
# q-expansion of newform 1944.2.a.f, downloaded from the LMFDB on 06 August 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 = 1944
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], [0], [1], [0], [-2], [-2], [-4], [-1], [1], [-9], [2], [0], [0], [4], [-9], [-5], [10], [4], [11], [-13], [1], [4], [-16], [0], [-13], [-9], [16], [0], [10], [18], [-8], [-10], [-18], [0], [-10], [14], [-20], [-9], [9], [9], [-18], [18], [-9], [2], [18], [10], [16], [-14], [18], [-20], [-14], [-1], [-18], [14], [-10], [16], [14], [-10], [22], [20], [11], [-9], [27], [-27], [-27], [18], [5], [9], [-18], [18], [20], [27], [22], [-16], [-25], [27], [25], [8], [-18], [27], [0], [8], [5], [1], [-36], [38], [4], [-22], [-2], [-16], [0], [20], [28], [-18], [9], [9], [-9], [32], [19], [36], [-13], [-14], [28], [-2], [-45], [-37], [-2], [4], [-27], [-5], [-22], [8], [-18], [4], [16], [-2], [-9], [-13], [18], [-36], [20], [-23], [25], [-32], [-31], [18], [50], [-32], [-36], [20], [44], [37], [-18], [-10], [-28], [29], [27], [-11], [1], [-40], [-53], [-45], [-34], [14], [10], [9], [32], [-54], [-9], [-41], [0], [14], [44], [8], [-23], [-47], [52], [-34], [-18], [-27], [-18], [-36], [38], [-36], [-54], [-36], [-44], [10], [-37], [46], [-18], [0], [-1], [1], [-52], [62], [5], [-63], [-8], [14], [-28], [-52], [-18], [18], [0], [-18], [-8], [-9], [-2], [45], [1], [54], [43], [-47], [-18], [44], [7], [40], [2], [-63], [-19], [14], [2], [-14], [18], [-9], [-46], [18], [38], [-41], [-27], [-63], [-14], [-52], [44], [-31], [2], [-2], [0], [45], [20], [0], [-54], [-28], [-18], [-58], [54], [-36], [18], [16], [18], [19], [-52], [-44], [9], [29], [-25], [13], [26], [40], [72], [-55], [70], [10], [72], [29], [10], [-36], [25], [68], [-54], [-36], [-73], [22], [18], [-18], [8], [67], [63], [10], [58], [0], [-38], [-54], [10], [-49], [41], [2], [-52], [17], [22], [47], [-59], [70], [-38], [80], [-36], [-76], [55], [70], [64], [52], [16], [-4], [20], [-32], [35], [9], [-4], [68], [61], [72], [-8], [-72], [-54], [-62], [50], [54], [72], [-76], [-43], [5], [-10], [82], [36], [-37], [-18], [-70], [-1], [-46], [81], [-54], [-34], [43], [31], [26], [54], [-27], [-31], [18], [13], [-38], [59], [-62], [-2], [63], [36], [-36], [16], [9], [-68], [-27], [36], [-36], [-63], [18], [-26], [18], [2], [-54], [-68], [18], [27], [54], [79], [-50], [0], [-5], [-81], [-49], [-45], [-82], [19], [-18], [-34], [-4], [-71], [-4], [-62], [-9], [-94], [-54], [-72], [-52], [91], [63], [-81], [82], [73], [2], [-36], [80], [-65], [-92], [-28], [-36], [-27], [19], [-90], [23], [-63], [26], [90], [36], [88], [-27], [-54], [-4], [-71], [9], [49], [-74], [-36], [-9], [45], [-74], [-18], [-72], [-43], [-98], [2], [-90], [-74], [-82], [58], [-26], [-32], [-90], [-7], [0], [86], [-83], [-40], [-68], [9], [-5], [-45], [14], [72], [80], [65], [-82], [-43], [26], [82], [26], [-92], [90], [-63], [-65]]
