# q-expansion of newform 6050.2.a.bd, downloaded from the LMFDB on 31 May 2026.

# 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 = 6050
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], [-1], [0], [1], [0], [4], [0], [-4], [0], [-6], [-10], [-8], [-3], [1], [-9], [12], [6], [11], [1], [-6], [-8], [14], [12], [-15], [-8], [-3], [16], [-3], [-7], [-6], [-17], [0], [0], [-22], [-15], [-10], [-8], [13], [-3], [-24], [-18], [5], [-24], [-26], [18], [2], [14], [7], [3], [23], [24], [12], [17], [-6], [6], [-12], [21], [-16], [-2], [6], [-17], [-18], [-32], [24], [-8], [-18], [-4], [4], [27], [-22], [-6], [-36], [19], [-14], [2], [24], [-3], [-20], [27], [-13], [0], [35], [6], [-14], [8], [21], [27], [10], [-15], [-5], [9], [-30], [16], [18], [14], [9], [15], [-33], [4], [-7], [28], [-12], [33], [-45], [-22], [-14], [15], [12], [24], [2], [-32], [10], [6], [8], [20], [18], [25], [45], [-30], [-30], [11], [22], [36], [-3], [-40], [-6], [-1], [0], [37], [4], [-46], [21], [2], [-20], [6], [26], [-42], [-23], [18], [42], [-34], [-45], [-23], [33], [-19], [42], [16], [-18], [20], [-3], [-26], [9], [52], [21], [31], [54], [20], [6], [-2], [21], [36], [60], [-56], [-36], [-12], [9], [20], [4], [29], [-6], [42], [-19], [18], [-26], [14], [-39], [8], [-15], [52], [-34], [-53], [60], [4], [-42], [3], [-30], [-2], [64], [-13], [-18], [34], [-36], [-40], [-54], [51], [-24], [-22], [40], [30], [-39], [-45], [14], [-38], [41], [-12], [0], [-4], [69], [9], [-28], [52], [-45], [28], [12], [6], [-55], [-17], [-18], [-33], [-54], [-58], [32], [-6], [37], [51], [53], [60], [-30], [-44], [12], [40], [26], [-46], [57], [13], [9], [-61], [30], [18], [24], [36], [-70], [-35], [-61], [-24], [-24], [-47], [60], [-4], [39], [52], [-42], [39], [38], [6], [-6], [65], [1], [-6], [-8], [-53], [-33], [11], [-50], [-30], [-76], [27], [15], [7], [-36], [50], [1], [4], [-58], [22], [-56], [-12], [35], [-79], [54], [9], [74], [9], [-19], [-44], [-54], [16], [-54], [50], [-30], [-54], [-12], [-12], [6], [28], [-18], [26], [-42], [-12], [13], [-38], [48], [32], [-81], [14], [-38], [3], [71], [78], [46], [-33], [18], [51], [31], [57], [2], [24], [12], [-56], [-45], [-58], [82], [-30], [-83], [60], [53], [-70], [-83], [51], [-54], [2], [48], [-16], [-81], [56], [60], [-43], [-78], [5], [-8], [46], [54], [42], [-28], [36], [90], [-70], [28], [12], [12], [-22], [-80], [51], [-71], [35], [-54], [0], [-54], [-90], [-36], [88], [87], [63], [-6], [28], [-62], [-66], [-29], [-43], [36], [50], [-69], [9], [14], [-74], [84], [90], [-32], [15], [-92], [-51], [72], [64], [42], [38], [-81], [-28], [-32], [16], [93], [62], [42], [48], [73], [18], [76], [74], [-75], [-4], [-21], [41], [-60], [31], [42], [93], [38], [34], [51], [4], [-30], [46], [66], [-36], [2], [-14], [-18], [-66], [-101], [-12], [0], [18], [-2], [-27], [-66], [-26], [60], [-75], [42], [74], [-96]]