# q-expansion of newform 5415.2.a.d, downloaded from the LMFDB on 27 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 = 5415
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], [2], [-4], [2], [6], [0], [6], [0], [4], [-6], [8], [6], [-2], [-14], [12], [-10], [-12], [12], [-10], [-4], [6], [0], [18], [6], [8], [-4], [0], [6], [8], [0], [-2], [12], [2], [16], [-18], [-10], [0], [-6], [20], [-8], [-12], [10], [-6], [4], [-16], [8], [-12], [-26], [26], [0], [28], [0], [18], [30], [20], [-20], [22], [-24], [26], [18], [-28], [-24], [-14], [30], [32], [10], [-6], [26], [14], [12], [10], [6], [16], [24], [26], [-10], [-4], [8], [20], [28], [-16], [2], [20], [-10], [36], [-14], [-22], [2], [-30], [-12], [-32], [24], [-20], [-34], [24], [-28], [-4], [34], [20], [38], [-4], [4], [-40], [-22], [-30], [42], [20], [28], [-24], [-34], [46], [44], [-4], [0], [18], [26], [14], [-12], [-28], [-10], [-30], [-36], [32], [2], [-30], [36], [-14], [-14], [-8], [-24], [-24], [-22], [10], [-34], [38], [-4], [2], [26], [-44], [-10], [-26], [12], [-8], [12], [42], [-22], [-24], [16], [-34], [-18], [-2], [-32], [4], [56], [20], [46], [-2], [-20], [26], [-54], [22], [-60], [42], [32], [32], [-18], [-44], [6], [0], [40], [-24], [26], [-40], [-18], [20], [-58], [-56], [-18], [-58], [-20], [34], [18], [22], [10], [30], [52], [48], [-52], [-18], [30], [68], [-40], [-26], [-18], [-62], [50], [-14], [2], [-8], [-56], [-38], [44], [-40], [-26], [48], [-36], [-26], [20], [-14], [-26], [-6], [36], [0], [20], [-30], [-12], [-24], [66], [60], [32], [-40], [-46], [4], [10], [34], [-24], [-8], [-32], [2], [-56], [0], [16], [-6], [22], [-34], [-22], [-72], [-24], [4], [20], [-26], [36], [6], [-48], [-30], [56], [-32], [50], [-78], [-46], [2], [-24], [-22], [52], [74], [4], [18], [-50], [-48], [-12], [26], [46], [-62], [20], [-16], [6], [44], [-30], [40], [-12], [-74], [-24], [58], [38], [-26], [-20], [-4], [-60], [24], [28], [-54], [-64], [78], [8], [10], [30], [44], [-68], [-2], [-2], [-10], [80], [38], [-74], [-12], [18], [-48], [34], [-74], [18], [48], [-68], [-36], [74], [28], [-52], [-88], [58], [66], [-6], [-60], [36], [74], [84], [8], [48], [-14], [66], [-28], [62], [16], [-56], [6], [60], [-44], [-20], [-8], [-34], [-26], [18], [-32], [-14], [-88], [38], [8], [-54], [54], [-18], [54], [30], [84], [52], [-94], [-44], [26], [-28], [-12], [38], [-4], [46], [-50], [80], [4], [18], [-48], [56], [38], [-88], [-18], [58], [-96], [40], [-10], [50], [54], [-40], [4], [-4], [-92], [-6], [4], [-32], [78], [-32], [-44], [74], [-34], [34], [-12], [66], [42], [18], [28], [-48], [-20], [6], [-74], [-32], [-16], [42], [-20], [22], [-24], [14], [52], [-32], [12], [10], [32], [-86], [-40], [-6], [-60], [-16], [-26], [-12], [-68], [20], [62], [-38], [-76], [72], [22], [-54], [-60], [-8], [-10], [-40], [12], [-66], [30], [-96], [-46], [34], [-36], [-6], [-4], [-36]]