
# q-expansion of newform 5445.2.a.f, downloaded from the LMFDB on 19 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 = 5445
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], [-1], [0], [2], [-6], [-7], [6], [0], [-1], [-7], [6], [8], [0], [6], [12], [-1], [-7], [-6], [-13], [11], [0], [18], [-1], [12], [-1], [-12], [-7], [12], [17], [-18], [12], [-4], [-6], [-16], [17], [-1], [6], [6], [12], [17], [-12], [5], [24], [11], [23], [29], [12], [14], [-24], [12], [-22], [-6], [6], [6], [24], [20], [11], [30], [-1], [-18], [-13], [30], [-22], [6], [29], [-13], [30], [35], [6], [-6], [-4], [-31], [8], [-6], [-30], [-7], [-18], [5], [0], [-10], [6], [5], [-19], [18], [6], [-10], [-30], [32], [42], [-24], [8], [-42], [23], [24], [-24], [0], [-25], [14], [8], [42], [-36], [0], [23], [-43], [-36], [42], [-42], [11], [32], [5], [6], [20], [8], [-30], [29], [24], [42], [-6], [-1], [29], [-48], [30], [-13], [6], [-34], [24], [8], [-22], [-19], [36], [-37], [-25], [6], [-25], [-12], [-28], [-18], [-30], [11], [42], [-13], [-12], [-7], [0], [5], [-54], [-49], [36], [5], [0], [-25], [6], [35], [12], [11], [-30], [-43], [-30], [12], [6], [23], [-18], [18], [30], [-40], [41], [14], [0], [-42], [-1], [24], [38], [-16], [-24], [-37], [6], [-28], [-25], [35], [54], [50], [-24], [36], [-24], [11], [17], [26], [66], [-61], [36], [-25], [-36], [-18], [6], [14], [11], [24], [-42], [24], [17], [-13], [29], [0], [66], [-7], [-6], [-36], [11], [-43], [-24], [35], [0], [-30], [-10], [17], [-54], [-54], [-24], [5], [-40], [-48], [53], [54], [-22], [-24], [-36], [-40], [30], [-46], [-19], [53], [72], [-25], [12], [-31], [-54], [-6], [-30], [24], [-28], [23], [11], [-42], [-6], [68], [42], [-40], [6], [47], [-48], [-24], [5], [-30], [0], [14], [-28], [18], [53], [-16], [-6], [53], [29], [36], [41], [-42], [48], [65], [60], [26], [65], [-13], [20], [-49], [47], [-36], [26], [47], [-12], [72], [-25], [18], [50], [-61], [24], [-61], [-24], [-7], [-54], [-12], [-36], [54], [42], [-10], [30], [-55], [60], [-36], [-7], [41], [-36], [-49], [-6], [-76], [-25], [-18], [5], [-30], [-79], [48], [-72], [-72], [8], [-12], [-7], [-42], [-12], [47], [60], [11], [29], [66], [-61], [42], [-31], [20], [11], [-30], [30], [5], [-84], [59], [18], [20], [6], [47], [-42], [-67], [71], [38], [-12], [-42], [-37], [78], [54], [65], [17], [66], [48], [44], [-25], [-66], [-64], [-61], [24], [12], [84], [42], [90], [-19], [-12], [12], [-66], [29], [2], [30], [56], [47], [-18], [47], [-24], [36], [-40], [47], [-48], [-12], [53], [42], [71], [-18], [-36], [41], [54], [5], [66], [8], [-94], [92], [-18], [53], [-18], [48], [-73], [90], [62], [29], [96], [-103], [30], [-94], [48], [104], [42], [-54], [20], [-73], [24], [56], [54], [26], [48], [6], [-31], [-82], [36], [84], [-73], [66], [-96], [-24], [-7], [42], [-12], [-19], [84], [48], [-48], [-43], [6]]
