
# q-expansion of newform 2940.2.a.m, downloaded from the LMFDB on 20 April 2024.

# 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 = 2940
weight = 2
poly_data = [-7, 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, 0], [-1, 0], [0, 0], [1, 1], [0, -1], [-1, -1], [-3, 2], [-1, -1], [5, -1], [-1, -2], [-2, -1], [3, 3], [2, 3], [-6, 0], [-2, -2], [-3, 3], [-8, 0], [-2, -1], [11, -1], [0, 5], [-3, -2], [3, -3], [1, -5], [-8, 0], [-3, -3], [-12, -1], [-9, 3], [-13, 0], [6, 0], [-10, -3], [-16, 2], [-7, 5], [-5, 0], [4, 4], [6, 4], [-4, -2], [-16, 0], [-17, 1], [6, 6], [-10, -4], [-1, -2], [-2, -8], [-16, -3], [3, 3], [2, -8], [-2, -4], [-2, 6], [-17, 1], [7, 0], [16, -2], [6, 0], [18, 2], [3, -3], [13, 7], [-2, 10], [-12, 6], [-18, -4], [-18, 1], [24, 0], [12, -1], [0, 0], [-20, 3], [-11, 1], [8, -5], [5, -7], [1, 8], [-20, 5], [-8, -8], [10, 0], [7, -11], [3, 9], [6, 5], [8, 3], [-21, -2], [22, -2], [-19, -7], [14, 1], [14, 2], [15, 8], [-16, 2], [-9, 4], [24, -6], [-6, -1], [0, 2], [-6, -6], [-2, 4], [-12, 1], [17, -1], [-8, -7], [6, -12], [-4, -4], [0, 1], [18, 6], [1, -10], [4, 10], [12, -6], [18, 6], [-18, 5], [5, -12], [-38, 2], [-18, 0], [26, -4], [-17, -5], [39, -2], [24, -7], [-9, 3], [33, 3], [18, 6], [27, -4], [30, 5], [-20, 8], [4, -2], [19, 0], [-4, 12], [17, 11], [-2, -9], [-15, 3], [31, 1], [-24, 0], [25, -6], [-8, -1], [25, 1], [-3, -3], [-13, 4], [27, -3], [-12, 4], [20, 2], [10, -9], [6, 5], [29, 0], [-21, -3], [3, -8], [-30, -8], [-17, -11], [9, 2], [7, 13], [10, 6], [3, 3], [-2, -8], [-12, 2], [-7, 17], [-8, -10], [-4, -10], [-25, -2], [-19, -1], [-30, 5], [11, -1], [0, -10], [-8, 10], [14, 0], [20, 8], [2, 9], [17, -1], [-18, 1], [13, 7], [21, 4], [-17, 7], [6, 17], [3, -3], [-43, -1], [-26, 10], [6, 13], [-32, -8], [-33, -3], [15, 15], [39, -2], [-30, 5], [-9, -14], [21, -15], [-10, 14], [6, 14], [5, -13], [-12, -11], [-11, 12], [-12, -12], [3, -14], [0, -6], [20, -8], [13, 12], [22, -1], [9, 3], [12, 1], [-14, 10], [18, -12], [-42, -6], [-16, 18], [-10, -5], [-25, 12], [-15, -3], [-14, -3], [-29, 1], [16, 8], [-60, 0], [-13, 11], [39, 3], [-1, 24], [32, -12], [40, -2], [15, -9], [0, 18], [21, 2], [16, -6], [0, -16], [3, 15], [-16, -10], [-36, -4], [-6, 0], [26, 2], [4, -18], [-28, 3], [-51, 3], [54, -2], [-44, 4], [-7, -1], [27, -4], [0, -8], [-24, 0], [-2, 10], [-6, 12], [29, -6], [5, -2], [-27, -15], [-24, 16], [-16, 14], [-8, -4], [0, -12], [-12, -6], [-18, 11], [-35, 1], [18, 13], [12, 8], [27, -8], [35, -13], [14, 1], [-7, -1], [5, -14], [39, 9], [20, -4], [-1, -13], [-27, -9], [-3, -10], [-34, 7], [2, -18], [-54, 6], [-18, -18], [44, 1], [-5, 19], [44, -6], [-20, -20], [24, 1], [-52, 8], [22, 4], [53, -8], [-14, 10], [23, 5], [-3, -26], [-54, -7], [-25, -19], [-14, -12], [-30, 13], [6, -18], [-53, 0], [-50, 3], [24, 18], [21, -4], [-13, 11], [6, 12], [56, 6], [33, 3], [15, 20], [24, 10], [28, 9], [35, 6], [-12, -22], [-30, 2], [11, 5], [-25, -12], [27, -2], [-8, -20], [10, -14], [-49, 0], [-6, -12], [-37, -8], [-26, 6], [-37, -1], [2, 9], [59, -7], [-17, 12], [11, -1], [-10, -10], [-26, -2], [39, 9], [-29, 1], [16, -7], [6, -24], [35, 16], [1, 1], [43, 7], [-14, -9], [30, 20], [8, -22], [4, 14], [-56, 4], [25, 14], [-30, 7], [-42, -12], [-19, -14], [54, 6], [28, -13], [21, -15], [-7, 29], [-27, -3], [-10, -6], [10, -2], [-23, 6], [57, -3], [-33, -3], [22, 9], [-33, 3], [-21, -10], [6, -5], [-40, 2], [20, -15], [10, 16], [71, -2], [69, -2], [0, -7], [-36, -12], [-52, 2], [3, 22], [77, -1], [37, 0], [39, -3], [-25, -12], [-4, 2], [24, -8], [57, 9], [-39, -4], [34, -3], [16, -1], [-7, -7], [-38, 10], [-45, -8], [3, 9], [2, -4], [11, 10], [40, 8], [33, -3], [-28, 2], [62, -8], [54, 4], [-48, -6], [-8, 0], [37, 20], [-26, 4], [42, 12], [-42, 6], [0, 12], [-53, 13], [-16, 12], [21, -3], [36, 18], [-18, 24], [46, -9], [-20, 20], [41, 17], [-40, -3], [-20, 8], [-66, -6], [39, -4], [87, -3], [15, -9], [5, -14], [24, 25], [39, -3], [-12, 12], [20, 10], [7, -11], [-50, 15], [8, 20], [-35, -23], [32, -24], [28, -8], [12, 2], [18, -18], [-35, -4], [-42, 20], [68, -3], [-23, 7], [-27, -8], [27, -3], [3, -15], [4, -3], [-61, 11], [8, -24], [-24, -28], [48, 6], [-15, -26], [-6, 24], [-14, -18], [-33, -21], [-16, 6], [-31, -1], [-23, -11], [35, 10], [36, -2], [14, -22], [-8, 21], [44, -16], [6, 8], [78, 6], [45, 21], [20, -24], [6, -17], [-5, 19], [77, -7], [0, -7], [40, -20], [18, 18], [10, -2], [-20, 27], [-33, 3], [4, -26], [12, -10], [23, -31], [-58, 8], [-12, 24], [24, 20], [41, 11]]
