
# q-expansion of newform 5202.2.a.m, downloaded from the LMFDB on 11 October 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 = 5202
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], [1], [4], [3], [6], [0], [4], [-6], [7], [-3], [8], [-2], [-4], [2], [-13], [-9], [-10], [-6], [12], [-7], [-5], [-12], [-4], [3], [-17], [13], [-15], [10], [10], [-16], [-20], [-12], [8], [-15], [-9], [12], [6], [8], [13], [0], [10], [-14], [9], [10], [-1], [8], [17], [20], [-14], [-24], [26], [15], [15], [0], [18], [-9], [5], [22], [0], [-4], [19], [-12], [0], [-14], [-2], [-22], [30], [-17], [12], [4], [30], [-17], [-32], [-10], [2], [5], [-22], [12], [19], [3], [22], [10], [-3], [-28], [12], [-4], [-17], [-35], [5], [3], [14], [25], [23], [8], [2], [31], [-20], [-26], [18], [-26], [-14], [-23], [-18], [20], [30], [3], [-6], [4], [-5], [13], [24], [12], [10], [-25], [-16], [-38], [-4], [22], [3], [22], [34], [22], [-27], [-8], [15], [8], [-12], [-41], [-12], [6], [10], [35], [38], [-34], [-55], [51], [24], [14], [44], [0], [-30], [-39], [5], [-12], [-30], [32], [-50], [-40], [30], [18], [46], [2], [-36], [-54], [22], [16], [-8], [19], [30], [3], [-4], [13], [19], [16], [14], [-4], [22], [6], [9], [41], [4], [-14], [-38], [47], [-14], [-38], [33], [-40], [44], [-48], [-19], [50], [12], [-66], [10], [-28], [-62], [51], [30], [-46], [-20], [-40], [21], [12], [60], [-37], [-58], [30], [18], [-30], [-8], [46], [69], [-4], [14], [20], [67], [-44], [34], [13], [-15], [-16], [-27], [-36], [-19], [3], [66], [-18], [-42], [-16], [-45], [-22], [-8], [-59], [-26], [74], [-50], [8], [5], [4], [34], [33], [-26], [-58], [56], [-5], [41], [-9], [-12], [5], [-34], [-55], [20], [48], [22], [-35], [3], [-40], [36], [-26], [70], [-78], [-27], [47], [-45], [-32], [-40], [-33], [-62], [23], [-20], [-60], [40], [-28], [20], [53], [20], [22], [78], [48], [64], [-46], [-4], [10], [80], [5], [4], [15], [-17], [30], [-11], [60], [46], [8], [52], [59], [21], [33], [82], [30], [72], [-44], [83], [30], [-3], [7], [74], [48], [-10], [61], [-39], [-43], [-19], [58], [7], [-71], [20], [-24], [-42], [32], [79], [-4], [8], [38], [-47], [-60], [-62], [21], [82], [-28], [10], [42], [56], [12], [27], [48], [-24], [-16], [58], [-50], [-74], [-53], [4], [4], [47], [-66], [-6], [-22], [-21], [82], [-28], [23], [3], [7], [-68], [-56], [22], [-4], [-57], [40], [63], [39], [5], [-56], [72], [-22], [-13], [-90], [-56], [76], [-14], [-12], [-28], [94], [6], [-57], [29], [1], [-32], [70], [-80], [-30], [95], [28], [69], [-78], [-9], [58], [62], [-15], [34], [-31], [-24], [-2], [80], [37], [-2], [62], [66], [-78], [-3], [-33], [-96], [-48], [39], [55], [6], [-32], [30], [-70], [-80], [-89], [-6], [-26], [57], [-86], [-50], [-50], [-80], [86], [-87], [-73], [14], [-67], [-29], [42], [13], [14], [-36], [-15], [18], [66], [85], [-47], [-61], [3], [-12], [-86], [66]]
