
# q-expansion of newform 3825.2.a.m, downloaded from the LMFDB on 12 September 2025.

# 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 = 3825
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], [0], [0], [4], [-6], [-4], [1], [4], [0], [0], [8], [-6], [12], [6], [4], [6], [-8], [10], [2], [-2], [6], [4], [-8], [6], [6], [-10], [14], [-4], [-14], [-2], [6], [10], [14], [-8], [-6], [-8], [4], [-4], [8], [6], [4], [18], [0], [-14], [-6], [20], [-20], [22], [-20], [26], [22], [-12], [-22], [-12], [14], [-4], [0], [-8], [-10], [6], [24], [26], [30], [2], [10], [-34], [-4], [14], [12], [14], [-6], [-20], [-8], [16], [0], [16], [34], [26], [4], [-6], [18], [-30], [14], [-16], [-8], [-4], [-36], [-28], [-26], [-22], [12], [-18], [-24], [0], [-32], [-40], [-10], [12], [30], [2], [20], [-2], [4], [-6], [12], [-20], [0], [14], [4], [-18], [32], [-28], [6], [-36], [-8], [-8], [44], [24], [-6], [32], [-10], [-22], [-26], [44], [0], [10], [-26], [18], [-38], [-20], [28], [-24], [-12], [-8], [-6], [-2], [30], [-20], [-22], [12], [32], [20], [12], [44], [-30], [22], [14], [30], [28], [-24], [34], [20], [22], [-48], [-24], [-6], [-24], [56], [-60], [16], [-44], [-26], [-6], [32], [18], [24], [-20], [46], [50], [58], [24], [14], [38], [-32], [-56], [16], [-28], [32], [-58], [10], [58], [-30], [-62], [22], [-12], [-30], [46], [14], [-18], [38], [34], [-44], [52], [-42], [-44], [6], [22], [46], [18], [64], [48], [-64], [-32], [-46], [-24], [66], [-24], [16], [-60], [28], [-26], [-50], [-8], [36], [2], [-58], [22], [14], [-8], [-18], [-70], [68], [6], [24], [40], [-6], [18], [-14], [26], [58], [-4], [44], [-16], [-38], [42], [-48], [14], [58], [-58], [-24], [-36], [-4], [6], [18], [34], [10], [20], [42], [12], [-8], [-8], [-36], [12], [-2], [46], [-36], [50], [24], [38], [16], [44], [-24], [-50], [-42], [-46], [44], [-38], [14], [60], [70], [14], [58], [-16], [-64], [-16], [34], [-76], [-46], [-38], [-24], [-36], [56], [72], [-22], [-72], [-32], [-26], [-22], [-40], [-18], [-68], [12], [-38], [62], [10], [-76], [64], [-46], [-46], [-70], [-28], [62], [-16], [60], [-72], [2], [72], [38], [4], [-36], [80], [40], [-56], [10], [-44], [-18], [0], [-74], [-38], [70], [-68], [38], [-90], [2], [30], [-10], [-24], [48], [0], [-54], [-2], [14], [16], [-12], [8], [36], [-94], [-78], [10], [-6], [-16], [-30], [60], [-88], [-90], [90], [-18], [-54], [-22], [-18], [-4], [-66], [-18], [80], [46], [82], [-36], [34], [74], [12], [38], [-84], [36], [90], [-38], [-20], [14], [14], [-26], [-52], [12], [-16], [70], [-40], [-26], [18], [-82], [-96], [-22], [28], [-4], [-42], [88], [-58], [44], [56], [64], [16], [-44], [12], [-50], [-54], [68], [22], [0], [10], [24], [54], [76], [42], [54], [6], [-34], [54], [-30], [20], [-32], [66], [-86], [-54], [-50], [2], [-32], [92], [68], [-36], [30], [52], [-18], [-24], [-34], [-22], [104], [-68], [-86], [-78], [60], [-44], [-68], [2]]
