
# q-expansion of newform 2142.2.p.f, downloaded from the LMFDB on 04 September 2025.

# 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 = 2142
weight = 2
poly_data = [1156, 0, 4428, 0, 5385, 0, 2400, 0, 450, 0, 36, 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, 0, 0, 0, 0, 0, 0], 1], [[-136816, 0, -40198, 0, 44682, 0, 15305, 0, 1528, 0, 47, 0], 32164], [[1199656, -664944, 2382410, -865206, 1025915, -282205, 173759, -39313, 12773, -2467, 335, -57], 128656], [[1199656, 664944, 2382410, 865206, 1025915, 282205, 173759, 39313, 12773, 2467, 335, 57], 128656], [[432820, 0, 116440, 0, -361295, 0, -131609, 0, -14197, 0, -463, 0], 128656], [[-1485732, 394020, -3340908, 350548, -1832787, 167211, -371977, 36465, -30753, 3289, -867, 99], 257312], [[-1485732, -394020, -3340908, -350548, -1832787, -167211, -371977, -36465, -30753, -3289, -867, -99], 257312], [[-2094400, 793600, -4492334, 865206, -2282203, 282205, -447755, 39313, -36509, 2467, -1023, 57], 128656], [[2094400, 793600, 4492334, 865206, 2282203, 282205, 447755, 39313, 36509, 2467, 1023, 57], 128656], [[0, -15318, 0, -33857, 0, -18414, 0, -3774, 0, -316, 0, -9], 1496], [[190400, -141872, 408394, -362608, 207473, -192970, 40705, -38932, 3319, -3238, 93, -92], 11696], [[0, -174542, 0, -372491, 0, -192621, 0, -38161, 0, -3129, 0, -88], 5848]]

hecke_ring_character_values = [[1667, [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]], [1837, [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]], [1261, [0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0]]]
aps_data = [[0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -1, 0, -1, 0, 0, 0], [0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0], [0, 0, -1, 0, 0, 1, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, -1, 0, 0, -1, 1, 0, 0, 0], [0, 0, -1, 1, 0, 0, 0, -1, 0, 1, 0, 1], [0, 0, 1, -1, 0, 0, 0, 1, 0, 0, 1, -1], [1, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0], [-1, 1, 0, 1, -1, 0, 0, 0, -2, 1, 1, 1], [1, 0, 0, 0, -1, 0, -2, 0, 1, -1, 1, 0], [-1, -1, 0, -1, 0, 0, -1, 0, 0, 1, 0, -1], [0, -1, 0, 0, 1, -2, 0, 2, -1, 0, 1, 1], [0, 0, 0, 0, 0, 0, 0, -2, -2, -3, 0, -1], [2, 0, 0, 0, -2, -3, -3, -1, 1, 0, 0, 0], [0, 0, 0, 0, 0, 1, -1, 1, -1, -1, 2, -3], [0, 0, -1, 1, 0, 1, -1, 1, 2, 0, -1, 3], [-1, 2, 2, 0, 1, 0, 0, 2, -1, -1, 1, -2], [-1, -1, 1, 1, -2, 0, 0, -3, 3, 0, 0, 0], [3, -2, 0, 0, -2, 0, 0, 0, 0, -3, 2, -2], [-3, 0, 0, 2, -1, 0, 1, 0, 2, 3, 1, 0], [4, -1, -2, 0, 1, 5, 0, -1, -1, 4, 1, 1], [0, 0, 0, 0, 0, -1, 1, 0, 1, -3, -1, -4], [-1, -5, -1, -1, -2, 2, 2, -2, 2, 0, 0, 0], [3, 2, 0, 2, -1, 0, -1, 0, 0, -3, 1, 2], [0, -1, -1, -1, 1, 2, 2, 1, -1, 0, 0, 0], [1, -5, -1, -1, -2, 1, 1, 1, -1, 0, 0, 0], [3, -2, 0, -1, -1, 0, -4, 0, 0, -3, 1, -2], [1, 0, 1, 0, -1, -6, 0, -3, 1, 1, -1, 0], [-3, -2, 0, 0, 1, -1, 0, 1, -1, -3, 1, 2], [0, 0, -2, 2, 0, 1, -1, -1, -1, -4, 0, 4], [-3, 0, 0, -2, -1, 0, 2, 0, 1, 3, 1, 0], [-4, -4, -2, -2, 2, -1, -1, -1, 1, 0, 0, 0], [-5, 0, 0, 0, -3, 0, -5, 0, -2, 5, 3, 0], [7, -3, 0, 0, 0, -7, -7, 1, -1, 0, 0, 0], [0, 0, 0, 0, 0, -2, 2, 0, 2, 4, -2, -2], [0, 1, 3, 3, 1, -3, -3, 3, -3, 0, 0, 0], [2, 0, -3, 0, 1, -6, 0, 1, -1, 2, 1, 0], [-4, -1, 0, 0, 1, 0, 3, 0, -2, 4, -1, -1], [8, 0, 0, -2, 2, 0, 0, 0, 2, -8, -2, 0], [0, 0, 1, -1, 0, -1, 1, 2, 0, 4, 2, 2], [4, -1, 2, 0, -1, 6, 0, 0, 1, 4, -1, 1], [-3, -3, -1, -1, -2, 5, 5, -4, 4, 0, 0, 0], [1, 2, 0, 0, 0, 6, 0, -2, 0, 1, 0, -2], [-2, -2, -3, 0, -1, 6, 0, -1, 1, -2, -1, 2], [6, 2, 0, 0, -2, 0, 4, 0, 0, -6, 2, 2], [3, 1, 1, 0, 1, -4, 0, 3, -1, 3, 1, -1], [0, 0, -2, 2, 0, -6, 6, -1, -3, 0, 2, 4], [5, 1, -2, 0, -2, 6, 0, 0, 2, 5, -2, -1], [0, 0, 0, 0, 0, 0, 0, -1, -2, -17, 1, -2], [1, 2, 0, 0, -1, 0, 13, 0, 0, -1, 1, 2], [-9, 3, 2, 2, -4, -1, -1, 1, -1, 0, 0, 0], [-10, 0, 0, 0, 2, 0, 4, 0, 0, 10, -2, 0], [-3, 4, 0, 0, 3, -9, -9, 2, -2, 0, 0, 0], [0, 0, 1, -1, 0, -2, 2, 0, 6, 1, -6, 3], [0, 0, 4, -4, 0, 5, -5, 3, 3, 1, 0, -3], [-6, -1, 0, 2, -1, 0, -2, 0, 1, 6, 1, -1], [-7, -1, -1, -1, -2, 3, 3, -3, 3, 0, 0, 0], [-5, 0, 0, -3, 1, 0, -6, 0, 0, 5, -1, 0], [0, 0, -2, 2, 0, 3, -3, -5, 1, 14, -6, 2], [9, -3, -4, 0, -2, 5, 0, -3, 2, 9, -2, 3], [-4, 1, 1, 1, 3, 10, 10, 1, -1, 0, 0, 0], [-2, -3, -1, -1, -5, 4, 4, -3, 3, 0, 0, 0], [6, 1, 0, 6, 1, 0, 1, 0, 2, -6, -1, 1], [6, 2, 0, 0, 2, -10, 0, 4, -2, 6, 2, -2], [-3, -1, -3, 0, -3, -12, 0, -3, 3, -3, -3, 1], [0, 0, 1, -1, 0, -2, 2, -1, 3, -11, -4, 1], [-6, -5, 0, 0, -2, 0, 2, 0, 0, 6, 2, -5], [-4, 2, 1, 0, -2, 11, 0, -8, 2, -4, -2, -2], [0, 0, 0, 0, 0, 4, -4, -1, 2, -11, -3, -4], [-1, 5, 3, 3, 0, -4, -4, 2, -2, 0, 0, 0], [0, 0, 1, -1, 0, 8, -8, 1, -1, 6, 2, -2], [-3, 0, -4, 0, -3, 2, 0, -4, 3, -3, -3, 0], [10, 4, -2, -2, 4, 1, 1, 1, -1, 0, 0, 0], [-5, 1, 0, -3, -3, 0, -4, 0, -2, 5, 3, 1], [0, 0, 4, -4, 0, -4, 4, 4, 6, 8, -2, -6], [0, 0, 1, -1, 0, 0, 0, -3, -3, -8, 0, -4], [12, 2, 2, 0, 2, 4, 0, 4, -2, 12, 2, -2], [2, -4, -4, 0, 0, 14, 0, 0, 0, 2, 0, 4], [18, 4, 2, 2, 0, -9, -9, -3, 3, 0, 0, 0], [-7, -2, 0, 0, -3, -6, 0, 0, 3, -7, -3, 2], [8, 4, -1, -1, 6, -10, -10, -1, 1, 0, 0, 0], [-3, 2, 0, 0, -2, 4, 0, 2, 2, -3, -2, -2], [0, 0, -2, 2, 0, 8, -8, -4, -2, -4, -2, 4], [-4, -5, 0, 2, 3, 0, -10, 0, 1, 4, -3, -5], [-10, -10, -3, -3, 0, -11, -11, -2, 2, 0, 0, 0], [-5, -4, -2, 0, -3, -13, 0, -1, 3, -5, -3, 4], [0, 0, 2, -2, 0, -6, 6, 2, 2, 1, 0, -5], [0, 0, -5, 5, 0, 2, -2, -1, 0, 2, -1, 1], [-2, -4, -4, -4, -2, 5, 5, -1, 1, 0, 0, 0], [0, 0, -1, 1, 0, 7, -7, -5, -6, 12, 1, 5], [12, 7, 0, 6, -1, 0, -1, 0, -2, -12, 1, 7], [-2, -3, 4, 0, -3, -5, 0, -1, 3, -2, -3, 3], [0, 0, -3, 3, 0, -2, 2, -7, -7, 0, 0, 0], [2, -2, 1, 0, -3, 16, 0, -1, 3, 2, -3, 2], [4, 0, 4, 0, -2, 4, 0, 0, 2, 4, -2, 0], [6, 1, -3, -3, -3, -1, -1, -6, 6, 0, 0, 0], [1, 1, -2, 0, 6, 14, 0, 2, -6, 1, 6, -1], [6, 3, 5, 5, -3, -13, -13, -2, 2, 0, 0, 0], [3, 3, 0, -1, -2, 0, -1, 0, 2, -3, 2, 3], [4, 6, 0, 7, 1, 0, 10, 0, -2, -4, -1, 6], [0, -6, -6, -6, 0, -6, -6, -4, 4, 0, 0, 0], [0, 0, -2, 2, 0, 1, -1, 4, 5, 5, -1, 4], [0, 0, 2, -2, 0, -6, 6, -2, -2, 18, 0, -4], [-5, -3, 3, 0, -1, 6, 0, -5, 1, -5, -1, 3], [-6, 6, 2, 2, -6, 1, 1, 1, -1, 0, 0, 0], [0, 0, -4, 4, 0, -1, 1, 0, 3, 11, -3, 12], [0, 0, 2, -2, 0, -4, 4, -1, 3, 4, -4, -8], [2, -4, -5, -5, -2, -18, -18, -5, 5, 0, 0, 0], [6, 8, 8, 0, 2, 12, 0, 6, -2, 6, 2, -8], [2, -2, 0, -4, -2, 0, 16, 0, 0, -2, 2, -2], [14, 2, -3, -3, 2, 4, 4, 3, -3, 0, 0, 0], [1, 2, 0, -4, -1, 0, -1, 0, -2, -1, 1, 2], [5, 1, 2, 0, 0, 5, 0, 3, 0, 5, 0, -1], [0, 0, 0, 0, 0, -4, 4, -4, -10, 4, 6, -2], [-5, -6, 0, -10, 1, 0, 9, 0, 4, 5, -1, -6], [-3, -3, 0, 2, 0, 0, -5, 0, 3, 3, 0, -3], [-2, 0, 2, 2, -2, 11, 11, -3, 3, 0, 0, 0], [-6, 6, 1, 0, -1, 4, 0, 1, 1, -6, -1, -6], [-8, -8, -2, -2, 0, 3, 3, -1, 1, 0, 0, 0], [0, 0, -1, 1, 0, 5, -5, -1, 4, 4, -5, -3], [2, -7, -10, 0, 2, 8, 0, -6, -2, 2, 2, 7], [4, 5, 0, -4, 5, 0, -9, 0, -4, -4, -5, 5], [13, 0, 0, -5, -1, 0, -2, 0, 6, -13, 1, 0], [-3, 11, 8, 0, 4, -3, 0, 5, -4, -3, 4, -11], [3, -1, -1, -1, 4, 1, 1, 4, -4, 0, 0, 0], [-5, -3, 0, -1, -2, 0, -5, 0, 2, 5, 2, -3], [0, -4, 0, -2, -2, 0, 10, 0, 6, 0, 2, -4], [19, 3, -1, -1, 4, -5, -5, 5, -5, 0, 0, 0], [0, 0, -3, 3, 0, 11, -11, 1, 2, 10, -1, -5], [0, 0, 2, -2, 0, -4, 4, 10, 8, 2, 2, -4], [-3, 0, 0, 4, 4, 0, 14, 0, -8, 3, -4, 0], [-12, 4, 0, 0, -2, 0, 14, 0, 0, 12, 2, 4], [0, 0, -7, 7, 0, -1, 1, -6, -2, 4, -4, 10], [-1, -1, 1, 1, 2, -18, -18, 0, 0, 0, 0, 0], [6, 2, 0, 0, 6, -6, -6, 2, -2, 0, 0, 0], [0, 0, 5, -5, 0, 2, -2, 1, -6, 4, 7, -5], [-7, 1, 0, 2, 0, 0, 1, 0, -3, 7, 0, 1], [0, 0, -5, 5, 0, 1, -1, 6, -1, 4, 7, -1], [6, -10, -4, 0, -2, 4, 0, -2, 2, 6, -2, 10], [17, 0, 0, 4, 3, 0, 3, 0, 4, -17, -3, 0], [-8, 2, 0, 5, 9, 0, -4, 0, -4, 8, -9, 2], [-6, 5, -2, 0, 3, -13, 0, 1, -3, -6, 3, -5], [2, 0, -1, 0, -2, 5, 0, -16, 2, 2, -2, 0], [-9, -10, 0, 0, 1, -10, -10, -3, 3, 0, 0, 0], [2, -5, 4, 0, -3, 3, 0, 3, 3, 2, -3, 5], [-9, 2, 0, -2, 9, 0, 16, 0, -1, 9, -9, 2], [3, 0, 2, 0, 1, -28, 0, 2, -1, 3, 1, 0], [0, 0, 1, -1, 0, 11, -11, 4, 1, -4, 3, 1], [-7, 7, 2, 2, 4, -9, -9, -1, 1, 0, 0, 0], [-1, 7, 3, 0, 6, -7, 0, 6, -6, -1, 6, -7], [11, 2, 0, 6, -5, 0, -2, 0, 5, -11, 5, 2], [-2, -4, 0, 0, -2, -12, -12, 0, 0, 0, 0, 0], [8, -7, 0, -2, -1, 0, 5, 0, -4, -8, 1, -7], [-1, 3, 3, 0, -5, 2, 0, 3, 5, -1, -5, -3], [7, 2, -4, 0, 2, -4, 0, -2, -2, 7, 2, -2], [16, -6, -2, -2, -6, 10, 10, 0, 0, 0, 0, 0], [-6, -10, -4, 0, 0, 12, 0, 4, 0, -6, 0, 10], [0, 0, 0, 0, 0, -11, 11, -1, 5, -8, -6, 0], [-2, -2, -10, 0, 2, 21, 0, 5, -2, -2, 2, 2], [-3, -2, 0, 1, -1, 0, 22, 0, 0, 3, 1, -2], [-2, 2, -4, -4, 6, 21, 21, 3, -3, 0, 0, 0], [0, 0, 0, 0, 0, 10, -10, -2, 6, 0, -8, 16], [0, 0, -2, 2, 0, -11, 11, -2, -9, 11, 7, -2], [0, 0, 2, -2, 0, -2, 2, 8, 0, 8, 8, -2], [-8, 0, 0, 6, 2, 0, 6, 0, 2, 8, -2, 0], [2, -7, 0, -2, 1, 0, -17, 0, 4, -2, -1, -7], [4, 4, -4, 0, 8, 8, 0, 2, -8, 4, 8, -4], [20, -5, -4, 0, -2, 16, 0, -4, 2, 20, -2, 5], [-2, 11, 4, 0, 5, 9, 0, 7, -5, -2, 5, -11], [28, -12, -7, -7, -4, 2, 2, -3, 3, 0, 0, 0], [-9, 14, 2, 2, 9, 8, 8, 5, -5, 0, 0, 0], [10, 5, 2, 0, 6, 0, 0, 16, -6, 10, 6, -5], [9, 3, -5, -5, 8, -8, -8, 3, -3, 0, 0, 0], [0, 0, 2, -2, 0, -8, 8, 5, -3, -2, 8, -16], [-6, -4, 0, 4, -6, 0, -14, 0, 2, 6, 6, -4], [-11, -3, 0, -3, 1, 0, -4, 0, 6, 11, -1, -3], [5, 1, -7, 0, 5, 12, 0, 9, -5, 5, 5, -1], [0, 0, 2, -2, 0, 4, -4, -1, -3, 0, 2, 0], [0, 0, 3, -3, 0, 7, -7, -1, -4, 6, 3, -1], [-2, -2, 2, 2, 4, -1, -1, 1, -1, 0, 0, 0], [-4, 6, 0, 14, 2, 0, 10, 0, -10, 4, -2, 6], [13, 0, 0, -1, 1, 0, 2, 0, 2, -13, -1, 0], [0, 0, 0, 0, 0, -4, 4, 5, 1, 6, 4, -6], [0, 0, -3, 3, 0, 19, -19, 4, 2, -17, 2, 9], [-8, 17, 5, 5, -1, 8, 8, -1, 1, 0, 0, 0], [-7, -1, 0, 1, 4, 0, -5, 0, 6, 7, -4, -1], [0, -5, -5, -5, 3, -21, -21, 0, 0, 0, 0, 0], [13, 2, 6, 0, -2, -24, 0, -6, 2, 13, -2, -2], [-12, 5, 0, 0, -1, 0, -9, 0, -4, 12, 1, 5], [-7, -2, 0, -2, -5, 0, 13, 0, -14, 7, 5, -2], [0, 2, -3, 0, 8, -15, 0, 14, -8, 0, 8, -2], [0, 0, 6, -6, 0, 4, -4, 12, 4, -5, 8, -3], [0, 0, 1, -1, 0, 5, -5, -2, 5, 8, -7, -11], [10, 0, 0, 3, 4, 0, -9, 0, 2, -10, -4, 0], [3, -10, 0, -2, -1, 0, 24, 0, 5, -3, 1, -10], [-3, -8, -6, 0, 2, -10, 0, 0, -2, -3, 2, 8], [1, 5, 1, 0, 10, 29, 0, 12, -10, 1, 10, -5], [1, 4, 6, 0, 3, 10, 0, 14, -3, 1, 3, -4], [-10, 14, 4, 4, 12, 4, 4, 6, -6, 0, 0, 0], [-5, -7, 0, -3, -9, 0, 6, 0, -4, 5, 9, -7], [-33, 1, -2, 0, 2, 6, 0, 2, -2, -33, 2, -1], [-7, -12, -4, -4, 1, -2, -2, 5, -5, 0, 0, 0], [0, 0, -4, 4, 0, 1, -1, -1, -7, -26, 6, 6], [13, -2, -4, -4, -11, 1, 1, -16, 16, 0, 0, 0], [0, 0, 3, -3, 0, 6, -6, -13, -6, -6, -7, 7], [-3, 5, 5, 5, -8, 13, 13, -1, 1, 0, 0, 0], [0, 0, -4, 4, 0, 6, -6, -4, -2, 0, -2, -2], [-12, 4, 0, 6, -4, 0, 22, 0, 6, 12, 4, 4], [-12, 1, -7, -7, 1, 18, 18, 1, -1, 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