# q-expansion of newform 5070.2.a.h, downloaded from the LMFDB on 17 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
    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 = 5070
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 = [[-1], [-1], [1], [0], [6], [0], [6], [-6], [-6], [-6], [0], [-6], [-12], [-8], [0], [-12], [6], [10], [0], [12], [-6], [-8], [0], [0], [6], [-18], [-14], [0], [6], [-18], [2], [12], [-18], [-4], [18], [12], [-4], [12], [0], [-12], [-12], [-2], [-12], [6], [-6], [20], [4], [0], [-24], [18], [6], [0], [-24], [-12], [-18], [6], [30], [12], [-8], [0], [-4], [6], [12], [-12], [10], [6], [18], [-22], [0], [6], [-6], [0], [26], [4], [-6], [24], [-30], [-18], [-24], [12], [-36], [6], [-12], [2], [-8], [24], [0], [6], [6], [0], [24], [24], [-24], [-12], [6], [18], [-6], [-6], [-16], [18], [-28], [30], [0], [-6], [-4], [-30], [-36], [-30], [-24], [-26], [14], [6], [-6], [-30], [-12], [-30], [0], [42], [0], [12], [42], [46], [24], [-24], [42], [6], [-30], [-12], [26], [-30], [-42], [48], [4], [16], [-48], [-24], [6], [-12], [-12], [6], [-18], [6], [22], [12], [2], [-12], [-54], [42], [4], [-48], [18], [-54], [56], [-30], [8], [36], [52], [0], [34], [18], [-12], [-54], [48], [36], [-18], [0], [20], [-28], [48], [-12], [42], [-6], [36], [-18], [-32], [30], [54], [-6], [34], [10], [48], [-12], [-32], [18], [-48], [30], [52], [24], [12], [24], [-34], [0], [20], [18], [-36], [54], [36], [-28], [-30], [-54], [6], [44], [-54], [-46], [30], [-48], [48], [-24], [48], [-4], [2], [30], [46], [60], [0], [-48], [22], [-18], [48], [-18], [-22], [-24], [-12], [-48], [12], [10], [-54], [-60], [-46], [30], [4], [-52], [-60], [30], [-28], [-24], [-36], [-54], [-12], [-48], [60], [-52], [-14], [-54], [-42], [-36], [-24], [66], [-54], [-42], [-18], [-36], [24], [62], [36], [78], [-38], [-12], [12], [42], [-10], [12], [-30], [-16], [-54], [28], [54], [-12], [48], [24], [50], [24], [30], [-68], [22], [24], [-48], [18], [-36], [12], [-18], [-48], [-6], [18], [36], [0], [70], [-42], [-24], [-18], [6], [-48], [6], [-78], [8], [-6], [8], [0], [60], [0], [58], [18], [52], [-84], [-76], [66], [-48], [-22], [48], [20], [18], [-6], [-42], [4], [-48], [-74], [-18], [-12], [18], [-66], [28], [42], [6], [72], [6], [-2], [-30], [-84], [-78], [36], [54], [48], [-44], [12], [30], [60], [-66], [18], [-60], [-38], [-42], [42], [-90], [-44], [54], [-60], [-74], [12], [-84], [24], [18], [-54], [-42], [62], [-58], [54], [-60], [6], [-54], [48], [6], [-6], [-6], [18], [52], [-82], [54], [72], [70], [60], [-92], [-72], [-54], [4], [56], [-54], [72], [42], [-18], [-98], [42], [-42], [24], [30], [-54], [-72], [-60], [32], [36], [78], [72], [-42], [30], [-8], [96], [94], [60], [-42], [-68], [30], [78], [-42], [72], [42], [-6], [-92], [30], [24], [36], [54], [-26], [-36], [12], [28], [-50], [42], [60], [-62], [-78], [102], [30], [-6], [72], [-12], [-6], [-78], [96], [-12], [102], [-24]]