
# q-expansion of newform 2800.2.a.f, downloaded from the LMFDB on 25 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 = 2800
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], [1], [-1], [4], [0], [-6], [-3], [-3], [0], [9], [2], [-9], [6], [6], [-8], [-10], [1], [7], [-2], [9], [-12], [-4], [16], [-14], [6], [-20], [-11], [-9], [1], [12], [-22], [6], [-17], [-17], [-14], [-12], [12], [-2], [-16], [22], [-20], [-1], [3], [18], [-4], [-26], [22], [-26], [-19], [16], [18], [-20], [0], [15], [4], [-4], [-22], [-7], [-14], [8], [12], [-4], [-22], [-33], [17], [34], [-17], [2], [0], [3], [-10], [15], [5], [24], [-35], [34], [-39], [14], [-12], [23], [4], [0], [4], [36], [-9], [11], [12], [16], [18], [-12], [-17], [39], [4], [-26], [-36], [30], [20], [3], [-11], [-33], [2], [37], [-15], [2], [22], [44], [-15], [8], [-20], [5], [-23], [16], [45], [49], [-8], [-42], [6], [-44], [14], [14], [18], [23], [-26], [-18], [22], [-26], [-8], [-4], [43], [0], [-4], [17], [32], [38], [40], [-38], [-28], [43], [-14], [2], [-23], [7], [-20], [-34], [28], [-38], [-26], [7], [-46], [-40], [25], [2], [-12], [-23], [29], [-42], [10], [-56], [-20], [11], [-40], [26], [-3], [24], [-23], [38], [19], [-36], [45], [10], [9], [-30], [26], [-36], [33], [-18], [-24], [44], [23], [-48], [-3], [-6], [-11], [-10], [11], [34], [-47], [-30], [-18], [9], [-29], [-26], [43], [18], [5], [1], [26], [40], [18], [-26], [52], [-2], [-54], [6], [-38], [-40], [-65], [42], [13], [-28], [39], [48], [-30], [48], [47], [10], [1], [45], [61], [20], [-45], [21], [32], [30], [34], [9], [56], [-65], [-51], [-60], [-60], [27], [64], [28], [50], [35], [-25], [48], [33], [-4], [4], [-21], [72], [56], [-62], [-18], [40], [-24], [-38], [-50], [-40], [-52], [-18], [-60], [-35], [20], [54], [-6], [-36], [-57], [12], [-8], [-42], [-32], [-5], [-50], [-9], [-31], [36], [52], [52], [41], [52], [44], [-33], [-30], [-49], [8], [70], [80], [-68], [-76], [-30], [-57], [47], [-9], [-60], [-30], [-21], [-28], [81], [46], [23], [72], [-22], [-68], [6], [42], [-14], [14], [61], [-25], [57], [-2], [-23], [36], [69], [58], [-54], [6], [29], [28], [-51], [-56], [26], [-32], [-70], [21], [16], [-22], [26], [-23], [-13], [-52], [45], [50], [85], [49], [34], [77], [-74], [4], [36], [-54], [85], [16], [-60], [-10], [58], [-75], [-2], [-75], [-15], [3], [38], [19], [62], [-42], [14], [-17], [10], [-38], [19], [62], [36], [-6], [-34], [76], [-29], [72], [-24], [-73], [66], [-46], [14], [79], [75], [-13], [-80], [9], [54], [92], [-49], [24], [71], [-72], [42], [18], [30], [-45], [-7], [-13], [62], [-28], [7], [20], [-61], [34], [-55], [-16], [-9], [-94], [-5], [17], [52], [-32], [76], [5], [36], [55], [13], [76], [6], [-52], [-17], [54], [46], [104], [-20], [73], [-84], [45], [-59], [-30], [-24], [-30], [-4], [-48], [-106], [-18], [32], [12], [-36], [46], [91], [-42], [-28], [42]]
