
# q-expansion of newform 6025.2.a.f, downloaded from the LMFDB on 24 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 = 6025
weight = 2
poly_data = [1, 0, -9, 1, 11, -3, -3, 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, 0, 0, 0, 0], 1], [[0, 1, 0, 0, 0, 0, 0], 1], [[-1, 2, 5, -4, -2, 1, 0], 1], [[1, 3, 4, -4, -2, 1, 0], 1], [[2, -4, -3, 9, -2, -3, 1], 1], [[-2, -3, 7, 6, -5, -2, 1], 1], [[0, 8, -1, -11, 3, 3, -1], 1]]

hecke_ring_character_values = None
aps_data = [[1, -1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, 0], [2, 0, -1, -1, 0, -1, 1], [-3, -1, -1, 0, -2, -1, -1], [-1, -1, -2, 2, -2, -1, -1], [-1, 1, 2, 1, 1, 3, -1], [-3, 2, 1, 2, -1, -1, 0], [2, -1, 0, 1, -1, -2, -2], [-2, 3, 1, 0, 3, 4, 2], [-3, 2, 1, 1, 2, 2, 1], [2, -1, 3, -4, 2, 0, 2], [-2, 3, 0, -3, -1, 3, 1], [-5, 5, 0, 4, 0, 3, 2], [2, 5, 2, -1, 3, 1, 4], [-1, 1, -1, 1, 5, 4, 1], [-4, 0, 3, 1, -2, -4, 0], [2, 3, 4, -1, 7, 5, 2], [-6, 0, -1, 0, -5, -4, -5], [-8, 4, 2, 1, 4, 3, 1], [1, -1, -4, 2, 3, -4, 0], [0, -5, -3, 1, 0, -4, 0], [3, 2, 2, -2, 1, 0, -1], [4, -9, -3, -4, -2, -2, 1], [-2, 0, 4, 0, -3, 2, -1], [1, -1, 1, 2, 4, -2, -2], [5, -2, -2, -3, 0, 3, 4], [7, -4, -5, -2, 1, -2, 6], [3, -2, -3, 1, 0, -3, 2], [-4, 0, -3, 6, -1, 3, -2], [-7, 5, 4, 3, -1, -1, -2], [5, -7, -6, -3, -2, -4, 2], [-5, 0, -8, 4, -4, -7, 1], [-5, 1, 2, -1, -7, 0, -4], [7, 2, 0, -1, 1, 4, 9], [-1, -6, -1, -3, -3, -5, -1], [1, -5, -3, -7, -2, -2, 4], [6, -9, 1, 0, -1, -3, -3], [3, 5, 6, -3, 9, 12, 9], [-6, 1, 6, -2, 1, 3, 1], [-5, -9, -2, 6, 1, 0, -7], [2, 0, -5, 0, 2, 5, -4], [-11, -9, -9, 1, -7, -5, -7], [7, -3, -2, -7, -6, 0, 3], [6, -2, -1, -4, -3, -1, 1], [-12, 16, 6, 4, 6, 7, 4], [5, -2, -2, -6, -9, -9, -2], [3, -2, 0, -3, 1, -2, 1], [-8, 10, 6, -3, 2, 1, 3], [6, -2, 2, 0, 5, 1, 0], [-1, -10, 2, 0, 5, -1, -6], [-12, -2, 1, 4, 5, -1, -9], [-1, 0, 0, 0, 0, 0, 0], [-10, 10, 4, 4, -5, 7, -1], [-8, -1, 0, 0, 1, 0, -9], [-3, -4, 1, 1, 1, 2, -9], [-1, -5, -8, 1, -4, -2, -4], [0, -1, -7, 6, 0, 0, -8], [-1, -1, -1, 8, 0, 6, -5], [-13, 5, 4, 7, -6, 3, -1], [5, -1, -2, -2, 1, 5, -4], [-4, -10, 2, -2, -1, -7, -6], [-7, -1, -5, 1, 2, -6, 2], [0, 2, 12, -10, 7, 10, 6], [-3, -8, -7, 7, 1, -6, -6], [7, 1, 6, 0, 1, 1, -2], [1, -5, 2, -3, 3, -4, -2], [0, 14, -2, 7, -1, -1, -3], [0, 1, 8, -6, -4, 5, 0], [5, -6, -6, -6, 2, 3, 2], [-1, 2, 13, 0, 6, 16, 1], [-5, 6, 3, -11, -5, -3, 0], [9, -1, 5, -10, -3, -4, 0], [-6, 7, 4, 7, -8, -11, -3], [10, 4, -1, -6, 7, -1, 7], [11, -16, -2, -10, 4, -4, -1], [2, -8, 4, -1, 7, 2, -2], [7, 7, 3, 10, 5, 3, 1], [-15, 12, 6, 6, 8, 2, -3], [-10, -7, -3, 10, -4, -3, -8], [-2, 9, -7, -2, 0, -4, -1], [-6, 0, 1, 7, 5, 5, 4], [-10, 4, 5, -7, 4, 7, 4], [-4, -4, -8, 8, -12, -11, -3], [3, -14, -6, -1, 3, -4, -7], [-2, 11, -1, -1, 5, 13, 0], [0, -2, 6, 0, 0, -5, 11], [15, -2, -9, -1, 0, -2, 4], [18, -15, -8, 4, 0, -1, -2], [1, 13, 8, -1, 2, 1, 12], [1, -13, -7, 4, -1, -9, -4], [-5, 9, 6, 1, 7, 7, 10], [18, -7, -1, -14, 5, -9, 6], [-9, 3, -11, 1, -11, -6, 6], [1, 1, 8, -12, -8, 0, -5], [-4, 8, 6, 2, -7, 2, -2], [14, 3, 2, 1, 4, 6, 15], [-6, -11, -13, 9, -14, -10, -19], [-11, -6, 7, -1, 2, 12, 0], [-4, -3, 9, 2, 2, 3, -7], [12, 3, -4, -12, 1, -5, 7], [11, -17, 1, -1, -1, -3, 0], [-15, -1, -2, 6, -12, -10, 3], [11, 7, 12, -5, 6, -1, 7], [0, -4, 17, -10, 0, -1, 1], [-1, 10, 14, -7, 6, 20, 8], [2, 0, -4, -2, -8, -8, 6], [-11, -13, -5, 3, -4, -3, -4], [-16, 2, -11, -1, -2, 2, -3], [6, -16, -7, 3, -7, -10, -7], [1, -2, 1, -12, -2, -2, -6], [-2, -11, 2, -4, -5, -14, -17], [-5, -1, -12, -1, -3, 1, -2], [6, 0, -2, -5, -3, 1, -7], [-3, -1, -5, -3, -7, -10, 6], [-10, 8, 0, 0, -1, 11, 7], [-16, 0, -5, 7, 0, -12, 5], [-11, -3, -12, 6, -10, -2, -8], [24, -16, -6, -2, -14, -14, 5], [-7, 5, -3, -5, -3, -3, -8], [5, -1, 6, -14, 6, -10, 15], [5, 1, 0, 0, -12, 1, -11], [-10, 10, -8, 4, -4, -1, -8], [7, 2, -3, 5, 10, 6, 7], [4, -16, -4, -8, -3, -7, -5], [-2, -6, -7, 4, 8, 5, -5], [10, 20, 16, -12, 12, 13, 20], [-14, -6, 5, -5, 1, 2, -3], [12, -20, -10, 9, -8, -9, -12], [-6, -5, 2, -1, 1, 6, -11], [-9, -5, -13, 15, -14, -9, -1], [5, -10, -16, 0, 5, -11, 0], [-10, 17, 4, 4, 6, 13, -11], [8, 8, -2, -4, 5, 12, 11], [0, -3, 13, -5, 1, -13, 0], [-19, -12, -10, 9, -22, -21, -9], [5, 0, -3, 3, -2, 0, 7], [-5, 16, 0, -2, 9, -2, 9], [-5, -1, 2, -5, 10, 11, 2], [-3, 20, -3, 4, 11, 10, 9], [14, 3, -14, 3, 11, 3, 11], [4, 5, 0, 13, 6, -2, -9], [2, -8, -2, 0, 13, 5, 1], [4, 5, 9, -15, 7, -7, 12], [11, 10, 5, -11, 7, 14, 16], [9, 5, 20, -5, 2, 0, 16], [-6, -8, -21, -5, -8, -6, 0], [-2, 0, 1, -7, -16, -16, -9], [15, 12, 16, -2, 19, 14, 6], [-3, -8, 5, 5, -6, -5, 4], [-21, 17, 9, 11, 6, 16, -11], [-5, 0, 1, -3, 1, -6, -10], [4, 11, 5, 1, 20, 13, 2], [8, 6, -6, -1, 10, 2, 9], [4, -6, -10, 8, -9, 1, -21], [-24, 9, 1, 17, -12, 7, -5], [0, -16, -11, 6, -20, -20, -3], [2, -13, -7, 5, -15, 5, -10], [-19, -17, -4, 11, -1, -11, -11], [-8, -4, 10, -5, -2, 2, 7], [-3, 4, 5, 3, -1, 11, -12], [-6, 18, 24, 6, 4, 7, 8], [-5, 11, 3, 1, -21, -9, -7], [15, -4, -1, -13, -8, -4, -3], [-5, -9, -11, 20, -8, -20, -18], [9, -11, -12, 7, -14, -1, -12], [5, -14, 5, 1, -8, -8, -26], [3, -22, 11, -18, 6, -5, -10], [14, -4, 4, -13, -6, -12, 18], [23, -16, -17, -5, -2, -14, 10], [18, -8, 3, -3, 2, -12, -6], [15, -4, -4, -17, -9, -10, 1], [-26, 4, 8, -4, 7, -1, -6], [-6, -4, 3, -6, 4, 8, 9], [9, -13, -15, 5, -18, -25, -12], [5, 9, -4, 12, -3, 3, -10], [9, 1, -5, 3, -4, 10, -6], [-27, 7, -3, 8, -9, -3, 5], [3, -7, 11, -16, -3, -4, 9], [-31, 2, -9, 0, -9, -5, 6], [-2, 14, 11, 25, 12, 10, 3], [21, -11, 3, -14, -3, -12, 7], [13, 7, 12, 4, 7, 7, 3], [12, -8, -9, -11, -13, -13, 13], [-5, 2, -4, 12, -13, 2, -14], [10, -10, -7, 8, 11, -3, 9], [25, 7, 10, -2, 13, 5, 9], [-11, 24, 6, 13, 24, 17, 2], [2, -2, 1, 5, -4, 1, -18], [1, 10, 0, -9, 12, 16, 9], [3, 29, 2, 3, 0, 10, 11], [-10, 3, 4, 7, 4, 13, -7], [1, 2, -2, -11, 4, -10, 11], [-30, -1, 13, 13, -7, 7, -12], [-19, 3, 2, 13, 3, -1, -7], [-19, 19, -1, -10, -4, -2, 1], [-18, -8, 3, -1, 17, 6, -1], [-18, 18, 23, 3, 8, 10, 3], [1, 7, 14, 2, 2, 11, -5], [-7, 22, -6, 10, 1, 9, 12], [-32, 13, 8, 7, 4, -6, 3], [-6, 30, 1, 10, -5, 0, 9], [-26, 18, -3, 14, 11, 22, 6], [1, 7, 9, -3, 5, 11, 2], [-34, 11, -1, 3, -6, -8, -8], [0, -29, -8, -18, -8, -3, -1], [19, -2, 1, -8, 11, 19, 17], [-14, 12, -6, 2, -2, -2, -11], [-19, 3, -7, -11, -11, -6, 1], [-2, 8, 9, 12, -2, 15, -7], [29, -24, -12, -15, 3, -1, 6], [0, -18, -12, 2, -19, -21, 11], [-7, 3, 21, -11, 16, 4, -3], [-5, -4, -11, 21, -7, -4, -28], [-4, -5, -13, 18, 0, -14, -12], [-2, 0, 2, -5, 1, -8, 10], [19, -1, 5, 0, 4, 4, 20], [-26, 7, 15, 11, -15, -8, -16], [11, 3, 1, -4, -5, 1, -8], [-1, -8, -1, 1, -5, -26, -11], [15, 4, 4, 1, -10, 13, -16], [-1, -6, -12, 7, -19, -22, -10], [5, 19, -5, 3, 0, 7, 6], [-1, 6, -2, 1, -18, -14, -8], [-5, 23, 27, -10, 18, 15, 18], [21, 7, -4, 8, 7, -1, -1], [-35, -9, 9, 7, -13, -15, -7], [-23, 3, 3, 14, 10, -2, -11], [16, 23, 13, 3, 17, 15, 24], [-24, -5, -25, 8, -9, -10, -13], [-2, -4, -12, -1, -23, -17, -11], [-16, 21, 13, 13, -8, 5, -2], [-6, -16, 11, 4, 0, 20, -11], [35, -2, -11, -8, 8, 6, -3], [-14, -14, -13, 1, -19, -9, -23], [-8, 14, 6, 20, 4, 14, 6], [-5, 2, -3, -6, -6, 14, 16], [-2, -8, 20, -13, -4, -12, -1], [-18, 24, 22, -1, 10, 23, -2], [-6, 0, 11, -11, 10, -17, 18], [2, -16, 8, -3, 0, -1, -13], [29, -18, -12, -14, 4, -22, 10], [-13, 14, -3, 9, -14, -5, -6], [-16, -1, 0, 7, -25, -9, -21], [-12, -11, -14, 3, -8, -3, 0], [-15, 6, -8, 10, -13, 6, 4], [-9, -1, 5, 3, -21, -15, 0], [11, 28, 7, -6, 7, 23, 25], [18, -6, 8, -12, -16, 10, -4], [2, -17, 13, 2, 2, 3, 1], [23, 28, 3, -19, 4, 23, 25], [3, -22, -18, -4, -14, -19, -6], [23, -11, 0, -22, -1, -19, 5], [8, -13, -15, -6, 5, 12, -4], [0, 7, 12, -10, 6, -1, 26], [-8, 25, 17, 0, 29, 23, 15], [-2, 28, 27, -17, 18, 19, 25], [17, 3, -10, -16, 0, 4, 13], [13, -20, -7, 1, -10, 1, -10], [11, -6, 7, -31, -14, -5, 5], [-18, 34, 18, 0, 2, 27, -3], [-42, 23, 9, 17, 8, 29, -3], [25, -36, -29, 9, -26, -29, -7], [18, 20, -3, -14, 6, 11, 20], [-30, 17, -16, 10, -13, -2, 11], [-24, 4, -2, -6, -5, -12, -17], [21, 2, 7, -15, 14, 11, 9], [14, -23, -5, -18, 4, -12, 2], [32, -26, -26, 7, -13, -25, -10], [-32, 27, 6, 16, -1, 16, -2], [-25, -12, 11, -5, 3, 3, -11], [30, 23, 8, -10, 21, 17, 26], [16, -8, -10, -11, 9, 8, 25], [-2, 9, -3, 6, 10, 18, 1], [12, -10, -16, 13, -1, -20, -14], [37, 6, -12, 1, -2, -5, 4], [16, 14, 2, 6, 3, -2, 6], [-22, -1, -10, 10, -18, -12, -14], [24, -20, -16, -11, 4, -17, 0], [-12, 4, 14, -1, 16, 9, -1], [-17, -2, 7, -9, 22, 9, -4], [-16, -7, 6, -1, -10, -4, 5], [0, -15, -11, 0, -38, -37, -9], [-9, -1, -18, 20, 14, 11, -3], [8, 14, 13, 26, 17, 0, -7], [-18, 0, -5, -12, -14, -15, 11], [3, -14, -15, -24, -6, 3, 5], [-18, -17, -13, -15, -3, -6, -7], [-7, -1, 7, -7, 9, -5, 11], [-25, 2, 13, 14, 1, 10, -15], [-16, -9, -12, 29, -7, 13, -32], [-34, -26, -12, 9, 10, -1, -21], [-9, 26, 11, -1, 23, 20, 32], [1, -10, 16, 1, 5, -3, 1], [32, -20, -7, 8, -8, 5, -3], [20, -25, 9, -17, -4, -4, -10], [-10, 33, -8, 12, -22, -8, 9], [-15, 9, 3, -17, -17, -9, 1], [3, 5, -13, 19, 15, 2, 4], [-6, 10, 15, 25, 14, 15, -9], [-25, 6, -6, 12, 3, 2, -29], [-1, -11, -9, 15, -1, -27, -4], [0, -16, -34, 16, -2, -25, -3], [-14, -7, 3, 1, -23, -31, -7], [7, -8, 11, 9, -5, -18, -22], [36, 8, -3, 1, 17, -3, 6], [-19, 9, -13, 0, -2, -19, 2], [14, 7, -12, 10, -4, -19, 0], [-5, -20, -30, 3, -36, -32, 6], [-1, 5, -10, 2, 16, -8, 4], [-47, 17, 24, -3, -7, 0, -17], [-6, 2, 3, 8, 1, -15, -19], [-1, 16, 4, -1, 14, 26, -6], [-6, -2, 12, -10, 5, 0, -16], [18, -28, -10, -40, -5, -10, 12], [-22, 10, 2, 18, 13, 22, -14], [-5, -29, -3, -12, -7, 1, -5], [-24, -13, -13, 3, -5, -27, -4], [-10, -16, -12, 10, -12, 10, -24], [15, 6, -5, -1, 7, 8, 8], [-4, 2, 25, 0, 11, 16, -14], [14, -11, -6, -30, 3, 14, 5], [8, -6, 1, 13, -6, 15, -2], [-33, 34, 4, -5, -3, 1, -3], [-45, 5, -12, -3, -12, -4, -16], [-9, 2, -14, 3, -31, -17, -5], [-46, 7, 11, -8, -3, -1, -5], [33, 4, -1, 2, -6, -13, -5], [6, -9, 9, 1, 14, 16, -10], [-7, -29, 11, -15, 20, 11, -6], [19, 7, 33, -18, 26, 19, 26], [0, 21, 12, 23, 10, 22, 23], [27, 4, 1, -27, -1, 13, 1], [-34, 10, 11, 1, -19, -25, -7], [10, 2, -10, -6, 20, -7, 14], [-24, 16, -6, -10, -8, 3, 7], [7, 0, -12, -5, 8, -10, 12], [-25, 3, -17, 6, -12, -9, 0], [11, 5, 20, -4, 10, 23, 6], [12, -43, 3, -5, -13, -21, -10], [29, 0, -15, -2, 16, 18, 19], [20, 3, 29, -11, 2, 9, 15], [17, 27, -11, -2, 6, 9, 17], [5, 3, -14, 10, -6, 12, -15], [9, -18, -17, 6, -15, -11, 2], [19, 4, 4, -23, 18, 6, 21], [16, 22, 8, 10, 8, 21, 12], [5, -9, -1, 22, 1, 7, -5], [20, 1, 4, -25, 6, 9, 17], [37, 33, 30, -20, 23, 18, 39], [-12, -14, 4, -16, 0, -17, -17], [-10, 11, 19, -7, -4, 16, -16], [9, -1, -28, 5, -15, -26, -1], [-40, -18, -2, 17, -9, -23, -6], [-8, 21, -11, 15, 6, 8, 6], [-24, -4, 2, 8, 13, -12, 2], [-10, -2, 11, 16, 14, -11, 2], [-7, -6, 6, 16, -8, -27, 1], [5, 10, 11, 0, 34, 20, -2], [-1, 12, -20, 10, -24, 4, 13], [14, 10, 12, -6, 26, 1, 14], [23, 11, 12, -23, 10, 0, 32], [-45, 4, 4, 18, -12, 0, 7], [-28, 0, 28, 2, -20, -22, -11], [7, -34, -37, -12, 3, 0, -14], [-13, 32, -18, 12, -21, -17, 14], [6, -10, -18, 5, -5, -19, -2], [-15, -30, -27, 8, 0, -21, -19], [19, -31, 8, 5, 5, 28, -22], [25, -13, 1, -7, 2, -36, 5], [-34, -19, -2, -15, -14, -14, -7], [2, 16, -2, -1, 6, -18, 15], [0, -17, -19, -3, -12, 5, -19], [26, -7, 3, -17, 18, 23, 9], [17, -33, -13, 6, -19, -1, -5], [7, -12, -19, -4, -6, -14, 18], [-38, 11, 19, 29, -4, 22, 7], [-11, 6, 15, 10, 9, -1, -21], [27, -36, 10, -17, -16, -27, -24], [-17, 36, 34, 5, 15, 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