
# q-expansion of newform 4900.2.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 = 4900
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], [-2], [0], [0], [3], [-4], [0], [-2], [3], [9], [-8], [-5], [6], [-11], [6], [-6], [0], [10], [-5], [15], [-10], [-7], [12], [12], [8], [-18], [-10], [12], [-7], [-15], [7], [-12], [6], [10], [-21], [-1], [2], [4], [-12], [6], [24], [10], [12], [-23], [-15], [22], [-28], [14], [-18], [-14], [-21], [0], [-2], [-12], [-24], [9], [-12], [-20], [-26], [-15], [-22], [0], [-4], [-12], [-22], [-3], [5], [-2], [-3], [-26], [-24], [3], [14], [13], [-7], [0], [9], [-22], [33], [-14], [-12], [-13], [-12], [32], [4], [-12], [-9], [37], [12], [16], [42], [-12], [1], [3], [-4], [6], [-36], [18], [20], [-25], [-17], [45], [-6], [-27], [5], [2], [-42], [12], [9], [-8], [20], [7], [39], [-8], [5], [9], [-40], [6], [-6], [-36], [-38], [-14], [-6], [45], [10], [30], [-10], [18], [8], [-4], [-25], [0], [-52], [43], [48], [34], [48], [-22], [-12], [-21], [-26], [-30], [-17], [-3], [-44], [-54], [-4], [18], [-14], [9], [-50], [48], [-29], [42], [52], [33], [5], [18], [2], [0], [-36], [-3], [-8], [-18], [-45], [48], [-7], [62], [11], [-36], [9], [-2], [33], [46], [-50], [-12], [29], [30], [32], [-20], [49], [0], [55], [42], [51], [18], [-23], [-22], [41], [54], [-34], [27], [-25], [18], [33], [-54], [29], [19], [-54], [48], [18], [-38], [-4], [10], [6], [-42], [70], [-48], [-33], [-26], [43], [60], [25], [0], [-42], [16], [-47], [54], [-57], [-9], [-31], [-68], [-45], [-29], [-72], [-34], [-6], [-15], [-40], [3], [31], [52], [-28], [-45], [56], [36], [-74], [-15], [51], [0], [3], [28], [-4], [-49], [48], [24], [2], [42], [-16], [72], [22], [42], [72], [-44], [-42], [12], [-7], [-28], [-66], [-70], [-44], [-51], [28], [-16], [-18], [-80], [-33], [42], [13], [3], [-68], [52], [-28], [-31], [20], [-52], [-3], [2], [47], [24], [-66], [32], [-12], [28], [-46], [15], [-47], [-27], [-20], [-66], [-9], [60], [63], [18], [-35], [24], [46], [-12], [-6], [2], [-22], [18], [-19], [-51], [-19], [34], [3], [4], [21], [-26], [18], [54], [45], [76], [27], [40], [-66], [0], [-46], [21], [-32], [-74], [30], [7], [-3], [76], [41], [26], [3], [-69], [-46], [15], [58], [-60], [-4], [-78], [-7], [-24], [-44], [-34], [-26], [-21], [-78], [-19], [75], [-33], [-38], [-23], [66], [-42], [-14], [25], [42], [-22], [23], [-66], [36], [6], [18], [-12], [73], [-48], [-24], [-45], [-22], [46], [-42], [-95], [59], [39], [88], [-9], [54], [-20], [-11], [72], [63], [-40], [6], [26], [-78], [-51], [79], [93], [-14], [84], [-73], [-4], [1], [42], [17], [72], [-45], [26], [75], [-41], [76], [-24], [-76], [-39], [4], [9], [-29], [36], [18], [-20], [37], [6], [14], [-96], [68], [-45], [12], [65], [-77], [6], [72], [26], [12], [24], [6], [14], [0], [84], [20], [-66], [-39], [54], [28], [102]]
