
# q-expansion of newform 4050.2.a.m, downloaded from the LMFDB on 27 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 = 4050
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], [0], [1], [4], [3], [8], [-3], [5], [0], [-2], [-2], [7], [10], [7], [-3], [-1], [-10], [-16], [-12], [-2], [12], [8], [-14], [8], [14], [-13], [0], [4], [-6], [19], [-15], [12], [19], [8], [4], [-13], [-4], [24], [-11], [-19], [-16], [-6], [-20], [7], [-6], [-15], [-8], [2], [8], [16], [22], [-2], [-3], [18], [11], [-12], [22], [13], [-15], [14], [3], [-18], [-18], [22], [5], [28], [-20], [-30], [10], [4], [30], [-32], [-6], [-1], [13], [10], [-2], [17], [-19], [16], [28], [-2], [28], [12], [-6], [-3], [10], [-24], [15], [6], [4], [3], [15], [39], [12], [12], [-7], [-4], [-20], [-2], [15], [-32], [-35], [-20], [4], [-12], [-36], [-12], [5], [32], [-47], [8], [-7], [20], [-26], [30], [-24], [-46], [5], [2], [-18], [11], [26], [-5], [22], [-12], [-8], [-33], [-10], [-8], [24], [26], [-23], [21], [6], [30], [16], [14], [-45], [28], [-34], [-48], [-22], [-36], [-4], [-22], [-6], [-28], [-21], [23], [-30], [18], [-39], [0], [18], [24], [18], [20], [46], [-6], [-26], [32], [15], [-54], [-32], [-20], [-57], [-37], [57], [15], [-60], [-38], [-34], [-14], [-3], [53], [-48], [16], [8], [-8], [59], [41], [30], [1], [-18], [19], [-28], [-31], [32], [36], [2], [-44], [30], [12], [-6], [-39], [27], [0], [-29], [-18], [48], [-69], [-47], [-44], [3], [30], [56], [2], [-44], [-28], [-18], [47], [42], [26], [-53], [8], [-70], [7], [-22], [-12], [-50], [39], [41], [36], [-60], [26], [62], [59], [7], [-61], [-5], [44], [21], [-56], [-25], [9], [46], [-15], [-6], [-54], [-19], [49], [-20], [14], [-54], [-21], [-39], [29], [44], [-42], [43], [-8], [-5], [61], [-15], [28], [8], [-35], [20], [-21], [62], [-8], [53], [60], [-55], [-36], [19], [-8], [39], [2], [56], [26], [-26], [-68], [-16], [34], [62], [29], [15], [-51], [14], [-23], [10], [-58], [40], [54], [43], [26], [34], [-68], [42], [18], [-12], [37], [-36], [60], [66], [-29], [-24], [-22], [39], [50], [-84], [-49], [-14], [16], [-76], [-64], [6], [-48], [24], [-15], [-6], [-16], [-55], [65], [44], [16], [6], [-43], [-58], [48], [-56], [-24], [-37], [-84], [-6], [77], [-43], [4], [11], [80], [70], [33], [42], [-62], [24], [73], [-11], [-33], [40], [10], [2], [-79], [-36], [-24], [-64], [38], [87], [11], [30], [-78], [33], [-58], [18], [-36], [33], [-22], [95], [-26], [15], [57], [73], [-30], [70], [9], [24], [3], [17], [41], [11], [6], [84], [-27], [1], [-64], [54], [42], [14], [18], [-26], [47], [22], [-60], [69], [-60], [-31], [-48], [65], [14], [-21], [-60], [68], [26], [82], [-70], [-1], [-43], [54], [-32], [-88], [53], [6], [-6], [50], [-75], [-15], [-34], [0], [-34], [-54], [60], [-65], [28], [40], [-74], [12], [-10], [-48], [-72], [-61], [-61], [9], [86], [39], [52], [-49], [24], [-8]]
