
# q-expansion of newform 2025.2.b.m, downloaded from the LMFDB on 25 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 = 2025
weight = 2
poly_data = [18, -24, 16, 2, 2, -2, 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], 1], [[701, 12, -2, -12, 48, -2], 131], [[-138, 226, 6, 36, -13, 6], 131], [[-7, -36, 6, 36, -13, 6], 131], [[-534, 772, 308, 14, -56, 46], 393], [[-267, 386, 23, 7, -28, 23], 131]]

hecke_ring_character_values = [[326, [1, 0, 0, 0, 0, 0]], [1702, [-1, 0, 0, 0, 0, 0]]]
aps_data = [[0, 0, 0, 0, 0, 1], [0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, 1], [1, 0, 0, -1, 0, 0], [0, 0, 1, 0, -1, 0], [0, 0, 1, 0, 0, 0], [-1, 0, 0, -1, 0, 0], [0, 0, 2, 0, 0, 1], [-2, 1, 0, -2, 0, 0], [3, -2, 0, 1, 0, 0], [0, 0, 1, 0, 1, 2], [5, -1, 0, -1, 0, 0], [0, 0, -1, 0, -1, 2], [0, 0, 1, 0, -3, -3], [0, 0, 0, 0, 0, 2], [-1, 1, 0, 0, 0, 0], [1, -1, 0, -1, 0, 0], [0, 0, 3, 0, 1, 1], [2, 1, 0, 3, 0, 0], [0, 0, 0, 0, 2, 4], [0, -2, 0, 0, 0, 0], [0, 0, 0, 0, -3, -3], [-3, 0, 0, 0, 0, 0], [0, 0, -4, 0, -2, -2], [-3, 1, 0, 2, 0, 0], [0, 0, -1, 0, -1, -2], [0, 0, 2, 0, -3, 1], [-2, -2, 0, 5, 0, 0], [0, 0, -4, 0, 3, -6], [0, 0, 1, 0, 1, -5], [6, 0, 0, 0, 0, 0], [0, 0, -2, 0, 3, 2], [-8, 0, 0, 0, 0, 0], [-2, -2, 0, -5, 0, 0], [7, 2, 0, -1, 0, 0], [0, 0, -2, 0, -2, 2], [0, 0, 1, 0, -7, 2], [0, 0, 1, 0, -6, -1], [0, 0, 2, 0, 6, 4], [-4, 1, 0, -3, 0, 0], [-4, 1, 0, 2, 0, 0], [14, 1, 0, -3, 0, 0], [0, 0, -5, 0, 5, 8], [0, 0, 4, 0, 3, 2], [5, -2, 0, -5, 0, 0], [5, 2, 0, 1, 0, 0], [0, 0, 3, 0, 2, 9], [0, 0, -1, 0, 0, 0], [12, -1, 0, -4, 0, 0], [0, 0, 5, 0, 3, -2], [-7, -3, 0, 4, 0, 0], [-8, 0, 0, -5, 0, 0], [12, 3, 0, -3, 0, 0], [0, 0, 0, 0, 9, 0], [0, 0, 1, 0, 9, 6], [2, -6, 0, 7, 0, 0], [-5, 0, 0, -5, 0, 0], [0, 0, 5, 0, -2, -4], [12, 2, 0, -5, 0, 0], [0, 0, 6, 0, -4, 5], [0, 0, -1, 0, 6, 4], [0, 0, 2, 0, 4, 9], [6, 1, 0, 5, 0, 0], [0, 0, -5, 0, 8, -10], [0, 0, 1, 0, 6, -2], [9, -3, 0, 2, 0, 0], [0, 0, 6, 0, -5, 2], [0, 0, -7, 0, 3, 2], [11, 2, 0, 0, 0, 0], [0, 0, -4, 0, 6, -8], [-6, 7, 0, -1, 0, 0], [0, 0, -1, 0, -5, 2], [0, 0, 4, 0, 2, 6], [-4, 4, 0, -2, 0, 0], [0, 0, -1, 0, 3, -2], [-9, 1, 0, 5, 0, 0], [0, 0, 4, 0, -5, 4], [3, -5, 0, 2, 0, 0], [10, -2, 0, 2, 0, 0], [-11, -2, 0, 13, 0, 0], [-2, 2, 0, 2, 0, 0], [12, -4, 0, -2, 0, 0], [0, 0, 9, 0, -4, 4], [5, 1, 0, 4, 0, 0], [0, 0, -5, 0, 0, 5], [23, 1, 0, 0, 0, 0], [0, 0, 3, 0, 1, 14], [-6, -5, 0, 8, 0, 0], [0, 0, -5, 0, 8, -8], [0, 0, -5, 0, 3, -14], [12, -4, 0, -2, 0, 0], [0, 0, 3, 0, -8, 0], [-7, 3, 0, -2, 0, 0], [-25, -1, 0, 0, 0, 0], [0, 0, -9, 0, 6, -5], [17, 2, 0, -4, 0, 0], [2, 7, 0, 6, 0, 0], [0, 0, -11, 0, 5, -3], [-3, -6, 0, 2, 0, 0], [0, 0, -6, 0, 10, -7], [0, 0, -4, 0, 3, 4], [0, 0, -5, 0, 6, -7], [18, 4, 0, 8, 0, 0], [20, 0, 0, 0, 0, 0], [0, 0, -8, 0, 4, 2], [0, 0, -2, 0, -12, -1], [0, 0, 3, 0, 3, 10], [-20, 4, 0, -2, 0, 0], [2, 2, 0, 10, 0, 0], [0, 0, -5, 0, -5, 7], [0, 0, 13, 0, -4, 0], [0, 0, 8, 0, 3, 10], [-5, -2, 0, 5, 0, 0], [19, 3, 0, 4, 0, 0], [0, 4, 0, -7, 0, 0], [0, 0, 3, 0, -10, -13], [0, 0, 1, 0, -6, 15], [0, 0, 1, 0, 9, -6], [-1, -5, 0, 0, 0, 0], [4, -2, 0, -12, 0, 0], [0, 0, 1, 0, -16, -2], [0, 0, 9, 0, 0, 6], [0, 0, 3, 0, -15, -6], [4, 0, 0, 10, 0, 0], [23, -1, 0, -1, 0, 0], [-22, 3, 0, 2, 0, 0], [-3, 3, 0, 0, 0, 0], [0, 0, 9, 0, -5, 3], [0, 0, 0, 0, 11, 0], [-13, 3, 0, 2, 0, 0], [0, 0, 12, 0, 3, -1], [-14, 5, 0, -1, 0, 0], [0, 0, 13, 0, 1, 16], [-6, 5, 0, 4, 0, 0], [-20, 5, 0, 4, 0, 0], [0, 0, -4, 0, 18, 4], [0, 0, 3, 0, 13, 2], [0, 0, -5, 0, 6, -14], [7, -1, 0, 6, 0, 0], [0, 6, 0, -4, 0, 0], [-3, -7, 0, 7, 0, 0], [0, 0, 5, 0, -1, 7], [0, 0, 5, 0, -18, 5], [9, 5, 0, 5, 0, 0], [-29, -2, 0, 1, 0, 0], [0, 0, -3, 0, 2, 2], [0, 0, -3, 0, 3, -8], [2, 7, 0, 1, 0, 0], [0, 0, 9, 0, 0, 7], [0, 0, -15, 0, -5, -8], [-5, 5, 0, 3, 0, 0], [0, 0, -3, 0, 2, 3], [0, 0, 3, 0, -15, 6], [0, 0, 5, 0, -2, -1], [-29, 7, 0, -2, 0, 0], [14, -2, 0, -14, 0, 0], [4, 4, 0, -14, 0, 0], [0, 0, 0, 0, 7, -14], [-4, -1, 0, -10, 0, 0], [0, 0, 5, 0, -3, 19], [0, 0, -5, 0, -6, -2], [0, 0, 6, 0, -2, 3], [-27, -4, 0, 1, 0, 0], [0, 0, 7, 0, -12, -6], [0, 0, -12, 0, 0, -3], [-31, 1, 0, 2, 0, 0], [0, 0, -8, 0, 10, -18], [-38, 5, 0, -2, 0, 0], [0, 0, -13, 0, -3, 2], [-15, 7, 0, 2, 0, 0], [25, 4, 0, -12, 0, 0], [27, 0, 0, 9, 0, 0], [0, 0, -2, 0, 11, 10], [-10, -2, 0, 16, 0, 0], [3, -7, 0, 4, 0, 0], [15, 5, 0, -18, 0, 0], [-9, -7, 0, 4, 0, 0], [0, 0, 0, 0, 8, -9], [-7, 11, 0, -9, 0, 0], [0, 0, -9, 0, -5, 6], [-22, 1, 0, -3, 0, 0], [0, 0, 9, 0, -22, -8], [0, 0, 11, 0, -9, -12], [0, 0, -5, 0, -9, 5], [17, -8, 0, -6, 0, 0], [0, 0, -8, 0, -2, -12], [0, 0, 12, 0, -1, 7], [-5, -5, 0, -1, 0, 0], [7, -9, 0, 14, 0, 0], [0, 0, -9, 0, -4, -14], [0, 0, 9, 0, -15, 11], [12, -3, 0, 11, 0, 0], [10, 6, 0, 11, 0, 0], [0, 0, 7, 0, -18, -7], [0, 0, -3, 0, 9, 16], [-8, 2, 0, 17, 0, 0], [0, 0, 10, 0, -7, -14], [0, 0, -15, 0, -9, -6], [0, 0, 0, 0, -6, 1], [0, 10, 0, -4, 0, 0], [-12, -3, 0, 5, 0, 0], [0, 0, 0, 0, 4, 4], [-16, -9, 0, 2, 0, 0], [-12, -2, 0, 14, 0, 0], [0, 0, -5, 0, -3, -16], [7, 10, 0, 11, 0, 0], [0, 0, -6, 0, -12, -1], [3, 8, 0, -14, 0, 0], [-13, -7, 0, 10, 0, 0], [0, 0, 4, 0, -5, -4], [-34, 8, 0, -4, 0, 0], [0, 0, -1, 0, 8, -15], [0, 0, 11, 0, -15, 2], [-9, 0, 0, -3, 0, 0], [-10, -10, 0, -5, 0, 0], [0, 0, 2, 0, -2, 3], [12, 6, 0, -12, 0, 0], [0, 0, 1, 0, 9, 1], [0, 0, 8, 0, -3, -2], [19, 1, 0, -9, 0, 0], [25, 2, 0, -11, 0, 0], [45, 0, 0, 0, 0, 0], [0, 0, 12, 0, 11, 13], [0, 0, -5, 0, -15, 7], [1, -1, 0, 4, 0, 0], [0, 0, 7, 0, -3, 14], [-20, -7, 0, 9, 0, 0], [0, 0, 5, 0, -20, -3], [-9, 14, 0, 1, 0, 0], [0, 0, -12, 0, 14, 14], [5, -4, 0, 3, 0, 0], [8, -7, 0, 1, 0, 0], [22, -10, 0, 9, 0, 0], [0, 0, -13, 0, 14, 19], [0, 0, -8, 0, 9, -1], [3, 1, 0, -3, 0, 0], [0, 0, 5, 0, 9, -4], [-3, 6, 0, -9, 0, 0], [3, 6, 0, -9, 0, 0], [0, 0, 12, 0, -15, -3], [-30, 3, 0, 5, 0, 0], [0, 0, 1, 0, 26, -3], [6, -9, 0, -2, 0, 0], [0, 0, 3, 0, -12, -24], [20, 5, 0, 11, 0, 0], [0, 0, 2, 0, -14, -5], [7, 8, 0, -9, 0, 0], [-2, -9, 0, -9, 0, 0], [0, 0, 10, 0, -3, 7], [0, 0, -5, 0, 7, 10], [-26, 13, 0, -8, 0, 0], [0, 0, -15, 0, -6, -12], [-9, -1, 0, 11, 0, 0], [0, 0, -6, 0, 0, 4], [60, 2, 0, -2, 0, 0], [-5, -3, 0, 16, 0, 0], [0, 0, 1, 0, 7, 5], [0, 0, -11, 0, 9, 14], [0, 0, 1, 0, -14, 8], [0, 0, -13, 0, 8, -7], [0, 0, -17, 0, -3, -5], [-35, 5, 0, 10, 0, 0], [0, 0, -8, 0, 14, -14], [0, 0, -5, 0, 15, 20], [20, 4, 0, -14, 0, 0], [53, 1, 0, 6, 0, 0], [-4, 1, 0, -18, 0, 0], [0, 0, -3, 0, -16, 3], [0, 0, 10, 0, 6, 22], [-13, 9, 0, -12, 0, 0], [0, 0, 8, 0, 10, -3], [0, 0, 3, 0, -4, 12], [45, -13, 0, 8, 0, 0], [0, 0, -8, 0, 16, -12], [0, 0, 1, 0, 2, -7], [0, 0, 5, 0, -9, -17], [15, -13, 0, 2, 0, 0], [3, 5, 0, -3, 0, 0], [-24, -5, 0, 11, 0, 0], [0, 0, -7, 0, 15, -10], [4, -11, 0, -3, 0, 0], [0, 0, -7, 0, -12, 0], [3, 12, 0, 14, 0, 0], [0, 0, -6, 0, 19, 7], [-12, 4, 0, 8, 0, 0], [0, 0, 3, 0, 2, 22], [0, 0, -14, 0, 6, -28], [-38, -6, 0, 0, 0, 0], [6, 8, 0, -17, 0, 0], [-37, 5, 0, 8, 0, 0], [0, 0, -19, 0, 3, 6], [0, 0, 9, 0, -3, -18], [-6, 17, 0, 1, 0, 0], [0, 0, 13, 0, -19, 14], [12, -4, 0, 19, 0, 0], [-10, 9, 0, 3, 0, 0], [0, 0, -18, 0, -3, -6], [-36, 0, 0, 18, 0, 0], [0, 0, -9, 0, -11, -6], [0, 0, 23, 0, 11, 6], [0, 0, -14, 0, 27, 18], [-40, -3, 0, 23, 0, 0]]
