
# q-expansion of newform 4730.2.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 = 4730
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], [0], [-1], [3], [-1], [1], [-4], [-7], [4], [3], [-9], [-4], [-3], [1], [-2], [-14], [12], [1], [-3], [6], [-7], [-11], [-16], [4], [-2], [12], [10], [-15], [8], [-13], [8], [0], [-17], [-4], [-3], [-14], [24], [-4], [16], [21], [-21], [16], [14], [-8], [-15], [-14], [-12], [0], [0], [-6], [14], [-15], [22], [-6], [-5], [-19], [-32], [-7], [4], [-3], [19], [30], [-29], [27], [-18], [5], [-20], [2], [10], [-6], [-2], [-9], [-8], [-4], [-10], [11], [14], [26], [3], [8], [29], [19], [40], [-31], [-16], [11], [20], [-26], [36], [13], [-16], [20], [-38], [8], [-39], [40], [-36], [-18], [-2], [-30], [12], [35], [17], [-15], [1], [-21], [44], [-21], [24], [-12], [36], [25], [4], [-4], [32], [18], [11], [5], [-34], [16], [22], [-11], [-6], [24], [-12], [44], [-30], [36], [40], [34], [-15], [9], [24], [26], [-22], [-45], [20], [5], [13], [33], [43], [6], [-14], [-1], [9], [2], [-46], [-32], [-21], [-24], [14], [11], [-11], [-57], [15], [14], [-3], [-18], [26], [-10], [15], [-7], [-10], [20], [-10], [-23], [-56], [-32], [-14], [15], [36], [-53], [40], [-36], [24], [-31], [-11], [29], [56], [-45], [-43], [-20], [16], [-6], [37], [23], [-2], [14], [-43], [-50], [6], [-14], [24], [-21], [-36], [49], [-5], [-17], [60], [69], [-12], [-12], [-44], [-56], [43], [-48], [50], [3], [-42], [22], [15], [68], [-34], [-17], [4], [3], [49], [20], [-48], [-27], [70], [-35], [-42], [-10], [-48], [-40], [6], [-40], [67], [-43], [-58], [14], [55], [-58], [-56], [-4], [24], [-39], [53], [-7], [22], [-52], [0], [-40], [-19], [-3], [-32], [28], [14], [-64], [54], [-71], [-72], [26], [-44], [68], [-34], [43], [-18], [2], [-55], [46], [-26], [-7], [69], [39], [-45], [6], [-58], [50], [-30], [32], [-39], [-50], [34], [-24], [65], [-54], [-6], [-63], [-56], [-13], [-36], [63], [8], [-30], [74], [50], [64], [-42], [30], [-1], [-6], [-43], [47], [-32], [56], [69], [-42], [1], [-44], [48], [-1], [52], [23], [-29], [-81], [89], [-16], [26], [-48], [-51], [-22], [26], [66], [-9], [6], [-40], [-32], [84], [-18], [14], [-53], [32], [-87], [-2], [25], [-7], [-40], [38], [-86], [-27], [-90], [-16], [-80], [36], [67], [-78], [-26], [61], [-26], [-67], [-21], [20], [17], [60], [-67], [-68], [-84], [10], [-84], [29], [18], [1], [-4], [58], [46], [-81], [-42], [72], [38], [-18], [-8], [-45], [-4], [-5], [-16], [18], [-20], [-47], [-23], [-10], [-10], [-1], [-40], [-12], [-13], [-56], [-42], [-29], [-10], [-6], [-92], [-58], [-14], [-66], [-10], [49], [11], [40], [41], [50], [29], [73], [-24], [42], [61], [4], [-68], [-45], [-10], [36], [-66], [-68], [5], [-92], [-98], [-59], [-44], [-30], [86], [102], [-74], [34], [-49], [92], [3], [-32], [-40], [0], [-93], [-34], [25], [-22], [10], [-32], [90], [-44], [-68], [-102]]
