# q-expansion of newform 6192.2.a.h, downloaded from the LMFDB on 15 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
    R = PolynomialRing(QQ, "x")
    f = R(poly_data)
    K = NumberField(f, "a")
    betas = [K.gens()[0]**i for i in range(len(poly_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 = 6192
weight = 2
poly_data = [0, 1]

# The basis for the coefficient ring is just the power basis
# in the root of the defining polynomial above.
hecke_ring_character_values = None
aps_data = [[0], [0], [-2], [2], [-4], [2], [2], [4], [2], [-10], [4], [-8], [-6], [-1], [2], [12], [4], [-8], [-4], [0], [10], [8], [8], [-6], [14], [-12], [12], [0], [2], [-6], [-16], [-12], [-18], [-4], [-22], [-14], [-12], [-24], [-2], [12], [-4], [-2], [-24], [-2], [-20], [-10], [8], [26], [12], [-10], [10], [-2], [-10], [-12], [30], [-20], [20], [20], [24], [6], [-4], [-12], [20], [2], [-26], [-24], [-32], [22], [12], [-28], [-10], [6], [-32], [-8], [-36], [0], [-14], [-34], [34], [-10], [28], [-32], [-26], [14], [-20], [24], [34], [22], [-28], [-14], [12], [-10], [36], [-36], [-20], [-28], [16], [22], [4], [-2], [-12], [-12], [40], [-6], [32], [10], [-12], [10], [-38], [-30], [22], [-14], [-34], [4], [10], [6], [-36], [-4], [34], [36], [38], [46], [14], [12], [40], [-4], [26], [2], [-2], [0], [12], [12], [-26], [-12], [-6], [-18], [6], [44], [12], [-30], [52], [-24], [-32], [20], [-32], [4], [54], [42], [-24], [-32], [2], [-34], [28], [56], [-28], [-12], [-20], [6], [2], [-28], [52], [-6], [12], [48], [10], [12], [14], [-52], [-34], [32], [-28], [-32], [0], [10], [46], [14], [-40], [-14], [36], [0], [-10], [40], [32], [6], [28], [18], [-20], [-56], [22], [-36], [66], [-44], [12], [-30], [28], [-10], [-14], [38], [-18], [40], [-40], [-14], [-8], [-38], [44], [6], [30], [24], [22], [-52], [38], [-40], [-56], [-52], [60], [26], [38], [46], [-24], [24], [-4], [20], [-30], [28], [-20], [26], [6], [52], [-54], [-20], [-48], [-76], [40], [-58], [28], [-30], [14], [-4], [-36], [-6], [76], [4], [-8], [18], [30], [22], [50], [12], [-20], [-66], [-34], [6], [-6], [10], [18], [-60], [64], [-60], [6], [66], [-2], [52], [-22], [-70], [-18], [12], [58], [70], [-16], [20], [10], [52], [-6], [-70], [22], [-10], [4], [4], [70], [52], [6], [-4], [18], [-80], [64], [-48], [-46], [30], [10], [-50], [-64], [-80], [-50], [-60], [-50], [-20], [-52], [-20], [80], [68], [-30], [-58], [56], [24], [32], [-26], [-8], [68], [48], [-64], [76], [18], [10], [-88], [-14], [-50], [-32], [18], [-82], [-18], [28], [62], [30], [32], [26], [-2], [80], [76], [18], [-30], [-40], [48], [38], [44], [4], [36], [-8], [-6], [-86], [-14], [6], [42], [-86], [14], [-56], [20], [12], [-12], [16], [-56], [52], [-2], [-28], [-46], [14], [-94], [36], [-12], [42], [-26], [28], [22], [28], [60], [60], [-62], [-70], [-4], [-58], [-92], [-80], [30], [-66], [-36], [-32], [28], [-62], [14], [-78], [-10], [72], [-94], [60], [-78], [28], [26], [-42], [70], [44], [-74], [86], [58], [68], [4], [-60], [42], [24], [14], [0], [-54], [98], [18], [28], [-14], [54], [2], [28], [-38], [-76], [28], [-26], [62], [-28], [20], [-30], [0], [-70], [40], [82], [-36], [102], [22], [12], [-40], [22], [-30], [60], [34], [108], [68]]