# q-expansion of newform 3920.2.a.bd, downloaded from the LMFDB on 18 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 = 3920
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], [2], [-1], [0], [-4], [-2], [0], [2], [4], [10], [4], [-2], [12], [4], [4], [2], [10], [-6], [-4], [12], [4], [4], [14], [-8], [8], [-14], [20], [4], [10], [-18], [12], [-22], [-18], [-2], [-10], [8], [14], [-20], [-4], [-18], [-12], [22], [-4], [-18], [-6], [20], [4], [8], [14], [14], [-10], [-16], [-4], [14], [28], [16], [6], [-16], [26], [-10], [-10], [-22], [14], [0], [16], [-22], [20], [2], [-4], [22], [-12], [16], [-8], [-30], [-20], [-12], [18], [22], [18], [20], [14], [14], [-24], [32], [16], [-28], [18], [6], [-42], [16], [-26], [36], [28], [36], [-4], [-24], [22], [-20], [-18], [30], [44], [-30], [2], [-6], [12], [12], [18], [44], [-12], [-8], [32], [-26], [42], [-30], [20], [-14], [-30], [12], [6], [28], [-14], [-26], [14], [36], [42], [10], [-6], [36], [4], [-14], [-20], [-36], [-16], [6], [12], [4], [-46], [-42], [-50], [22], [26], [-2], [8], [-36], [-46], [-20], [14], [-32], [14], [-24], [-18], [-8], [-36], [-12], [-28], [0], [28], [-36], [12], [10], [-28], [18], [20], [-6], [6], [36], [-36], [-6], [30], [-34], [44], [-26], [28], [54], [-28], [-48], [-44], [-6], [8], [-14], [-44], [-58], [-34], [-36], [-12], [-50], [2], [-26], [6], [60], [36], [12], [-60], [6], [-60], [-36], [-22], [-26], [12], [16], [-18], [-20], [34], [-8], [18], [10], [12], [-36], [30], [-10], [-46], [-14], [-68], [-42], [16], [52], [32], [-36], [-16], [-42], [66], [64], [66], [-12], [54], [-70], [-36], [40], [-56], [68], [22], [30], [-28], [26], [14], [-48], [-20], [-38], [-12], [-24], [-4], [10], [-24], [-34], [4], [-60], [0], [26], [44], [0], [-14], [40], [72], [-16], [42], [-36], [42], [-62], [18], [48], [-60], [-36], [34], [-54], [0], [6], [-10], [-48], [-12], [54], [14], [-44], [-12], [64], [44], [-24], [-52], [-34], [62], [-22], [-40], [12], [-28], [10], [-26], [72], [10], [58], [76], [56], [30], [62], [10], [-22], [50], [-10], [-48], [6], [-34], [58], [-40], [-2], [-68], [-60], [-68], [-26], [-68], [-22], [56], [-46], [-8], [26], [-18], [-60], [40], [56], [22], [-12], [44], [-18], [-74], [46], [18], [8], [-26], [92], [-4], [70], [-68], [6], [-2], [54], [28], [50], [20], [-42], [-6], [12], [-16], [-12], [-6], [-22], [54], [-84], [78], [36], [-26], [68], [40], [74], [54], [-14], [14], [20], [-10], [-12], [-12], [74], [-38], [4], [-78], [-64], [44], [-36], [38], [-54], [-26], [-4], [-10], [20], [30], [48], [-90], [-80], [-30], [90], [-48], [-44], [-44], [-64], [-42], [-74], [64], [-94], [-26], [-80], [20], [-70], [-4], [-76], [-22], [42], [60], [82], [-40], [30], [20], [-44], [-4], [2], [-6], [90], [80], [-40], [70], [12], [98], [-66], [-46], [14], [44], [-58], [-60], [-4], [-58], [-90], [84], [84], [-80], [88], [58], [34], [-64], [-50], [96], [42], [36], [-90], [62], [-44]]