# q-expansion of newform 4752.2.a.bj, downloaded from the LMFDB on 26 June 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 = 4752
weight = 2
poly_data = [3, -5, -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], [[-4, 0, 1], 1]]

hecke_ring_character_values = None
aps_data = [[0, 0, 0], [0, 0, 0], [-1, 0, -1], [1, -1, -1], [-1, 0, 0], [1, -2, 1], [-2, 1, -2], [1, 2, -1], [0, 0, -1], [-3, -3, 1], [-1, -4, 1], [4, -2, -1], [-3, -1, 1], [4, -1, -2], [1, -2, 0], [-4, 2, -2], [2, -2, -3], [-3, -4, -1], [1, 4, -1], [8, -2, 2], [0, -2, 2], [8, 1, -2], [7, -2, 3], [-6, 0, 0], [-6, 4, 1], [-4, 7, -2], [-5, 4, 3], [5, -4, 3], [0, 6, -4], [-7, -6, 1], [5, 2, 3], [2, 2, -2], [-8, -2, -2], [4, 3, 0], [-5, -2, -5], [5, 0, 5], [17, -2, 4], [22, 0, 0], [-5, -2, 3], [1, 0, 7], [11, -2, 4], [7, -4, 0], [-1, -2, -4], [5, 0, -5], [13, -5, 7], [-5, -4, -3], [3, -11, -1], [-5, 0, -3], [-7, -2, 5], [6, -4, 9], [1, -4, 3], [-7, 8, 1], [-8, -4, -2], [9, -8, -2], [5, -4, 1], [-2, -8, -4], [11, -4, 1], [10, -3, -2], [-4, 8, -8], [0, 5, 6], [-1, -4, 3], [-25, 2, -1], [11, 1, -5], [-2, -10, 1], [8, 0, 2], [14, -6, 2], [15, 4, 1], [-3, 12, -5], [-12, 10, 2], [-7, -4, -5], [-10, 6, -10], [-9, 4, -3], [-8, -6, -2], [15, 2, 1], [5, -6, 3], [-16, 2, -2], [10, 0, -4], [-4, -12, -2], [0, -2, 0], [10, -8, -6], [-20, -2, 2], [13, -6, 10], [3, 2, -1], [-2, 8, -9], [-14, -7, 2], [-14, 8, -2], [-5, -6, -5], [-21, 8, -3], [9, 4, -1], [-22, -4, 2], [-8, 4, 1], [18, -8, -2], [-18, 12, 6], [-4, -2, -12], [1, -10, -1], [6, 2, 6], [-15, 10, -1], [3, -2, 1], [20, 3, -2], [-28, -2, -4], [14, -3, 14], [24, -3, -6], [-20, -2, 4], [1, 4, 7], [-13, 13, 1], [19, 0, 6], [-16, 0, -7], [18, 5, -4], [14, 4, -1], [7, -6, 5], [-13, -2, -7], [15, 18, -5], [-9, 12, 9], [-21, 10, -9], [-15, 2, 3], [-9, -8, 3], [-4, 10, 0], [22, 4, -2], [8, 8, 4], [19, -2, -3], [-9, 14, 10], [-13, -8, -3], [25, -1, 7], [-1, 14, -6], [8, -10, 6], [2, -7, 4], [20, 2, 1], [22, -2, 12], [-3, 8, 7], [-11, -4, -3], [-18, 11, 4], [14, -4, 2], [-19, -6, -5], [-21, -12, 2], [31, -4, 13], [-24, 8, -10], [-13, -4, 5], [19, -10, -11], [5, 2, 9], [1, -17, 7], [-9, 5, -5], [0, 5, 8], [12, -10, 14], [17, -10, 3], [-24, -2, -1], [-4, 6, -1], [-20, -2, 6], [3, 11, -3], [9, -8, 11], [2, -10, -3], [-14, 10, 2], [-25, 10, 1], [-6, -2, 2], [-9, -6, 1], [-15, -2, 7], [15, -12, 12], [3, -17, -5], [-18, 18, -4], [1, 18, -3], [-7, -13, 3], [-42, 6, 4], [4, -19, 12], [4, -1, 8], [-8, 4, 8], [48, 0, -2], [-4, 20, 4], [8, 0, -12], [4, -18, 2], [3, -14, 15], [-16, -18, 2], [13, 4, 1], [-26, 12, -7], [15, 2, -1], [-5, 6, -15], [-10, 8, -4], [-23, -8, 5], [27, -2, 5], [29, 0, 9], [-2, -5, 14], [1, -6, -13], [-18, 12, -8], [-16, 12, -10], [-3, -24, 0], [15, -17, -7], [-11, -10, -4], [-23, -2, 1], [-19, 16, -3], [-15, 0, 1], [-19, 2, 11], [-35, 6, 7], [-23, 6, -12], [42, -12, -2], [-8, 0, -6], [10, 0, 0], [4, -16, -8], [-3, -14, 13], [-14, -8, 10], [6, 4, 16], [-38, 13, -2], [33, -4, -5], [-26, 7, -10], [-35, 14, 3], [-8, 2, 8], [-22, 8, 0], [-23, 6, -8], [-10, 16, -10], [-41, 12, 5], [25, 0, 7], [-18, 13, -6], [-3, -8, 1], [6, 6, -4], [-30, 6, -18], [-29, 0, -13], [-10, 10, -15], [39, 2, -3], [-8, 18, 4], [-11, 14, -9], [-9, -8, 5], [-14, 10, -20], [-26, 8, 8], [13, 0, -7], [15, -21, 5], [-15, 0, -21], [-25, 6, -5], [2, 10, -14], [-16, 10, 8], [-22, 8, 8], [40, -10, -2], [-14, 17, -14], [42, -6, 0], [-22, 0, 8], [-7, 15, -5], [0, 1, 2], [-35, 4, -9], [23, -20, -5], [-25, 0, -11], [-3, 14, 8], [17, 5, -1], [8, 26, -10], [-8, 22, -6], [-4, -18, -1], [5, 3, 5], [-14, 0, -6], [10, -10, 6], [59, -6, 3], [27, -12, 23], [-12, 6, 6], [-7, 20, 8], [37, -3, 7], [7, 2, 11], [-28, 12, -6], [19, -8, 1], [-45, 10, 0], [-2, -26, -3], [-27, -3, -1], [18, -10, 12], [-8, -16, -11], [-39, 7, -21], [-16, -26, 8], [16, -2, 0], [23, -13, 11], [16, 26, -4], [-9, 4, 5], [16, -26, 20], [5, -24, 5], [-18, 8, -16], [-28, -12, 10], [17, -24, -3], [47, -13, 13], [0, 27, -6], [12, -16, 6], [10, -20, -12], [47, -6, -6], [11, -2, 1], [-31, 0, 1], [30, 6, -2], [39, -12, 24], [-16, 16, -10], [9, 22, -11], [-29, -4, -11], [9, -4, -19], [10, 0, 2], [-54, 10, 0], [34, 8, -4], [7, -11, -3], [10, 10, 7], [40, -4, -13], [-56, 13, 0], [-37, 8, -9], [-19, 0, -17], [15, 26, -9], [18, -20, 3], [23, 1, 9], [22, 12, -2], [45, 6, -3], [-19, -4, 15], [2, 0, 10], [40, -10, 16], [-15, -4, -7], [16, 15, -4], [20, -14, 6], [62, -3, 0], [-25, -2, -3], [-4, 18, 2], [6, 10, -16], [18, -18, 17], [-28, -16, 0], [-26, -10, -6], [-24, 20, 1], [-4, 18, 4], [3, 8, -1], [-50, -12, 4], [28, -5, -8], [-16, 8, 8], [44, -20, 2], [19, -6, -3], [-14, 16, -13], [28, 2, -8], [15, -22, 10], [-31, 3, 15], [5, 11, -5], [45, -4, 16], [35, -13, 3], [6, -38, 8], [29, -9, -7], [-33, 14, -16], [-12, 18, 6], [-47, 2, 3], [-17, -8, -11], [-70, 3, -4], [35, 2, 7], [37, -20, 25], [38, -3, 24], [4, -24, 20], [-27, -6, -21], [-68, 2, -2], [14, 32, -11], [25, -27, 7], [21, 6, 10], [27, -2, 5], [-2, 8, 13], [1, 2, 25], [6, 11, -2], [8, 4, -18], [-1, 28, 3], [-36, 16, -12], [7, -4, 8], [45, -6, 5], [-8, 22, 16], [34, 10, -10], [-22, 19, 22], [5, -20, -12], [-25, 16, -23], [34, 25, -8], [-11, 22, 11], [-7, 34, -19], [-62, 10, 4], [27, 6, 9], [55, -11, 7], [28, 16, 4], [22, 8, 12], [19, -6, -17], [-32, 28, -11], [14, 20, 14], [-13, -14, 9], [33, -10, -6], [9, -7, 7], [-3, -17, -1], [19, 2, -7], [2, -4, 5], [-16, 14, -26], [14, 14, 18], [-1, -23, 11], [27, 6, 9], [16, 14, 8], [77, -6, 2], [-3, -6, 15], [-39, 8, -17], [-22, 5, 6], [-14, 14, 8], [-37, -28, 5], [2, 16, -8], [-43, 10, 3], [-32, 9, -28], [-15, 23, 1], [-3, -30, 18], [30, -24, 24], [-12, 14, -28], [17, 14, -1], [7, 8, -2], [-24, 20, -2], [47, -8, -11], [-25, 4, -22], [-3, -6, 1], [-6, 0, 10], [43, -16, 17], [-21, 29, -5], [-57, 16, -21], [-18, -18, 4], [13, -28, 3], [-28, -2, 18], [-45, -6, 5], [45, -11, -11], [9, 16, -21], [30, -23, 0], [5, 21, -7], [-68, 0, 2], [-25, 29, 21], [-5, 0, 1], [21, 18, 2], [-35, 14, -11], [19, -5, 19], [8, -28, 8], [-15, -3, -1], [12, -28, 6], [-78, 2, 0], [-58, 8, -4], [16, -34, -10], [-13, -30, 1], [47, 6, -13], [18, -12, -14], [8, -22, -24], [3, 6, 14], [-11, 16, -9], [27, -28, 8], [-2, 30, 12], [-7, 22, -24], [1, -17, 5], [41, -26, 21], [18, 2, -10]]