
# q-expansion of newform 1824.2.a.f, downloaded from the LMFDB on 21 February 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 = 1824
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], [3], [-3], [-3], [0], [1], [-1], [0], [-8], [2], [-4], [-12], [1], [-9], [6], [6], [-1], [10], [10], [-11], [-6], [0], [-6], [-8], [14], [-10], [-18], [-10], [-4], [-8], [-13], [1], [9], [5], [-18], [2], [4], [4], [18], [-24], [22], [13], [-14], [18], [5], [-22], [16], [4], [21], [15], [13], [-24], [-13], [28], [3], [-22], [0], [-23], [6], [13], [16], [6], [7], [-26], [18], [18], [-12], [23], [1], [18], [-19], [4], [16], [-6], [14], [23], [17], [10], [14], [4], [-38], [0], [-4], [20], [15], [-20], [39], [-3], [27], [15], [-32], [-34], [20], [31], [-24], [-24], [26], [-14], [-5], [-8], [-3], [24], [-12], [-32], [-33], [-13], [-42], [-42], [-2], [-4], [9], [17], [-28], [37], [16], [25], [45], [21], [10], [-40], [34], [12], [20], [-41], [-42], [46], [1], [-37], [50], [45], [26], [-28], [23], [-47], [51], [-10], [-44], [2], [39], [-28], [13], [-11], [-2], [-4], [42], [-46], [18], [-7], [8], [34], [-3], [29], [-30], [-46], [-18], [-16], [30], [-43], [-6], [-52], [26], [32], [-60], [-14], [-52], [-52], [13], [-18], [-39], [-32], [-14], [-60], [-26], [-40], [-25], [-28], [45], [0], [27], [-13], [-48], [-10], [-26], [17], [-39], [-66], [10], [4], [52], [-24], [45], [-38], [-24], [-52], [12], [-19], [-14], [6], [35], [-44], [20], [-6], [40], [12], [-47], [-13], [-20], [45], [62], [46], [26], [49], [8], [-8], [4], [-40], [60], [42], [43], [-50], [28], [-20], [15], [38], [23], [-62], [46], [-2], [57], [-45], [6], [-62], [-30], [-53], [41], [1], [-1], [-52], [-22], [42], [-29], [-16], [-14], [-48], [-45], [35], [54], [-56], [43], [-34], [-49], [-68], [-6], [-41], [52], [13], [-24], [-32], [-54], [40], [22], [59], [42], [-21], [30], [54], [71], [-14], [39], [-6], [-64], [-7], [49], [-50], [8], [-23], [66], [-18], [-75], [-22], [32], [-61], [-58], [-39], [21], [19], [-60], [-35], [12], [-2], [-3], [4], [10], [10], [-22], [8], [83], [70], [-65], [-71], [46], [5], [6], [27], [-54], [36], [52], [25], [-22], [69], [-30], [28], [52], [24], [10], [-9], [-34], [-65], [21], [-4], [13], [-42], [82], [26], [38], [44], [40], [-76], [-39], [7], [66], [16], [-11], [-75], [-13], [-10], [-24], [23], [-68], [-46], [-3], [52], [-82], [-14], [24], [-70], [68], [-84], [-3], [64], [-44], [-33], [42], [-66], [40], [8], [-96], [97], [0], [89], [6], [-72], [-42], [7], [-52], [46], [62], [70], [-20], [-16], [-1], [-12], [-15], [-79], [32], [0], [11], [-49], [-68], [90], [-1], [43], [15], [-52], [-60], [-28], [-10], [-1], [84], [10], [46], [52], [-85], [6], [54], [-50], [60], [60], [10], [54], [11], [-76], [-78], [54], [-77], [-27], [36], [0], [15], [22], [-17], [62], [-13], [34], [30], [-72], [38], [10], [6], [42], [16], [47], [-92], [36], [-40], [-8], [-69], [-23], [59]]
