
# q-expansion of newform 5700.2.a.m, downloaded from the LMFDB on 21 September 2025.

# 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 = 5700
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], [-1], [-4], [0], [-6], [-1], [4], [9], [4], [2], [-5], [4], [6], [-7], [-1], [7], [6], [5], [7], [8], [-6], [15], [8], [18], [-4], [-3], [18], [2], [14], [-18], [2], [21], [8], [-10], [13], [-25], [-3], [-9], [9], [-20], [4], [16], [-24], [-7], [10], [-10], [5], [6], [10], [-2], [8], [-2], [-3], [-16], [26], [19], [-17], [26], [-28], [2], [-8], [4], [6], [19], [22], [2], [-28], [1], [-28], [14], [4], [4], [-2], [-3], [16], [22], [-34], [36], [2], [10], [-17], [26], [8], [-34], [13], [-35], [6], [-28], [22], [-6], [8], [-6], [29], [-24], [27], [7], [32], [38], [18], [-32], [-5], [1], [-5], [18], [14], [4], [-24], [8], [42], [-23], [-14], [-11], [-8], [-26], [41], [12], [24], [-12], [38], [40], [-51], [21], [28], [-24], [-21], [-12], [-17], [49], [-25], [-29], [-38], [-19], [12], [-33], [-17], [38], [43], [6], [-50], [-24], [37], [-12], [-38], [9], [-37], [-57], [25], [33], [-24], [-20], [-7], [-56], [42], [-17], [3], [28], [-3], [10], [-48], [21], [-35], [-17], [-2], [12], [-2], [46], [16], [44], [3], [-38], [2], [18], [-16], [58], [-41], [40], [-30], [19], [-24], [-29], [12], [-11], [0], [0], [-14], [32], [8], [-36], [44], [-44], [-24], [54], [42], [14], [-3], [5], [-6], [30], [-14], [-36], [-2], [46], [-54], [-16], [64], [-61], [24], [-32], [-13], [0], [39], [39], [-9], [40], [-27], [-34], [-12], [44], [12], [-6], [30], [11], [-20], [-61], [73], [-13], [-62], [40], [29], [-10], [0], [-57], [-43], [-74], [-5], [24], [12], [7], [-39], [13], [28], [48], [-54], [-52], [5], [-24], [40], [-38], [37], [42], [8], [-22], [72], [-38], [-73], [-26], [5], [-14], [24], [3], [17], [-20], [14], [-2], [-23], [-22], [-66], [48], [-64], [76], [-74], [71], [40], [8], [6], [-30], [18], [-28], [-9], [47], [46], [72], [1], [38], [-14], [74], [-39], [9], [-34], [-78], [-15], [-69], [22], [-24], [-20], [40], [4], [4], [-23], [18], [-52], [9], [36], [22], [76], [20], [-36], [38], [-24], [10], [75], [30], [-4], [-42], [36], [63], [-70], [56], [60], [37], [9], [16], [-12], [-36], [74], [-52], [-80], [-88], [-46], [29], [69], [40], [-77], [24], [-22], [-61], [37], [-53], [16], [70], [-7], [-58], [21], [37], [17], [6], [-80], [-48], [-30], [-26], [-34], [-58], [40], [58], [12], [28], [20], [68], [-43], [-66], [-5], [12], [-20], [10], [8], [-68], [-90], [-80], [86], [96], [-3], [-8], [50], [9], [18], [-13], [70], [-10], [9], [-6], [-1], [38], [-56], [48], [28], [11], [3], [-23], [64], [46], [80], [-37], [45], [18], [-10], [9], [-38], [52], [58], [-10], [82], [-35], [39], [84], [-38], [-27], [82], [66], [-88], [26], [-44], [85], [-65], [-30], [-81], [38], [-2], [-87], [-99], [56], [76], [-33], [56], [87], [-73], [18], [105], [0]]
