# q-expansion of newform 7440.2.a.ca, downloaded from the LMFDB on 27 July 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 = 7440
weight = 2
poly_data = [9, 8, -8, -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], 1], [[0, 1, 0, 0], 1], [[1, -7, -1, 1], 2], [[-9, 5, 3, -1], 2]]

hecke_ring_character_values = None
aps_data = [[0, 0, 0, 0], [1, 0, 0, 0], [1, 0, 0, 0], [0, 0, 1, 0], [-1, 1, 1, 0], [3, -1, 0, 0], [3, 1, -1, -1], [-1, -1, 1, 0], [0, 0, 0, -1], [2, -2, -1, -1], [1, 0, 0, 0], [1, -1, 0, 0], [0, 0, 2, 2], [1, 3, -1, 0], [1, -3, 1, 1], [2, 0, 2, 3], [2, 0, -1, -1], [-2, 4, 0, 0], [1, 3, -2, 0], [1, 1, 0, 3], [2, 0, 1, -2], [0, -2, -2, 1], [-1, -1, 1, 3], [1, -1, 0, 3], [0, 0, 2, 4], [1, -3, 1, -2], [-5, 1, -2, -2], [2, -2, -2, -1], [-1, -1, 3, 1], [5, -1, -1, -2], [2, -2, -4, 0], [0, -2, -3, 1], [9, -5, 1, -1], [-6, 2, 0, 2], [5, 1, -3, 0], [7, 1, -3, 1], [3, 3, -5, -4], [3, 1, -4, -2], [-3, 1, -3, -2], [4, 2, 4, 0], [-2, 6, -2, -2], [-7, -5, 1, 2], [0, 2, 3, -3], [14, -2, 0, 2], [7, -3, -1, 1], [-8, 2, 0, -1], [5, 1, 1, 0], [-12, -2, 2, 2], [-2, -6, -2, 3], [-1, 1, 5, 2], [8, 6, 0, -1], [0, -2, -2, 4], [2, 0, -6, -6], [6, -2, -6, -6], [20, -2, 0, 1], [-16, 8, 2, 2], [4, 4, 1, 3], [11, -1, -1, 4], [-1, -7, 0, 4], [2, 8, -2, 4], [3, -5, -2, 0], [11, 1, 5, -1], [-5, -1, 4, -2], [6, -6, 1, 1], [-3, 5, 0, 0], [15, 1, -3, 1], [21, -1, 1, 1], [-1, 1, -2, -4], [-6, 2, 0, 0], [-3, 9, 1, -1], [5, -1, 5, -1], [-13, 5, 0, 1], [2, 6, -2, -4], [-1, -3, 1, -4], [-7, 5, -1, 0], [-13, -9, 1, 3], [2, -12, 3, -1], [-15, 3, 1, 0], [9, -1, 2, 5], [4, -6, 6, 0], [2, 2, 1, 5], [-13, 3, -7, 1], [2, 6, 1, -7], [-8, -6, 3, 6], [-2, -2, 6, 0], [-6, -2, -2, -3], [8, -6, 7, -1], [-9, 1, -4, -2], [-2, -2, 1, 1], [-2, -4, -2, 0], [-3, 9, 1, 3], [3, 3, -6, -3], [0, 4, -2, -2], [23, -1, -5, 0], [0, -4, -4, 4], [-29, -1, 3, 1], [4, 10, -1, -3], [10, -2, 4, -2], [21, -3, 5, 0], [-1, -5, -3, -3], [7, -3, 4, -2], [-20, 6, 2, -1], [-5, 7, 1, 3], [-5, 7, -8, -3], [2, 6, 4, -2], [-16, 4, 0, 4], [8, -4, 10, 6], [22, -4, -6, -2], [-5, -9, -4, 1], [-20, -6, 4, 0], [2, 0, -9, 0], [-15, 5, 8, 8], [3, 1, -1, -6], [1, -13, 5, 3], [15, -5, 1, 2], [6, 2, 1, -1], [15, 3, -5, -4], [3, -1, 5, -4], [1, -1, -5, -3], [20, -4, -9, -9], [-8, -6, 4, -4], [-7, 9, -6, -2], [-26, 4, -2, -1], [4, 0, 4, -3], [15, -5, -5, -6], [5, 1, -9, -8], [-5, 9, -1, 2], [6, -4, -2, 2], [-2, -12, 5, -2], [6, -8, -4, 8], [-5, -3, 9, 5], [-15, 5, 9, -4], [6, -2, -4, 6], [-23, 5, 2, -2], [9, -7, -6, -5], [-6, -6, -8, 3], [12, 10, -6, -3], [-15, 3, 1, -2], [-27, -5, 1, -1], [7, 5, -8, -1], [31, 3, -1, 2], [8, -6, -7, 3], [-12, -10, -4, 4], [35, -9, -7, -5], [-11, 7, 11, 4], [3, -7, 4, -3], [5, -3, -5, -4], [16, -10, -4, 0], [8, 4, -12, -8], [-11, 5, 7, 6], [14, -10, -6, 4], [18, 4, 3, 7], [-9, -11, 7, 2], [-15, 5, 9, 1], [13, 1, 2, 4], [32, -10, -8, -2], [-6, 18, -2, 0], [-25, 11, 2, 1], [-2, -10, 8, -4], [6, -8, 7, 9], [11, -1, 5, 1], [-3, -5, -3, -3], [-26, -8, 0, 6], [2, 10, -3, -5], [10, 8, 4, -8], [-14, 6, -6, -6], [-19, 1, 5, -4], [6, 0, -10, 2], [-20, 18, -4, 0], [4, -2, -14, -9], [-6, 0, 12, 10], [31, -11, -11, -8], [-8, -4, 13, 9], [-12, -4, 2, 6], [-16, -4, 6, 0], [-23, 5, -2, 5], [14, -6, -4, 8], [9, 15, 3, -6], [-6, -10, 5, 6], [7, -3, -3, -4], [-41, 3, 4, 2], [25, -7, 7, -2], [16, 4, 2, -8], [3, 9, 1, -1], [-9, 3, -15, -11], [-10, 18, 3, 1], [-17, 15, -3, -2], [-33, 5, 8, 4], [14, 8, 6, 0], [-5, 5, -6, -3], [-22, 0, 5, -10], [0, 8, -6, -11], [3, 5, -11, -15], [30, -6, 1, -3], [-17, -1, 3, 5], [15, 1, 1, 5], [-6, -4, 2, -6], [17, -5, -7, -4], [-3, 7, -3, -2], [-31, 9, -17, -5], [-5, 9, 11, 2], [-1, 3, 3, 4], [-1, 3, 1, 6], [7, 1, -9, 6], [-16, 2, 3, -1], [-27, 3, -5, -7], [6, 2, 4, -4], [21, -3, -9, -9], [-2, 6, -14, -10], [14, -14, 14, 2], [-34, 6, -1, 0], [-6, 4, 3, 9], [-19, 1, 0, 2], [14, 6, -4, -5], [2, 4, -10, -6], [-10, 2, 4, 13], [3, -13, -4, -2], [14, 10, -4, -6], [-33, -5, 5, 3], [2, 4, 6, 10], [1, -1, -3, 0], [-22, 2, 8, 0], [-4, -4, -4, -12], [4, -10, -3, -2], [6, -2, 0, 1], [19, -7, 3, 4], [3, -7, 5, -5], [-14, 8, 10, 8], [10, 6, 4, -6], [4, 0, 13, 7], [29, -19, -2, -2], [-24, -8, -6, 6], [-34, 6, 2, 4], [45, -13, -4, -5], [18, -20, 0, 2], [1, -7, 7, 6], [1, -5, -1, 2], [-8, 18, 6, -2], [5, 19, 5, -6], [26, -4, -12, -6], [-23, 13, -13, -3], [34, -6, 18, 4], [-18, 8, -14, -6], [24, -6, 0, -8], [-3, 3, -9, -1], [-9, 1, -14, 5], [0, -6, -14, -10], [25, -9, -7, -6], [-16, -12, 6, -2], [-23, 1, -7, -5], [15, -3, 1, 4], [18, 2, 16, 2], [-25, 15, 7, 6], [34, -6, -10, -5], [-14, 24, 0, 2], [-32, -2, 7, 3], [-2, 4, -16, -4], [-27, 11, 15, 12], [-31, 11, -5, -3], [-28, 6, 4, -2], [-38, 8, 3, -2], [11, -5, 9, -7], [15, -11, -13, -12], [13, -11, 7, 4], [-9, 5, -3, -9], [17, -3, 3, -4], [-1, -7, 11, 10], [10, -6, 6, -2], [3, 7, -8, 0], [-28, -6, -2, 0], [5, 9, 3, -7], [24, -6, 0, 6], [-48, 6, -1, 0], [-19, 5, 15, 14], [-6, -4, -18, -6], [-9, 11, 10, 0], [-8, 12, -2, 1], [26, -4, 4, -4], [2, 20, 4, -8], [39, -5, -1, -4], [-5, -17, -1, -7], [-44, 4, -10, 0], [7, 3, -13, 3], [16, -2, -4, 12], [-19, -5, -6, -8], [-12, 8, 8, -10], [24, -14, -7, -2], [-17, 5, 3, 9], [-20, -12, -6, 2], [6, 14, 7, 1], [13, 9, 7, 2], [-28, -12, 2, 7], [-27, -9, -9, 3], [4, 0, -5, -3], [17, 5, 12, -6], [24, 6, 5, 7], [-21, 17, -5, -1], [-29, 9, 9, -1], [27, -1, -9, -10], [-33, 11, -9, -4], [-12, -8, -4, 2], [-38, -4, 0, -1], [-12, 4, -8, -8], [-14, -18, 10, 7], [37, -21, -1, 1], [-24, 18, 0, 6], [19, 15, 1, 1], [44, 2, -4, 4], [-20, -12, -2, 4], [14, 0, 12, 0], [-5, -25, 1, 7], [-38, 8, 1, 3], [68, -14, -8, -4], [-7, -3, 13, 0], [-27, -3, -5, -5], [-12, 18, -10, -8], [30, -2, -2, -2], [-44, 2, 10, 4], [-22, 24, 6, 4], [-43, -9, 2, -1], [-3, -9, -9, -9], [10, 22, -1, 4], [1, 5, -7, -8], [-20, -2, 13, -6], [6, 0, 6, 11], [0, 2, -4, 2], [-37, 11, -17, 0], [-1, 5, -14, -8], [-22, 10, -18, -10], [-18, 0, -16, -11], [6, 12, 0, -12], [25, 19, -5, -3], [0, -12, 4, -6], [5, -3, -1, 11], [15, 19, -7, 12], [30, -26, -2, -3], [-2, 0, 6, -6], [-12, 2, -16, -13], [-20, -16, 2, 5], [-28, -10, 4, 6], [-5, 19, -14, -8], [-1, -19, 5, 15], [-4, -8, 17, -3], [-19, 1, -5, -6], [-20, 14, -2, -6], [-30, 0, 15, 11], [9, -3, 5, 11], [7, -1, 15, 4], [34, -6, 12, 2], [-39, -5, 3, 7], [15, 1, 1, -3], [-50, 6, -1, -8], [19, 5, -7, 0], [0, 2, 4, -4], [8, 18, -8, 12], [-35, -1, -17, 1], [48, 8, -6, 8], [19, 9, -9, 2], [-13, 9, 9, -7], [-11, -15, 1, 13], [7, -1, -7, -16], [15, 5, -10, 9], [2, 2, -18, -23], [-10, -8, 13, 9], [-13, -5, 8, 0], [-4, 6, 15, -4], [-15, 23, -9, -3], [30, -4, 14, 6], [-10, -10, 8, 13], [26, -12, -15, -13], [33, -11, -19, -12], [13, 1, -17, 7], [22, -18, -8, -2], [-32, -16, 4, 2], [13, -11, -8, 4], [-25, -11, 13, 10], [3, 5, 14, 3], [0, -28, 4, -8], [-14, -18, 4, 8], [-6, 20, -9, -4], [18, 12, 1, 11], [1, -1, -7, -5], [-6, -14, 2, -4], [-3, 3, -21, 3], [-62, -2, 10, 12], [29, -15, 13, 1], [-4, 8, 0, 0], [17, -3, -8, 6], [27, 15, -5, -2], [10, -10, 10, 13], [40, -6, 2, 4], [40, -22, 2, -1], [-6, 6, 19, 7], [-7, -3, -16, -20], [-5, -21, 2, 7], [16, -2, -6, -10], [9, -11, 5, 4], [-16, 20, 4, -2], [-1, 21, 7, -3], [-20, -12, 5, -7], [-21, 5, 9, -12], [-30, -16, 6, 1], [36, -6, -3, -12], [-29, -23, 5, 3], [-30, -6, 13, -7], [-16, 0, 2, 16], [36, 4, 4, 2], [10, -8, 7, -5], [-5, -3, 11, -14], [39, -3, 3, 0], [-23, -27, 6, 6], [22, -8, -10, 8], [40, -8, -4, 0], [-5, -3, -19, -19], [54, -12, -12, -2], [19, 1, -3, 2], [18, -16, -14, 2], [-4, 20, -1, 0], [-8, -14, 16, -3], [-29, 3, 19, 7], [36, -14, -3, 1], [13, 15, -20, -16], [38, -22, -10, 3], [4, -6, -9, -15], [-32, 0, -4, -12], [62, 0, -2, 9], [10, -26, 10, -1], [11, 7, -14, 3], [20, 0, -12, 4], [40, -20, -12, -6], [-38, 22, 8, 1], [42, 10, -7, 9], [-20, 16, 18, -2], [11, 7, -13, 3], [-11, -3, 14, 24], [49, -5, -2, 9], [-2, 16, 2, 2], [9, -25, -1, -8], [-37, 15, -3, 8]]