
# q-expansion of newform 6045.2.a.f, downloaded from the LMFDB on 18 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 = 6045
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], [-1], [-1], [-3], [1], [-3], [2], [3], [0], [1], [-7], [-9], [-10], [12], [-3], [12], [-1], [-4], [-9], [2], [-1], [-18], [15], [17], [18], [-4], [9], [2], [6], [20], [18], [12], [5], [21], [8], [-4], [11], [-12], [-6], [0], [11], [-6], [23], [12], [-16], [-4], [8], [-6], [14], [9], [-15], [-10], [-12], [-18], [0], [0], [-16], [8], [30], [-4], [12], [11], [24], [-10], [24], [8], [-4], [3], [-10], [12], [24], [-10], [14], [2], [-12], [-12], [11], [30], [-22], [-6], [8], [0], [-16], [-1], [9], [-15], [-37], [-3], [5], [9], [-15], [11], [0], [32], [-12], [27], [30], [-34], [-16], [44], [-12], [-3], [6], [-13], [-7], [36], [-36], [48], [-19], [14], [11], [12], [26], [20], [-42], [5], [33], [30], [6], [32], [-28], [9], [-24], [-10], [-24], [-46], [6], [44], [-49], [-52], [48], [23], [38], [-6], [-40], [-24], [-4], [-21], [36], [38], [-27], [-40], [-12], [38], [15], [-19], [9], [5], [-48], [14], [-24], [2], [15], [-46], [-30], [-7], [15], [2], [39], [-6], [-51], [-40], [36], [48], [-18], [-7], [-10], [38], [-15], [48], [32], [36], [17], [-13], [24], [26], [21], [-34], [-46], [35], [42], [-22], [-6], [-36], [54], [2], [53], [50], [21], [38], [-12], [-25], [-51], [-36], [21], [50], [-28], [48], [-15], [-30], [-37], [5], [53], [45], [-9], [-16], [12], [-3], [23], [20], [-30], [26], [-24], [48], [-64], [-4], [0], [18], [48], [5], [68], [18], [-16], [21], [-1], [-51], [72], [-28], [-21], [-58], [11], [14], [-36], [-10], [-24], [-58], [-24], [60], [-66], [18], [-7], [-58], [38], [30], [-36], [-16], [45], [14], [-12], [-58], [-6], [-24], [-19], [-15], [-3], [2], [-55], [-42], [-79], [62], [57], [-4], [-34], [0], [-55], [-15], [-51], [-31], [-6], [-58], [11], [-1], [68], [-58], [29], [6], [80], [2], [12], [-3], [32], [0], [-46], [-4], [24], [74], [-36], [8], [-24], [-66], [-60], [0], [60], [-46], [60], [8], [-39], [12], [17], [86], [-48], [-61], [63], [32], [-25], [-39], [-13], [-12], [-88], [-45], [-30], [12], [68], [-72], [-61], [39], [-63], [-46], [90], [-28], [14], [66], [-61], [24], [-7], [74], [-73], [39], [54], [8], [57], [80], [48], [-52], [18], [56], [90], [50], [-34], [-1], [-9], [-21], [44], [-84], [-24], [-43], [89], [60], [-15], [86], [-43], [87], [26], [59], [-6], [-15], [24], [42], [-72], [-82], [54], [-48], [60], [20], [44], [-18], [-79], [-7], [30], [20], [-66], [-66], [95], [-52], [27], [18], [-79], [-54], [-10], [-102], [-36], [-88], [54], [-40], [-18], [44], [-22], [20], [-57], [8], [-36], [63], [8], [63], [-4], [56], [66], [5], [54], [-4], [9], [-67], [30], [-51], [-61], [98], [-78], [95], [63], [8], [45], [-81], [-19], [-94], [60], [3], [62], [90], [-21], [-30], [65], [96], [54], [74], [12], [-99], [-15], [56], [-66]]
