
# q-expansion of newform 5415.2.a.f, downloaded from the LMFDB on 01 May 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 = 5415
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], [2], [-3], [4], [0], [0], [-6], [-3], [-5], [-8], [6], [-4], [6], [6], [9], [-7], [-2], [9], [-4], [7], [0], [-3], [10], [-15], [-8], [18], [-11], [-18], [-2], [12], [-18], [20], [9], [19], [-16], [-22], [6], [-6], [-21], [-2], [-21], [-14], [0], [5], [-17], [10], [24], [-19], [-24], [15], [-17], [15], [-24], [12], [27], [-25], [-28], [18], [-4], [0], [16], [-24], [-10], [18], [-20], [-14], [30], [14], [-6], [-24], [-4], [4], [-11], [0], [-27], [20], [-9], [13], [-9], [-17], [-15], [-26], [1], [-36], [15], [-10], [-15], [26], [12], [-3], [-20], [-33], [-4], [-6], [-18], [3], [22], [-25], [-26], [30], [18], [-9], [-31], [38], [6], [-18], [-36], [-5], [-20], [2], [18], [20], [5], [-3], [2], [24], [12], [-36], [-41], [-8], [18], [-30], [35], [-30], [41], [21], [38], [32], [11], [-24], [7], [32], [6], [41], [18], [46], [-42], [-45], [7], [27], [-22], [30], [34], [24], [14], [48], [-43], [18], [-50], [-33], [44], [12], [-8], [45], [32], [57], [-34], [3], [-30], [48], [-10], [-60], [18], [18], [-32], [50], [-29], [-6], [-9], [22], [15], [8], [-8], [6], [-37], [-42], [16], [14], [-28], [63], [46], [48], [-54], [-45], [-38], [-56], [-14], [-3], [-8], [60], [-65], [45], [-36], [-54], [50], [-46], [12], [12], [54], [-35], [-26], [-26], [0], [-60], [-61], [-6], [-21], [28], [50], [27], [14], [-36], [-24], [70], [32], [-30], [0], [54], [-65], [-44], [-3], [2], [-60], [2], [42], [24], [28], [45], [44], [25], [13], [66], [44], [60], [23], [54], [-3], [-9], [-24], [56], [-46], [-62], [-6], [48], [-16], [-12], [49], [-12], [38], [-27], [18], [10], [-78], [-33], [-25], [-8], [-12], [-16], [-68], [12], [-19], [58], [60], [19], [-51], [-39], [10], [30], [46], [-2], [-10], [-49], [28], [-52], [-18], [-2], [22], [15], [-60], [-49], [-42], [58], [32], [-12], [-4], [48], [35], [-9], [51], [-12], [6], [-45], [-8], [-27], [-8], [72], [24], [26], [32], [0], [-19], [-24], [11], [-8], [18], [-17], [0], [-52], [-48], [3], [75], [-74], [90], [7], [45], [-39], [-64], [-78], [-44], [-10], [21], [70], [-60], [-17], [-20], [46], [12], [30], [-67], [-54], [53], [42], [56], [54], [-74], [66], [-43], [-28], [82], [72], [3], [-8], [-30], [57], [-79], [-80], [-39], [12], [49], [34], [-33], [82], [-5], [-48], [-24], [9], [72], [-66], [8], [33], [-36], [96], [2], [46], [78], [52], [43], [12], [13], [30], [39], [8], [26], [69], [-75], [38], [45], [64], [87], [48], [-16], [-12], [-41], [-48], [-91], [-4], [56], [-84], [82], [-84], [-57], [20], [45], [76], [-23], [-27], [-20], [-33], [-14], [-42], [40], [-24], [66], [104], [32], [3], [-14], [36], [28], [18], [36], [5], [-76], [9], [75], [-44], [42], [-72], [27], [-8], [-78], [81], [-56], [-36], [6], [-51], [41], [39]]
