
# q-expansion of newform 3675.2.a.m, downloaded from the LMFDB on 24 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 = 3675
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 = [[1], [1], [0], [0], [0], [-3], [2], [1], [2], [-8], [-8], [7], [0], [-8], [-10], [-14], [10], [7], [-5], [-12], [-11], [-7], [14], [-6], [9], [-16], [-13], [6], [-15], [-6], [11], [14], [-10], [3], [-4], [-1], [7], [-9], [2], [18], [-12], [22], [-8], [6], [12], [11], [-5], [-15], [20], [13], [24], [-28], [21], [0], [0], [16], [-16], [-12], [5], [6], [-19], [-16], [-12], [-30], [14], [14], [21], [-2], [10], [-22], [24], [30], [-8], [11], [35], [10], [-36], [14], [-12], [5], [24], [-35], [-2], [-2], [-11], [6], [6], [-3], [-2], [33], [-2], [2], [8], [-6], [-19], [40], [-8], [34], [-16], [-3], [-12], [-4], [28], [12], [-3], [-2], [-6], [42], [26], [-45], [-43], [6], [-24], [-12], [45], [24], [-29], [12], [44], [-24], [11], [-1], [-14], [0], [-15], [34], [1], [-4], [43], [-31], [-49], [6], [23], [-43], [-6], [-10], [48], [-17], [30], [-6], [3], [-48], [5], [-24], [-15], [-42], [-6], [18], [-52], [6], [23], [12], [31], [58], [1], [54], [24], [-44], [22], [42], [-22], [-20], [-23], [8], [-40], [-22], [0], [-37], [-6], [-28], [8], [-61], [10], [-35], [21], [44], [-28], [-36], [29], [-38], [-64], [-38], [30], [22], [-22], [-26], [-19], [-4], [-22], [40], [29], [44], [29], [14], [-36], [-12], [-45], [-26], [50], [-34], [-18], [49], [-50], [10], [38], [8], [-9], [-22], [40], [-15], [43], [22], [-35], [64], [-48], [35], [-43], [-18], [-20], [26], [10], [17], [6], [64], [-66], [-37], [-14], [10], [8], [-42], [-31], [-52], [-8], [-20], [-9], [14], [45], [2], [-6], [72], [62], [-36], [-35], [7], [-28], [62], [40], [-6], [57], [-40], [-19], [60], [-24], [-35], [-42], [10], [-42], [49], [-52], [-50], [19], [-52], [49], [-34], [-32], [19], [36], [-34], [-44], [-10], [-21], [44], [-23], [71], [-75], [-23], [10], [-46], [-53], [-34], [-48], [25], [6], [-1], [-17], [20], [45], [18], [-5], [-76], [-48], [18], [-58], [18], [11], [-66], [-56], [-38], [36], [-19], [-22], [-84], [-71], [24], [27], [9], [34], [41], [14], [-57], [-36], [24], [-84], [-75], [38], [-26], [32], [42], [61], [-6], [-25], [70], [54], [-29], [-82], [65], [-36], [-71], [-18], [24], [-83], [38], [-56], [-78], [56], [-72], [-25], [-24], [-70], [31], [50], [72], [-74], [73], [-54], [-26], [1], [67], [28], [12], [43], [-62], [6], [11], [-77], [-66], [-36], [54], [26], [34], [3], [-18], [72], [42], [-1], [-17], [42], [-29], [19], [60], [76], [-6], [-34], [-3], [-10], [-24], [36], [67], [-4], [77], [-90], [-50], [64], [-18], [40], [-24], [-1], [-86], [45], [36], [89], [34], [-18], [43], [74], [-47], [21], [-100], [15], [-64], [26], [-54], [-41], [-16], [-42], [43], [93], [32], [5], [-102], [-89], [32], [18], [7], [38], [68], [34], [1], [10], [-18], [-24], [26], [-28], [-44], [58], [-54], [56], [12], [25], [-48]]
