# q-expansion of newform 1944.2.a.q, downloaded from the LMFDB on 06 August 2026.

# 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 = 1944
weight = 2
poly_data = [-9, 9, 36, -1, -12, 0, 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], [[3, -11, -7, 2, 1, 0], 2], [[-7, 0, -4, -7, 1, 1], 4], [[3, 26, 6, -11, -1, 1], 4], [[7, -28, 4, 11, -1, -1], 4], [[-11, 1, 9, 0, -1, 0], 2]]

hecke_ring_character_values = None
aps_data = [[0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0], [0, 0, -1, 0, 0, 0], [1, 0, 0, 0, -1, 0], [0, -1, 0, -1, 0, -1], [1, 1, -1, 0, -1, 0], [1, -1, 0, -1, 1, 0], [2, 0, 1, 0, 0, -1], [-1, 1, -1, 1, -1, 1], [0, 3, -1, 2, -1, 0], [3, 0, 1, -1, 1, 1], [2, 2, -2, 3, 0, 1], [3, -1, 1, 0, 1, 0], [3, -3, 1, -1, 1, -1], [-1, -1, 0, -2, -1, 1], [2, 1, 1, -2, 0, 0], [-2, 1, 2, -2, 1, -1], [4, -2, -1, -1, 0, -2], [4, 0, -1, -3, 0, 0], [-1, -3, 1, -2, -1, -2], [2, 5, -1, 4, -2, 0], [3, 0, -2, 1, 0, 1], [-5, 0, 1, 0, 3, 1], [5, 3, 0, 2, -3, -1], [4, -2, 1, -1, 0, 1], [3, 0, 2, -2, 1, -1], [3, 2, 0, 0, -2, 0], [-5, 3, 0, 2, -1, 0], [5, -3, 2, -2, 3, -1], [6, 0, 0, -3, -2, 1], [4, -2, -2, 0, 0, -1], [-4, -4, 1, -2, 4, 0], [10, 4, 0, 3, -2, -1], [1, 3, -1, 0, -1, -2], [1, 2, -1, 7, -1, 2], [4, -2, 2, 1, 0, 0], [2, 4, -4, 1, -2, 3], [0, 2, -4, 3, -2, -1], [-3, 3, -1, -2, 1, 2], [0, 5, -3, 7, -2, 1], [-9, -4, 1, 0, 3, 2], [2, 2, 3, 0, 2, 1], [-1, 3, 1, -2, -3, -2], [-2, 0, 3, 0, -2, 0], [-2, -5, 1, -3, 4, -2], [-1, 0, -1, -4, 1, 0], [0, -6, 0, 1, 2, 1], [-2, -5, -2, -6, 3, -1], [-4, -5, 1, -1, -1, -3], [-2, 0, -2, 4, 4, 0], [13, -1, 3, -2, -1, 2], [-4, -2, -2, 0, -2, 0], [-3, -2, -1, 0, 2, -3], [0, 0, -2, 5, -4, -1], [0, 2, -2, 3, 4, 1], [-1, -7, 3, -8, 3, -2], [-4, -1, 3, -7, -1, -1], [-2, 5, -1, 2, -4, -1], [1, 5, 3, 2, 3, 0], [2, -4, -2, 4, -2, 0], [-5, 1, 5, -2, -1, 2], [-7, -5, 1, 3, 3, 1], [-8, -2, 3, -6, 2, -3], [4, 0, 6, 0, 0, -2], [-2, -3, 4, -9, 2, -1], [-10, 5, 2, 0, 0, 3], [-1, -1, 4, -1, 1, 0], [-4, -1, 0, 5, -1, -2], [7, -2, 6, 0, -1, 0], [-10, 2, 0, 4, 0, 2], [4, 2, -1, 4, 0, 3], [-11, 7, 1, 8, -1, 2], [-5, 4, -5, 7, -3, 2], [-13, 9, -1, 6, -5, 2], [-2, -8, 9, -6, 6, -1], [0, 0, 3, 0, 6, 3], [-3, 1, 4, -1, 1, -1], [3, -1, -2, -6, -1, -1], [-4, 2, 2, 0, -2, 4], [-8, -3, -8, 2, 0, 0], [-10, -3, 2, 1, 5, 2], [1, 3, 2, -2, 1, 1], [-1, -1, 3, 0, 1, 4], [-11, -5, -1, -3, 1, 2], [-7, 3, 3, -4, -3, -1], [-1, -1, 6, 0, -1, -1], [6, -2, 1, 1, 2, -6], [3, 1, 0, 2, -1, 0], [-2, 6, -1, 1, -2, -1], [-14, 4, -2, 4, 2, 3], [8, 4, 0, -2, -4, -5], [-6, 0, 3, -3, 4, -4], [2, -2, 1, 1, 6, 0], [15, 2, -6, 3, -1, -1], [-4, -6, -1, -7, 6, 0], [-9, 3, 1, -3, 3, -1], [-2, -1, -6, 6, 0, 4], [-4, 4, 1, 8, 2, 1], [-11, 1, -7, 5, -1, 3], [0, -6, 4, -7, 2, 1], [0, -6, -7, -1, 2, -2], [1, -7, 5, -3, 1, 2], [-11, 0, -5, 0, -3, -1], [-1, -11, 5, -7, 1, -3], [4, 4, -1, 8, 2, 1], [-5, 0, 0, 3, 4, 3], [16, 5, -2, 4, -4, 1], [-7, 5, -4, 8, -5, -3], [-4, -8, -2, -5, 0, 1], [3, 8, -1, 7, -1, -4], [10, -6, 4, -2, 2, 0], [-9, 5, 1, 6, -5, -2], [8, 4, 0, -3, 4, 3], [-9, -3, 3, -4, 9, 2], [-3, 7, -4, -3, 1, 7], [-13, -9, 6, -3, 3, -2], [-16, -4, -5, -2, 2, -3], [-6, 6, 0, 3, 2, 3], [1, -4, 6, 3, 3, -2], [3, 11, -7, 9, -5, 2], [8, 0, 0, 2, -4, 2], [-7, -8, 1, -6, -3, -3], [-14, 3, -4, -4, -1, 3], [-1, -6, -1, -7, 2, -2], [19, 5, -3, 8, 3, 2], [-9, -4, 3, 0, 1, -2], [8, 6, 1, 2, -6, -1], [21, 7, -6, 3, -7, -2], [6, 5, -3, 9, 0, 0], [1, 1, 2, -3, 3, 0], [23, -1, -3, 2, -3, 0], [-21, -9, 6, -7, 1, 0], [13, 7, -4, 6, -3, 2], [14, 0, 8, -5, 2, -3], [-14, 0, 5, -2, 4, 1], [-9, 2, -2, 4, -1, 5], [-12, -1, 5, 0, -1, 0], [5, 5, 1, 3, 1, 1], [-7, 0, -2, 2, 3, -3], [12, -6, 3, 3, 2, 0], [18, 4, -7, -2, -2, 1], [-12, 2, 7, 0, 6, 5], [1, 15, -3, 9, -3, 2], [-8, 7, 3, 3, -5, 3], [4, 10, 4, 4, -2, 0], [-5, -13, 9, -1, 1, -1], [-15, -9, -1, -4, 1, 6], [8, -6, 1, 0, 2, 3], [11, 1, 0, 5, -5, 6], [22, -2, -6, 1, -4, -1], [16, 0, 3, 1, 4, 2], [11, 15, 0, 11, -5, -2], [-5, -1, 0, -1, 5, -8], [-13, -11, -1, -10, 5, 0], [18, 0, 2, -2, 4, -2], [28, 8, -7, 5, -2, 2], [2, 11, -2, 12, -4, -1], [-20, 6, -10, -1, -2, 7], [-3, 10, -1, 9, 1, 5], [-18, 15, -5, 6, -8, 0], [9, -7, 0, 5, 5, -1], [21, 3, -8, 1, -7, -2], [-6, 1, -3, -2, 4, 5], [22, -9, 3, 2, -2, 2], [19, 11, -4, 12, -9, 5], [12, 8, 4, 6, -4, -2], [20, -9, -1, -1, 2, -2], [-2, -8, 0, -1, 0, -9], [3, 17, -6, 13, -5, 5], [0, -11, 4, -5, 9, 4], [4, 11, 0, 10, 4, -1], [10, -4, -3, -8, 0, 1], [-3, 1, -2, 5, -5, -2], [-11, -12, 0, -12, 12, 0], [2, -17, 12, -8, 12, -3], [-15, 11, -1, 5, -3, 5], [21, 3, -2, -4, -9, -9], [-23, 2, 1, -2, 3, 3], [-4, 7, 5, 3, -2, 0], [18, -6, 0, -5, 2, -3], [2, -1, 2, -5, 4, -1], [-22, 5, 1, 0, 2, -5], [-16, -4, 9, -15, 4, -6], [7, 5, -7, 8, 3, 2], [5, 21, -8, 17, -7, 6], [18, -7, -4, -4, -4, 0], [35, 1, 0, 0, -1, 3], [-17, 9, -7, 6, -5, 6], [26, -1, -1, -2, -10, 1], [3, -9, 4, -4, 1, -3], [-4, -1, -4, 8, 3, 5], [-4, 4, -1, -6, 0, 0], [15, -5, 8, 0, 1, -5], [-11, -4, -2, 5, 7, 0], [-9, 10, -8, 8, -2, 6], [4, -4, -11, 4, -2, -1], [30, -5, 2, 1, 6, -6], [-11, -5, 1, -14, 3, -4], [-10, 2, -9, 0, -8, 1], [-23, -1, -6, -5, 5, 0], [-20, -5, 3, -1, -2, 0], [-7, -10, -1, 4, 3, 1], [7, 3, -10, 6, -5, 1], [18, -15, 10, -8, 6, -2], [6, 1, 5, 6, 2, -6], [31, -2, -9, -8, -6, -5], [5, -10, 0, -15, 2, -7], [-1, -17, 6, -23, 7, -3], [-5, -5, 11, 3, 5, -1], [-7, -17, 4, -11, -1, -4], [-6, 1, -4, 1, 2, -3], [10, -4, -3, 3, -8, -4], [-22, 0, 0, -6, 2, 3], [22, -3, -7, -8, 2, -4], [-10, 4, -2, 1, 0, 5], [6, 8, 2, 7, -2, 6], [27, -1, -7, 1, -5, 3], [8, -4, 10, 0, 4, -2], [14, 10, 7, 3, 6, 4], [-21, 8, 3, 4, 4, 1], [12, 0, -1, 10, 2, 1], [-1, 4, -2, -3, -4, -3], [-17, -1, -1, -2, 3, -10], [4, -15, 9, -1, 1, -3], [-13, 2, 2, -11, -9, 0], [3, 5, -2, 7, -3, -6], [30, 4, 4, -3, 2, 1], [15, 11, 0, 8, -11, 1], [-1, -5, -1, -9, 3, 3], [0, 1, -5, -8, 1, 2], [0, 16, 0, 19, 4, 1], [-4, -4, 6, -8, -6, 4], [4, -9, -7, 6, -5, -2], [13, 3, -3, 0, 3, 0], [-21, -11, -7, -12, 3, 4], [-17, -5, 12, -3, 11, 4], [-4, -11, 11, -6, 1, -4], [-8, -6, 7, -6, -2, 3], [5, 4, 7, 7, 1, -6], [1, 1, -5, 12, -1, -2], [-10, 3, -3, 9, -1, 7], [-5, 7, 8, -9, -3, 0], [-37, -6, 1, -8, 5, -1], [-23, 1, 0, -2, 5, 5], [-8, -10, 2, -8, -8, 0], [21, -3, -6, 11, -1, 0], [-14, 7, -4, -2, 0, 7], [-22, -2, 7, 5, 6, 7], [22, 20, -12, 13, -10, -1], [0, 8, -13, 5, -8, 6], [8, -4, -5, 4, -4, 9], [24, -12, 7, -21, 8, -2], [-24, -6, 3, -13, 8, 0], [1, -8, 2, -4, 13, -4], [-9, -5, 3, 0, 3, 3], [-13, 3, 0, 2, -1, 4], [-1, 5, 13, 1, -1, 7], [-8, 8, 0, 19, 0, 3], [-2, -2, 2, -8, -10, -5], [-3, -8, 5, -9, -3, 6], [-4, 3, -4, -12, 2, 4], [-20, -11, 12, -2, 8, -6], [-8, -20, -6, -6, 0, 0], [-27, -3, 9, -1, 9, -3], [-3, -9, 11, -4, 13, -8], [23, -13, 7, -19, 7, 3], [14, 7, 4, -3, 7, 0], [3, -5, -3, 5, 11, -3], [-1, -23, 9, -18, 17, -4], [-10, -5, -8, -1, -5, -6], [-5, -3, -9, 15, 9, 9], [18, 6, -3, -7, 0, -2], [-21, 2, -1, 6, 7, 1], [-32, -17, 12, -9, 2, -1], [3, 2, 13, 5, -9, -2], [-8, -14, 4, 0, 8, -3], [20, 11, -13, 11, -8, 10], [-1, 3, -3, -5, -5, 1], [12, -7, 3, -11, -4, -2], [-8, -4, -3, -10, -4, -1], [5, 11, -18, 5, -3, 4], [43, 8, 3, 10, 1, 1], [-16, 0, -7, 13, 2, -2], [-10, 16, -1, 16, -6, 1], [-6, -6, -2, 3, 4, -11], [2, -14, -13, 1, 0, -4], [-18, -5, -9, 3, -6, 1], [8, 5, -15, 1, -6, -1], [-5, 0, 7, 0, -4, 9], [-33, -17, -7, -15, 3, 5], [-19, 2, -7, 9, -10, -2], [-26, 5, 3, -2, -2, -1], [-3, 13, -13, 20, -9, 12], [31, 14, -2, 12, 3, 0], [-13, -5, -4, 5, -7, 2], [44, 1, 2, -8, -3, -1], [4, 21, -6, 3, -6, -7], [-11, 1, -5, -1, 9, 0], [-4, -8, 1, -8, 4, -10], [20, 10, 12, 8, 0, 0], [-23, 11, -11, 17, -9, -5], [36, 4, -5, 0, 0, 0], [23, 2, -4, 5, -3, 10], [-34, -5, -7, 0, 6, 2], [-16, -2, 8, 7, 6, 9], [6, -1, 0, -12, -14, -4], [-6, 1, 6, -6, -11, -5], [2, -2, 5, 0, -4, -3], [-25, -3, 3, 0, 1, 10], [14, -4, 1, -1, 2, 4], [-29, 1, 0, 7, 7, -4], [30, 3, 0, 3, -8, -5], [11, 25, -1, 22, -7, 6], [-18, 18, -15, 20, -12, -1], [-24, 10, 0, 7, -2, -3], [-8, 3, -15, -3, 7, 3], [-43, 2, -6, 10, -3, 2], [30, 10, -1, 12, 4, -7], [-12, -14, -6, -12, 2, 0], [39, -3, -8, 0, -17, 3], [-7, 7, 3, 17, -5, -1], [-11, 16, -14, 15, -7, -5], [-31, 0, -4, 5, 7, -1], [-12, 3, 9, 4, 2, 10], [9, 5, 1, 0, 11, 12], [11, -4, -10, 9, -1, 5], [4, 10, 9, 3, 4, 8], [-10, 0, 1, -20, -2, -1], [-2, -6, -4, -5, 14, 9], [-1, 26, -4, 28, -7, 11], [36, -18, 7, -5, 6, -8], [18, 4, 2, 5, -4, 3], [-15, -1, 1, 6, -11, -9], [-8, 31, -14, 21, -17, 4], [11, -13, -8, 6, -5, -1], [17, 8, -13, 7, -11, 1], [-2, -4, 1, -1, 0, -10], [19, 1, 0, 4, -7, -9], [-12, -25, 11, -31, 16, -6], [1, -1, 5, -3, 13, 0], [-1, -1, 12, -11, 9, 4], [14, 8, 5, -5, 0, 0], [40, 1, 10, 7, -2, -5], [29, -10, -4, -7, -6, -3], [10, -17, -5, 2, 4, -3], [3, -2, -5, -1, 2, 0], [15, 19, -11, 17, -3, 11], [19, -1, -2, 7, 3, 2], [14, 2, 5, -11, 8, 2], [-4, -1, -9, 12, -1, 14], [42, -8, -5, 3, 2, -2], [58, 3, -10, 2, -10, -1], [-23, -8, 8, -23, 12, -5], [-38, -4, 11, 3, 10, 9], [-4, -2, -4, -13, 2, 3], [5, -15, 1, -28, 3, -4], [-4, -18, 4, -10, 2, 2], [36, -15, 2, 4, -4, -2], [18, -8, 6, 5, -2, -9], [-32, -6, 8, 3, 8, 7], [0, -6, 0, 0, 22, 8], [-9, 11, -7, 22, -3, -2], [-21, -19, -9, -17, 9, -3], [-18, 2, -4, 8, 2, 1], [-19, 15, 0, 26, -1, 9], [-36, -14, -7, -9, -2, 2], [18, -19, 17, -21, 9, 1], [8, -15, -1, 6, 2, 4], [3, 10, -2, -8, 3, 10], [26, -6, 2, -14, -2, -1], [-18, 4, -20, 4, -14, 6], [15, 7, -11, 6, -17, -2], [-1, -16, 6, -7, 9, -2], [2, -21, 4, -10, 8, -1], [0, -18, 11, -4, 0, 5], [-5, -5, -2, -3, 3, -12], [-6, -8, -8, 5, 6, 5], [-25, 1, 9, 10, 3, 1], [-26, -20, 18, -3, 6, 1], [-19, -14, -1, 7, -2, -4], [-12, -4, 3, 6, 0, -6], [0, 8, -9, 8, 2, 3], [41, 14, -3, -1, -6, -4], [-2, 4, -5, 4, 0, 9], [-9, -11, -1, -22, 3, -6], [-15, 5, -1, -7, -13, -3], [17, 0, 11, 0, 4, -11], [14, -2, 9, -4, -6, -3], [16, 12, 2, 17, -2, -7], [-25, 13, -7, 10, -5, 4], [30, 5, -7, 12, -6, -3], [1, -17, 1, -17, -1, 1], [-16, -4, -14, -14, 0, 4], [5, 1, 12, -4, -5, -3], [16, -2, -4, 7, -10, -5], [1, 6, -8, -11, -2, 3], [50, 5, 1, 3, -7, -3], [-10, 16, -13, 6, -8, 1], [24, -2, -4, -9, -12, -7], [14, -2, -3, -17, 2, 3], [14, 10, 2, -1, 8, 3], [0, 4, -3, 1, 14, 14], [44, -1, 3, 7, -13, -3], [0, -12, -6, 9, 0, 1], [21, 8, 2, 15, 11, -1], [23, 5, 5, 1, -13, -8], [13, 15, 2, 8, -11, 5], [-11, -5, -2, -5, -17, -8], [-9, 3, 13, -16, -5, -2], [-7, 0, 15, 13, 1, 0], [15, 4, 9, -13, -9, -5], [-9, 12, -15, 10, -11, 9], [-5, -18, 11, -7, -5, 1], [16, 6, 7, 3, -8, 12], [-9, -22, 0, -14, 7, -2], [-38, 19, -6, 19, -8, 1], [16, -24, 6, 0, 10, -4], [-5, -13, -8, 7, -1, 0], [7, -21, 7, -2, 3, 6], [34, -10, -9, -5, -18, -4], [-8, 6, -12, 2, 10, 0], [23, 3, 10, -3, -5, -6], [-3, 25, -4, 25, -13, -2], [30, -23, -8, -15, -2, -5], [28, -5, -1, -1, -12, -8], [28, 21, -14, 10, 2, 6], [-7, -6, -11, 1, 1, 4], [5, 3, -15, 5, -1, 1], [24, 0, 5, -11, 0, -2], [-2, 2, 0, 23, 0, -3]]