
# q-expansion of newform 9800.2.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 = 9800
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], [0], [0], [-5], [-7], [3], [-2], [-8], [-5], [-10], [-4], [-6], [-2], [7], [10], [-10], [-12], [2], [0], [2], [-7], [-4], [-8], [-17], [10], [-1], [-2], [13], [12], [6], [-8], [-12], [-16], [2], [1], [-2], [-8], [11], [-9], [12], [10], [-17], [0], [-18], [10], [-19], [11], [-1], [-10], [-8], [25], [20], [-20], [14], [22], [0], [-20], [28], [11], [-31], [-5], [-7], [-30], [5], [-26], [28], [16], [-28], [-26], [-21], [-16], [7], [0], [-16], [0], [33], [-13], [-3], [14], [-26], [29], [-21], [-2], [-28], [-10], [15], [6], [-12], [-32], [-35], [12], [-20], [-27], [-17], [9], [-8], [-14], [0], [15], [-28], [12], [-4], [18], [-12], [33], [12], [-11], [39], [10], [29], [-22], [12], [36], [19], [-26], [39], [12], [6], [23], [4], [32], [-11], [12], [36], [-5], [-21], [0], [-32], [5], [-25], [-6], [-23], [24], [-48], [34], [49], [-45], [-1], [13], [40], [37], [14], [24], [2], [-4], [-38], [26], [-44], [-6], [-6], [-32], [36], [48], [-28], [-48], [-7], [-6], [3], [-44], [2], [42], [50], [6], [42], [-3], [-12], [39], [-39], [-14], [-20], [34], [-13], [24], [-38], [-36], [-13], [33], [-52], [16], [40], [-56], [-26], [-6], [-12], [-56], [22], [-29], [-27], [0], [6], [-66], [-15], [-18], [-48], [-63], [-37], [-34], [-33], [43], [-18], [-58], [38], [-2], [30], [-25], [36], [-48], [-54], [40], [-50], [4], [-2], [-33], [-34], [-68], [-4], [-6], [-22], [-36], [-6], [-46], [-39], [54], [-57], [-10], [-21], [-1], [19], [16], [-40], [30], [-8], [41], [35], [27], [58], [44], [-63], [0], [68], [-4], [39], [38], [5], [-6], [72], [-2], [-47], [-54], [50], [22], [6], [44], [6], [3], [-42], [-44], [-3], [-27], [-42], [-60], [10], [-39], [-18], [0], [-19], [-6], [-30], [-18], [-22], [-74], [-21], [80], [-14], [-57], [-16], [14], [53], [-2], [-85], [45], [56], [28], [55], [27], [-46], [-22], [50], [2], [-15], [17], [-22], [-26], [46], [74], [48], [-63], [-8], [55], [31], [56], [-25], [36], [-19], [8], [-26], [-4], [4], [22], [-44], [-61], [-78], [-54], [10], [-64], [14], [-9], [-65], [-15], [-70], [14], [-72], [-46], [-22], [74], [73], [52], [10], [-54], [-50], [42], [-52], [-7], [4], [-9], [-87], [-6], [-82], [-33], [36], [-26], [-24], [15], [-12], [-84], [-80], [32], [56], [-17], [-30], [-88], [-43], [35], [-43], [-66], [-90], [64], [62], [-56], [84], [-82], [-10], [27], [68], [90], [-1], [-58], [65], [7], [0], [78], [-21], [-34], [42], [-36], [75], [-7], [64], [1], [18], [-100], [22], [52], [-22], [31], [-77], [73], [-16], [-7], [-55], [87], [-27], [88], [-65], [96], [-8], [94], [79], [86], [-72], [-18], [20], [79], [4], [-42], [-44], [39], [-4], [-8], [-17], [-98], [24], [52], [4], [0], [3], [24], [2], [-89], [37], [-22], [62], [56], [57], [15], [-42], [42], [70], [80], [32], [20], [-4], [9], [29], [8], [-29], [7], [-6], [28], [-35], [91], [-34], [-3], [72], [22], [5], [-96], [-96], [-16], [-20], [24], [-24], [70], [-86], [-71], [-55], [-12], [67], [42], [-77], [83], [-15], [-93], [-3], [-22], [56], [-11], [66], [44], [28], [48], [6], [25]]
