
# q-expansion of newform 2340.2.a.f, downloaded from the LMFDB on 02 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 = 2340
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], [0], [1], [-1], [-3], [1], [-3], [2], [-3], [-6], [2], [-7], [-9], [8], [6], [-3], [0], [-7], [-4], [-3], [-10], [-1], [0], [-3], [-1], [12], [8], [-3], [-4], [-6], [-10], [-18], [18], [17], [-3], [-4], [-22], [-13], [12], [6], [6], [-19], [-24], [5], [12], [20], [-4], [-16], [30], [-10], [-9], [-27], [-22], [-12], [18], [0], [12], [14], [2], [6], [-16], [12], [11], [-12], [26], [0], [8], [32], [-15], [20], [-6], [24], [8], [-4], [2], [6], [-12], [-7], [-18], [26], [6], [32], [0], [32], [-19], [21], [-9], [-13], [15], [23], [-3], [39], [29], [-12], [14], [12], [-15], [-30], [-28], [-46], [-4], [42], [9], [36], [-1], [35], [18], [24], [0], [-13], [32], [-1], [-42], [-10], [20], [12], [-49], [15], [-6], [24], [44], [-10], [-3], [-12], [-34], [-12], [-28], [36], [26], [-19], [2], [6], [-31], [8], [-30], [-40], [0], [-28], [39], [30], [-16], [3], [32], [-30], [-34], [45], [23], [-57], [-43], [-18], [-10], [-30], [-16], [-21], [-10], [-24], [-25], [-27], [14], [21], [24], [21], [32], [-48], [-24], [60], [-19], [-28], [38], [-15], [12], [-58], [-60], [59], [41], [12], [50], [45], [-52], [-10], [59], [36], [14], [6], [-30], [-12], [8], [-31], [-4], [-45], [-52], [-30], [-61], [3], [-12], [-63], [-22], [-10], [-24], [21], [-30], [35], [-13], [-67], [3], [-45], [38], [-60], [-27], [29], [-40], [-66], [-28], [-54], [-12], [8], [-64], [-6], [-18], [-42], [23], [14], [18], [-4], [-63], [-55], [63], [6], [2], [57], [-16], [-55], [20], [60], [-22], [-6], [-4], [54], [-6], [-12], [-24], [41], [20], [44], [24], [72], [-16], [-39], [-4], [-72], [-22], [42], [18], [-55], [-21], [-45], [-10], [-73], [54], [47], [26], [15], [-16], [20], [30], [-49], [-51], [69], [5], [18], [-58], [-67], [35], [80], [-34], [-55], [-48], [56], [20], [54], [-27], [68], [72], [50], [80], [36], [-16], [-12], [14], [24], [-60], [-12], [-12], [-36], [62], [-30], [80], [-57], [-72], [-73], [14], [-60], [71], [39], [-16], [29], [-81], [11], [48], [-16], [-27], [-54], [-78], [68], [-48], [5], [3], [57], [26], [-36], [-52], [14], [54], [35], [6], [-49], [74], [35], [33], [78], [74], [-21], [-16], [-24], [-16], [42], [50], [24], [74], [2], [71], [-33], [-27], [68], [-30], [-12], [29], [59], [-48], [-9], [-70], [29], [-69], [26], [71], [54], [-51], [-36], [18], [6], [26], [84], [-60], [36], [2], [44], [-96], [83], [-43], [-18], [44], [-60], [-54], [-67], [-52], [15], [-6], [11], [12], [26], [6], [90], [-16], [-66], [-40], [-84], [-10], [80], [-88], [93], [-16], [-66], [-87], [-76], [21], [-46], [-10], [-90], [29], [78], [38], [-9], [-67], [-60], [-81], [53], [62], [18], [-7], [63], [-46], [3], [69], [29], [-34], [-30], [39], [-82], [-48], [87], [18], [29], [12], [96], [-82], [-60], [-39], [99], [-34], [-6]]
