
# q-expansion of newform 2352.2.a.m, downloaded from the LMFDB on 21 February 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 = 2352
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], [3], [0], [-3], [-4], [0], [4], [0], [9], [1], [8], [0], [10], [6], [-3], [-3], [-10], [10], [6], [2], [1], [9], [6], [-1], [-18], [-8], [3], [14], [0], [-5], [9], [18], [-2], [18], [1], [-4], [16], [-6], [18], [-12], [8], [0], [-19], [6], [-20], [-14], [19], [27], [-4], [-24], [24], [-1], [-27], [6], [6], [21], [-11], [8], [6], [-14], [33], [-8], [-24], [-31], [9], [-20], [-7], [-12], [26], [24], [-30], [19], [8], [-8], [-18], [-6], [-4], [24], [-25], [0], [-22], [-12], [-34], [-35], [33], [12], [-1], [30], [-8], [-36], [18], [-41], [33], [-2], [12], [-3], [-18], [4], [26], [-8], [3], [-39], [-36], [34], [23], [-21], [24], [18], [11], [7], [-16], [-6], [34], [7], [-30], [34], [18], [3], [24], [14], [29], [-33], [-33], [-8], [-15], [-10], [18], [13], [-10], [-50], [-42], [7], [38], [12], [-19], [6], [-50], [-33], [0], [-2], [-3], [40], [-15], [-4], [24], [-10], [-42], [-50], [-6], [32], [-6], [-32], [24], [-8], [-6], [-8], [-6], [35], [-9], [0], [-6], [1], [39], [42], [-36], [13], [14], [-49], [-33], [27], [-58], [60], [5], [-41], [-36], [-2], [15], [16], [26], [49], [-45], [-34], [-18], [54], [3], [-34], [22], [-13], [-12], [-31], [51], [-20], [-6], [33], [-54], [-10], [62], [30], [-42], [-21], [-8], [14], [26], [-48], [33], [16], [27], [24], [58], [-46], [-9], [1], [-69], [36], [2], [-5], [0], [-18], [30], [-40], [-56], [-30], [-11], [15], [-64], [42], [36], [13], [36], [74], [16], [49], [6], [58], [-60], [35], [15], [-60], [72], [-12], [-56], [-5], [-70], [12], [60], [25], [-60], [-32], [-24], [2], [-54], [-30], [2], [69], [-51], [-22], [52], [-3], [29], [16], [9], [62], [44], [-6], [70], [-81], [-12], [4], [39], [-16], [34], [-43], [-17], [5], [16], [3], [-16], [38], [-48], [54], [37], [-12], [20], [-50], [24], [-25], [-9], [7], [-12], [-6], [-45], [-36], [60], [-4], [-9], [-80], [42], [-36], [46], [-58], [39], [7], [-63], [-2], [50], [-39], [-58], [-6], [14], [-6], [30], [60], [-32], [-48], [38], [21], [-48], [53], [-24], [64], [-46], [33], [16], [48], [-37], [-62], [22], [-18], [75], [2], [-81], [-32], [24], [10], [12], [20], [54], [41], [19], [-82], [-36], [-87], [-77], [9], [-9], [-10], [46], [6], [-15], [4], [11], [-42], [55], [-10], [42], [30], [45], [12], [-6], [-46], [24], [30], [-12], [-56], [35], [-42], [-56], [-7], [51], [64], [66], [9], [-80], [-16], [-24], [-12], [11], [66], [62], [-6], [30], [16], [-84], [-50], [-42], [-32], [-4], [4], [12], [-55], [-6], [9], [58], [-78], [23], [79], [42], [-80], [-33], [-76], [48], [4], [-6], [21], [-29], [62], [90], [-26], [81], [-1], [-21], [-9], [-26], [-70], [-75], [84], [-56], [72], [6], [51], [-106], [78], [33], [-70], [78], [27], [48], [-38], [12]]
