -15, 12], [25, -9, -9], [-26, 24, 10], [-23, 11, 19], [49, -5, 1], [20, 6, -14], [-5, -31, -10], [34, -6, 10], [-26, -18, 18], [-28, 28, 8], [-16, -12, -9], [0, 8, 4], [-18, 4, -8], [8, 22, -5], [24, 2, 12], [-39, 11, 10], [-14, -2, 3], [-23, 19, -15], [13, -21, -8], [-9, -35, 0], [41, -5, 9], [-64, 6, -8], [-13, -29, 5], [45, 9, -6], [41, -31, 24], [10, 0, -10], [-8, -8, 15], [14, 8, -3], [52, -12, 30], [-22, -34, 3], [59, 3, 3], [14, -46, 10], [18, -26, 6], [38, 16, -12], [-21, 1, 3], [12, -24, 7], [-18, 8, 30], [-9, 1, -11], [-3, -13, 13], [-26, 16, 0], [27, -23, -10], [-60, -6, -1], [-44, -6, -4], [-13, 7, 10], [20, 32, -10]]
