# q-expansion of newform 4158.2.a.bw, downloaded from the LMFDB on 04 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 = 4158
weight = 2
poly_data = [-6, -10, 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], 1], [[0, 1, 0], 1], [[-7, -1, 1], 1]]

hecke_ring_character_values = None
aps_data = [[1, 0, 0], [0, 0, 0], [0, 1, 0], [-1, 0, 0], [1, 0, 0], [0, 1, -1], [1, 0, 0], [0, 0, 0], [1, 1, 0], [0, -2, -1], [4, 1, 1], [1, -2, 1], [-1, -1, -1], [1, 0, 0], [4, 0, 0], [5, 0, 2], [0, 1, 1], [1, 1, 1], [3, 1, -2], [2, -4, -1], [-7, -2, -1], [5, 0, -1], [2, -4, -2], [5, 1, 0], [-2, 3, 0], [7, 0, 1], [2, -1, 1], [11, 0, 1], [-6, 1, 0], [2, 2, 0], [-1, 0, -3], [-3, 1, -2], [5, -4, 1], [-2, 4, 0], [3, -1, -2], [-8, -2, -4], [-1, -5, 2], [-2, -2, 1], [10, 2, 4], [4, -4, -2], [10, 4, 0], [-4, -2, 1], [1, 1, 1], [-4, 0, 3], [6, -6, 0], [0, 3, -1], [8, -3, 1], [-3, -3, -1], [-8, 6, -1], [-6, 2, 0], [12, 1, -2], [3, 3, 1], [-6, -2, -2], [-1, -1, 1], [-6, -6, 0], [-5, -2, -4], [-4, -3, 2], [-2, -4, 5], [10, 1, 2], [4, -2, 4], [-4, -5, 0], [-5, 6, 1], [-4, 4, 4], [-12, 0, 2], [-10, 1, -4], [11, -3, -1], [12, 0, 5], [-17, -1, 0], [5, -4, 3], [-3, 6, 4], [12, 4, 1], [10, 9, -3], [21, 2, 4], [-18, 4, 2], [4, -4, -4], [-7, -6, 1], [-19, 5, 1], [-12, -2, -2], [3, 2, -1], [0, 4, 0], [1, 0, -4], [5, -8, -1], [5, 7, -5], [-8, -3, -4], [7, -3, 0], [18, 6, 0], [9, 4, 1], [0, -4, -3], [3, 0, 3], [-2, 0, 6], [-7, -5, 3], [-2, -1, -4], [-12, 6, -2], [-24, 6, 0], [-15, 1, -7], [20, -1, -2], [-8, -4, 2], [-26, -4, 1], [-16, -6, 0], [-14, -6, 2], [20, 2, 4], [9, 7, 0], [-16, -8, -1], [16, -3, -2], [0, -3, -1], [6, 3, 4], [-17, 10, 2], [-17, -7, 3], [-6, -10, -1], [-14, 4, -6], [2, -4, 5], [-18, -3, 2], [4, 8, 0], [24, 2, 0], [18, -1, 2], [-3, 10, -5], [12, -4, 4], [-8, -10, 0], [-2, -7, 1], [-9, -2, -1], [7, -5, -3], [33, -1, -2], [0, -8, 0], [16, 3, 4], [-10, 10, 0], [-3, -7, -1], [29, -6, -1], [-15, 4, -1], [-3, 0, -8], [-33, 0, -2], [-17, -7, -3], [16, -3, 1], [-26, -4, 2], [16, 6, 2], [-22, -3, 5], [1, -16, -1], [-16, -4, -2], [14, -11, -2], [16, 1, -2], [-28, 6, 2], [-8, -9, 8], [14, -8, 3], [10, -12, 2], [8, 10, 2], [8, -4, -6], [-9, -10, 5], [-14, -13, 7], [12, 1, 3], [-5, -10, 5], [7, 5, 4], [-2, 11, 8], [20, 6, 5], [3, 11, 4], [-32, 1, -6], [16, -2, 0], [4, -6, 4], [-2, 12, -2], [0, 6, -6], [-4, -8, 6], [-5, 8, 1], [22, -5, -2], [2, -6, 4], [-7, -8, -5], [-36, -5, 3], [-30, -2, 0], [-23, -2, -1], [4, 10, 4], [-5, -4, 0], [-36, 4, -3], [10, -5, -2], [-12, -12, 6], [30, -6, 1], [7, -2, 0], [-4, -2, 5], [32, 0, 0], [-8, 14, 1], [15, -8, -4], [-34, 1, -7], [1, 1, -7], [8, -3, -5], [8, -5, -6], [-4, -10, -9], [-1, 12, 1], [-21, 10, -4], [-4, 2, 5], [6, -12, 0], [-26, -5, -2], [-17, 10, -1], [-1, -5, 10], [-20, 11, 4], [-2, 12, 0], [35, 6, 9], [15, 15, -8], [-20, -4, 4], [3, 2, 9], [5, 15, 2], [18, 12, 1], [-22, -6, -6], [36, -3, -3], [-4, 0, 2], [-7, 3, 0], [-7, -1, 6], [38, 8, 0], [-6, 4, 2], [19, -14, 2], [-8, -9, 4], [-10, 9, 3], [-4, 2, 4], [30, 8, 2], [26, 2, -4], [11, 11, -4], [-8, 4, 6], [26, -1, 2], [-6, 1, -9], [0, 13, 10], [-12, 7, 6], [26, 2, -4], [15, 5, -1], [37, 3, 4], [2, 11, -3], [-2, 0, -8], [11, -13, -9], [34, -2, 8], [20, -7, 6], [49, -3, 1], [4, 10, 4], [23, 9, 6], [6, 0, 6], [-1, 13, 8], [8, 2, -6], [-16, 14, -4], [6, -4, 0], [-5, 0, -9], [-30, -10, 2], [-8, -4, 6], [28, 2, -6], [-26, -6, 4], [-8, 0, -5], [-6, -9, -6], [-4, -18, -2], [-12, -4, 2], [6, -10, -2], [-2, 7, -7], [45, -2, 5], [1, 18, 4], [-8, 6, 10], [43, 8, -4], [-29, -13, 1], [-14, 5, 3], [18, -7, -7], [22, -6, -8], [-42, 11, 1], [33, -7, 0], [-42, -7, -2], [14, 10, -10], [18, -6, -6], [0, -10, 0], [0, -10, -8], [8, 10, -10], [-9, -14, -1], [16, 6, 0], [-10, 0, -6], [-9, 7, -5], [35, -13, -1], [12, 10, -5], [21, 10, 1], [9, 13, 13], [46, 12, -4], [-20, -3, 3], [18, 12, -4], [-1, -5, 14], [15, 15, -1], [-12, -17, 4], [11, -4, 5], [35, -2, 11], [43, -6, -4], [36, 7, 4], [-46, 3, -6], [-24, -10, -2], [16, 0, 13], [-8, 3, -2], [34, -14, 0], [18, -9, 6], [27, 16, 0], [2, 5, 0], [18, -16, 2], [-25, -13, 3], [15, -11, 2], [-33, -10, -4], [21, 10, 16], [-9, 21, 3], [-15, -8, 0], [-10, -3, 2], [57, 7, 0], [-28, 3, -8], [-16, 4, 2], [5, -15, -1], [-20, 2, -8], [-45, 1, -6], [-38, 18, 4], [-56, 0, -4], [13, 5, 10], [-23, -4, 7], [24, 9, 12], [22, 4, 3], [-6, -23, -1], [-2, 5, -14], [60, -6, -1], [18, 12, -6], [24, -1, -4], [10, 7, 14], [48, 1, -3], [70, -5, -2], [16, 4, 5], [0, -21, -3], [-13, 10, 9], [5, -6, 12], [-63, 5, -1], [-32, 11, 0], [66, 4, -2], [-14, -13, 4], [-25, -9, 3], [-25, 10, -5], [-16, 17, 0], [6, 9, 6], [34, -19, 0], [-14, -12, -1], [-13, 18, -1], [25, 18, -8], [-24, 12, 6], [12, 16, -2], [11, 21, -5], [-6, 6, -2], [23, 3, 7], [16, -1, 5], [18, 13, -5], [11, 18, -5], [18, -1, -1], [8, 5, -2], [34, 6, -2], [-34, 12, 0], [10, 8, -6], [6, -10, 10], [-54, -7, 4], [12, -9, -3], [10, -28, 0], [24, -8, -7], [-17, 23, 2], [-8, 11, -12], [20, -4, 10], [-22, 7, -6], [-14, 1, -15], [19, -20, 0], [-11, -7, 4], [14, 20, -8], [-5, 9, -5], [34, 8, 4], [14, 0, 6], [22, 3, 8], [-2, -10, 6], [40, 0, -4], [-33, 4, -10], [25, -9, -4], [-21, -6, -9], [16, -20, 6], [27, 1, 7], [48, 10, -4], [-21, 20, 1], [34, -2, -3], [-18, 3, 0], [63, 6, 7], [-50, -7, -11], [-11, 15, -6], [47, -11, 4], [64, -4, 0], [36, -3, -3], [18, 21, -1], [-24, 22, 2], [-7, -29, 5], [-28, -8, -12], [26, -6, -2], [-7, -8, 1], [-6, -12, -8], [-54, -7, -14], [50, 11, 6], [34, -16, -2], [-26, 9, 8], [21, -11, -8], [-12, -21, -1], [3, -2, 3], [-36, -19, 7], [34, -14, 6], [52, -9, -4], [38, -12, -6], [-39, 11, 2], [-49, 11, -2], [17, 1, -8], [-41, 1, 10], [41, 17, -4], [-34, 18, 7], [8, -5, -9], [-27, -18, 3], [-14, 4, 2], [69, -3, 6], [45, -4, 5], [-2, -8, 10], [17, 6, 13], [-16, -18, -16], [-42, 14, 0], [2, 1, -11], [-41, 10, -1], [-1, 11, 4], [22, 15, -2], [26, -19, -3], [11, 15, 16], [-28, 9, -10], [32, 11, 9], [-3, 13, -1], [-16, -4, -1], [-60, 15, 2], [17, 14, 14], [14, -1, 13], [-50, 0, 2], [4, -4, 1], [16, 8, -2], [-46, -8, 6], [-28, -12, 2], [-12, 14, 2], [26, -12, 6], [-24, 22, 2]]