# q-expansion of newform 3648.2.a.c, downloaded from the LMFDB on 28 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 = 3648
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], [-1], [-2], [0], [-4], [-2], [-6], [-1], [4], [2], [-4], [-10], [10], [4], [4], [10], [12], [-14], [-12], [-8], [-6], [4], [12], [-6], [10], [-2], [12], [-4], [6], [2], [12], [12], [-14], [12], [6], [-20], [-22], [20], [0], [-6], [12], [14], [-4], [-6], [22], [16], [12], [28], [28], [10], [-6], [-12], [10], [-28], [2], [12], [-6], [8], [26], [10], [12], [18], [-12], [-4], [-22], [-6], [-4], [-14], [-20], [26], [18], [-12], [8], [-26], [-36], [-16], [-18], [10], [-14], [-14], [-12], [-26], [24], [26], [4], [-20], [34], [26], [-2], [8], [-20], [-4], [-4], [20], [-20], [-36], [-6], [26], [-4], [-38], [4], [-34], [-12], [-38], [20], [2], [12], [34], [0], [10], [4], [2], [42], [4], [40], [18], [4], [4], [-18], [-4], [38], [26], [26], [4], [-36], [22], [42], [-28], [8], [-6], [20], [-32], [-44], [2], [-30], [-14], [-30], [-20], [-6], [-30], [12], [30], [-8], [-28], [-42], [16], [-6], [42], [52], [24], [-34], [18], [-20], [32], [44], [-24], [-8], [-22], [26], [-22], [12], [-22], [-48], [-28], [-46], [24], [-4], [2], [10], [6], [-12], [-26], [-36], [26], [-12], [-6], [-52], [54], [-28], [-62], [8], [4], [-34], [58], [36], [54], [22], [-20], [50], [60], [-14], [20], [-44], [18], [52], [-6], [2], [42], [2], [36], [-30], [-28], [38], [2], [44], [-26], [64], [-36], [-6], [-28], [50], [-2], [48], [12], [-48], [-30], [0], [18], [32], [6], [-18], [4], [-46], [64], [68], [42], [26], [72], [28], [60], [-6], [-52], [-60], [10], [-4], [60], [66], [46], [68], [24], [4], [52], [56], [-74], [18], [12], [-64], [20], [20], [-36], [-22], [-46], [-68], [26], [-2], [52], [42], [-4], [-54], [-38], [44], [-60], [-30], [6], [42], [28], [66], [42], [20], [-18], [6], [28], [-54], [-56], [-46], [24], [36], [-10], [-38], [-12], [48], [32], [60], [-18], [28], [12], [66], [18], [32], [50], [-66], [12], [42], [20], [70], [-26], [-44], [-26], [28], [36], [58], [34], [-16], [68], [-12], [26], [-60], [-26], [60], [-38], [-52], [54], [-14], [60], [12], [-6], [-36], [32], [2], [-62], [52], [42], [-78], [28], [-30], [-54], [4], [4], [32], [22], [74], [18], [16], [-44], [-36], [-28], [30], [-78], [-70], [80], [-10], [18], [-54], [52], [34], [-20], [-30], [-36], [-8], [30], [4], [2], [46], [-36], [-26], [-38], [52], [-60], [-38], [24], [98], [26], [-72], [20], [44], [74], [6], [-12], [-62], [12], [84], [76], [-22], [-80], [82], [-52], [28], [50], [90], [66], [-30], [98], [-64], [-30], [-84], [-32], [32], [-70], [-44], [56], [-46], [-86], [-12], [4], [-72], [-46], [-100], [-70], [-20], [30], [22], [-86], [60], [-38], [18], [88], [-46], [50], [28], [92], [-14], [30], [76], [-76], [-86], [-42], [24], [-20], [42], [32], [-30], [22], [52], [100], [34], [90], [68], [10], [-52], [4]]