
# q-expansion of newform 2499.2.a.m, downloaded from the LMFDB on 12 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 = 2499
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 = [[2], [-1], [-1], [0], [1], [-1], [1], [-1], [-3], [-2], [0], [-6], [1], [5], [-12], [0], [0], [2], [-8], [0], [-6], [-4], [-6], [-16], [12], [-2], [-11], [13], [-18], [7], [15], [-13], [12], [6], [18], [-16], [-1], [10], [17], [-5], [2], [14], [10], [20], [-17], [-6], [4], [-1], [27], [-26], [9], [6], [4], [-10], [22], [4], [21], [11], [-2], [-16], [-16], [-8], [-4], [0], [10], [-30], [11], [-8], [-4], [29], [8], [2], [36], [-34], [0], [18], [-36], [-6], [-3], [13], [12], [-37], [0], [-25], [-30], [12], [30], [-23], [-8], [-8], [-6], [1], [-2], [26], [4], [-9], [-8], [-3], [-40], [32], [34], [28], [-34], [-8], [-40], [41], [38], [-36], [-30], [-12], [-20], [19], [42], [-16], [35], [27], [-32], [46], [-45], [-36], [5], [-38], [-27], [15], [22], [-12], [-6], [39], [-20], [-6], [-45], [48], [-32], [17], [-18], [1], [-26], [-34], [-28], [-3], [-38], [11], [-18], [51], [-14], [35], [-44], [14], [-32], [-6], [-20], [-30], [-1], [31], [26], [-51], [47], [-29], [-54], [-50], [20], [-10], [-47], [42], [42], [39], [-40], [-58], [4], [10], [12], [9], [-39], [-1], [7], [19], [-26], [-23], [9], [23], [-19], [36], [-52], [52], [54], [-26], [22], [17], [-20], [-4], [56], [0], [59], [30], [-17], [-42], [14], [12], [-18], [6], [41], [16], [-63], [-7], [-2], [-8], [48], [48], [-3], [68], [24], [24], [16], [-64], [-65], [8], [29], [-28], [7], [-42], [29], [68], [20], [68], [10], [26], [47], [-55], [49], [-36], [-41], [8], [-16], [-32], [-36], [-12], [-70], [-31], [-15], [16], [-27], [0], [-33], [-17], [-45], [29], [2], [31], [25], [-36], [-7], [18], [22], [-16], [-10], [50], [44], [-66], [3], [-2], [-58], [-28], [-2], [-56], [25], [-53], [0], [-58], [16], [40], [-10], [-31], [71], [-40], [38], [-41], [16], [69], [1], [-56], [-14], [-6], [-8], [-10], [-8], [12], [52], [-54], [-5], [-84], [-65], [5], [-54], [-69], [-62], [-25], [-16], [4], [-11], [28], [45], [-12], [16], [73], [48], [54], [60], [-54], [28], [34], [48], [-82], [15], [27], [20], [67], [66], [-9], [62], [-6], [82], [10], [36], [-1], [-66], [-25], [-14], [76], [8], [21], [16], [-51], [40], [-42], [-62], [53], [-67], [27], [-8], [-7], [85], [2], [-81], [91], [56], [13], [-70], [-5], [69], [34], [-19], [-66], [-26], [-90], [55], [-14], [-42], [-60], [47], [-54], [60], [65], [42], [-76], [37], [-35], [37], [4], [-72], [-42], [48], [39], [54], [-32], [28], [47], [52], [79], [96], [21], [-53], [18], [22], [83], [0], [49], [-55], [-58], [86], [72], [40], [69], [-30], [55], [20], [46], [20], [-36], [-90], [-72], [-26], [-46], [-71], [-66], [-12], [38], [-42], [98], [1], [33], [70], [-10], [44], [4], [-87], [66], [81], [-50], [-93], [36], [-30], [56], [59], [-20], [-50], [-36], [-92], [-3], [-3], [-33]]
