
# q-expansion of newform 1452.2.a.f, downloaded from the LMFDB on 25 April 2024.

# 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 = 1452
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], [2], [0], [2], [-4], [6], [0], [8], [-8], [10], [-8], [2], [-8], [-2], [12], [-10], [12], [8], [-6], [2], [-16], [-14], [-2], [16], [4], [0], [10], [6], [10], [-12], [-2], [6], [-20], [10], [6], [16], [-12], [0], [-12], [-10], [16], [22], [-24], [-16], [-18], [-4], [24], [-14], [-24], [-4], [18], [4], [-30], [4], [-2], [-10], [-14], [8], [26], [-20], [-2], [-16], [-18], [-14], [8], [-14], [0], [14], [-6], [8], [-4], [-34], [16], [0], [-14], [18], [-18], [10], [-12], [6], [4], [-18], [-34], [12], [42], [-22], [-12], [4], [-20], [-28], [-8], [0], [4], [20], [-2], [6], [34], [18], [2], [-20], [36], [-20], [10], [34], [44], [16], [24], [30], [-14], [-2], [-10], [-32], [0], [-6], [-16], [-16], [6], [20], [30], [22], [-8], [44], [-8], [36], [34], [40], [-8], [26], [34], [40], [-32], [54], [20], [-34], [-2], [-14], [-38], [20], [-18], [-16], [-12], [48], [18], [0], [6], [28], [-8], [-48], [38], [46], [44], [48], [0], [8], [-50], [6], [38], [24], [52], [-36], [-14], [20], [38], [8], [-56], [-2], [-62], [46], [-36], [-62], [24], [42], [-28], [-18], [26], [30], [-14], [-42], [-32], [12], [-42], [20], [-16], [6], [50], [32], [58], [-60], [58], [-56], [4], [-30], [-4], [-14], [-34], [-34], [-12], [8], [-20], [-54], [-42], [26], [-28], [2], [20], [-52], [-36], [-4], [22], [30], [8], [20], [0], [14], [-50], [0], [56], [54], [-14], [-30], [-34], [-36], [28], [-34], [14], [0], [6], [24], [50], [38], [-22], [12], [-4], [-8], [-22], [52], [-60], [0], [-60], [-38], [56], [22], [-8], [16], [-8], [36], [-38], [72], [-14], [-32], [-8], [-10], [-36], [72], [10], [70], [-30], [-34], [38], [28], [42], [46], [66], [36], [30], [-18], [-58], [-56], [18], [56], [66], [-26], [-6], [72], [-12], [-2], [-74], [-20], [-68], [-72], [-72], [22], [66], [48], [-18], [56], [72], [-24], [18], [-4], [-16], [-20], [-18], [36], [-8], [54], [16], [46], [54], [-36], [-54], [36], [-60], [-50], [36], [14], [48], [6], [4], [-46], [32], [44], [12], [26], [-28], [0], [74], [-36], [78], [26], [76], [48], [24], [2], [-60], [4], [52], [72], [18], [18], [-70], [48], [2], [36], [66], [-24], [10], [58], [-22], [74], [4], [-4], [30], [-8], [70], [40], [64], [-2], [14], [42], [30], [-30], [-58], [4], [-96], [-8], [-12], [-64], [6], [-64], [96], [-24], [-60], [-6], [44], [34], [54], [-84], [-32], [-72], [-64], [-26], [14], [-36], [52], [14], [-36], [2], [42], [26], [2], [4], [-6], [-48], [-24], [-102], [-14], [-88], [-82], [-42], [12], [44], [-24], [26], [-14], [58], [-88], [0], [-82], [54], [34], [-78], [-12], [98], [-30], [-16], [-68], [68], [22], [-76], [44], [-54], [-54], [-14], [-36], [-44], [-70], [-20], [86], [14], [24], [-48], [-54], [70], [-20], [-24], [-40], [-4]]
