
# q-expansion of newform 1620.2.i.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, ZZ
    R = PolynomialRing(QQ, "x")
    f = R(poly_data)
    K = NumberField(f, "a")
    betas = [K([c/ZZ(den) for c in num]) for num, den in basis_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 = 1620
weight = 2
poly_data = [1, 0, -1, 0, 1]

# The entries in the following list give a basis for the
# coefficient ring in terms of a root of the defining polynomial above.
# Each line consists of the coefficients of the numerator, and a denominator.
basis_data  = [[[1, 0, 0, 0], 1], [[0, 0, 1, 0], 1], [[0, 1, 0, 1], 1], [[0, 2, 0, -1], 1]]

hecke_ring_character_values = [[811, [1, 0, 0, 0]], [1541, [-1, 1, 0, 0]], [1297, [1, 0, 0, 0]]]
aps_data = [[0, 0, 0, 0], [0, 0, 0, 0], [-1, 1, 0, 0], [0, 1, -1, 0], [0, 0, -1, 0], [-2, 2, -2, 2], [3, 0, 0, -1], [-1, 0, 0, 2], [0, 0, 2, -2], [0, -6, 1, 0], [1, -1, -4, 4], [-1, 0, 0, 3], [-6, 6, -3, 3], [0, -5, -1, 0], [0, -3, -1, 0], [9, 0, 0, 1], [-6, 6, -1, 1], [0, 4, 0, 0], [4, -4, -6, 6], [6, 0, 0, -3], [5, 0, 0, 3], [0, 4, -6, 0], [0, -3, 7, 0], [0, 0, 0, 3], [0, 1, 1, 0], [0, -6, -7, 0], [-8, 8, -6, 6], [0, 0, 0, 2], [-1, 0, 0, 4], [-9, 9, 5, -5], [-1, 0, 0, -3], [12, -12, -1, 1], [0, -6, -6, 0], [-5, 5, -6, 6], [0, 0, 0, 0], [0, 7, -6, 0], [-5, 5, 1, -1], [17, 0, 0, -1], [-9, 9, -7, 7], [0, 0, 14, 0], [0, 0, 0, -7], [-13, 0, 0, 2], [0, 0, -11, 0], [-5, 5, 9, -9], [0, 0, 0, -8], [2, 0, 0, -8], [13, -13, 4, -4], [0, -8, -8, 0], [0, 21, -3, 0], [4, -4, 8, -8], [-9, 0, 0, 9], [12, -12, -2, 2], [0, 7, 10, 0], [-24, 0, 0, -2], [12, -12, 2, -2], [0, 0, 14, 0], [18, 0, 0, 1], [-4, 0, 0, -12], [0, 7, 7, 0], [0, 12, -10, 0], [-20, 20, 2, -2], [9, -9, 9, -9], [-7, 0, 0, -3], [6, -6, -13, 13], [0, -8, 4, 0], [0, -9, -7, 0], [0, 19, -6, 0], [10, -10, 14, -14], [27, -27, -3, 3], [0, 25, 0, 0], [0, -3, 15, 0], [-18, 0, 0, -5], [0, 16, -6, 0], [-5, 5, 11, -11], [8, 0, 0, 2], [-6, 6, 14, -14], [0, 24, -2, 0], [20, 0, 0, -10], [6, -6, -12, 12], [4, -4, 8, -8], [18, -18, 2, -2], [0, 7, 12, 0], [18, 0, 0, -9], [8, 0, 0, -2], [0, -17, -6, 0], [0, 39, 1, 0], [-12, 0, 0, 7], [0, -17, 3, 0], [0, -6, -9, 0], [-8, 8, 6, -6], [-21, 0, 0, 3], [0, -12, -5, 0], [-34, 0, 0, -2], [-6, 6, 1, -1], [-17, 17, -6, 6], [18, 0, 0, -10], [-6, 6, -12, 12], [0, 0, 0, -6], [14, 0, 0, 6], [-1, 0, 0, -2], [0, 16, 12, 0], [-6, 0, 0, -6], [21, -21, 1, -1], [0, -24, 11, 0], [13, -13, -8, 8], [17, 0, 0, 3], [0, 9, 3, 0], [30, 0, 0, 4], [-30, 30, 9, -9], [0, 7, -10, 0], [13, -13, -9, 9], [14, 0, 0, -6], [-12, 12, 0, 0], [0, 10, 0, 0], [-7, 0, 0, -4], [0, -18, -3, 0], [1, -1, -9, 9], [24, 0, 0, 14], [-6, 6, 2, -2], [0, -6, -2, 0], [-5, 5, -6, 6], [0, 28, -6, 0], [0, 6, -20, 0], [-18, 0, 0, 2], [0, 10, 0, 0], [30, 0, 0, 7], [0, -14, -2, 0], [12, 0, 0, -3], [0, -2, -18, 0], [-2, 2, -24, 24], [5, 0, 0, 18], [-21, 21, 9, -9], [-20, 20, -12, 12], [-10, 0, 0, 6], [-18, 18, -7, 7], [-11, 11, -14, 14], [-9, 0, 0, 5], [31, -31, 3, -3], [6, -6, 10, -10], [24, 0, 0, 9], [-1, 0, 0, 14], [0, 12, 5, 0], [10, -10, 4, -4], [42, 0, 0, -2], [-31, 0, 0, 12], [0, -24, -9, 0], [0, -5, -3, 0], [0, -36, -4, 0], [13, -13, -12, 12], [27, 0, 0, -13], [10, -10, -14, 14], [-12, 0, 0, -3], [-7, 0, 0, 23], [15, -15, -1, 1], [0, 22, 12, 0], [0, -12, -3, 0], [41, 0, 0, -6], [0, 24, 13, 0], [26, 0, 0, 4], [-12, 12, 2, -2], [0, -18, 20, 0], [6, 0, 0, 2], [-23, 23, 9, -9], [-24, 0, 0, -1], [-6, 6, 6, -6], [0, -51, 1, 0], [35, 0, 0, -12], [0, -47, 3, 0], [-13, 0, 0, -6], [15, -15, -7, 7], [0, 18, -17, 0], [19, -19, -6, 6], [-48, 48, 1, -1], [0, -2, -4, 0], [-50, 50, 4, -4], [30, -30, 17, -17], [0, -2, 28, 0], [-6, 0, 0, 5], [-28, 0, 0, -8], [0, 19, -22, 0], [0, 37, -11, 0], [0, 24, 10, 0], [19, -19, 15, -15], [-24, 0, 0, -22], [3, -3, 11, -11], [0, 24, 1, 0], [20, 0, 0, 6], [0, -5, 13, 0], [19, -19, -22, 22], [18, 0, 0, 22], [-28, 0, 0, -10], [0, -21, -23, 0], [-4, 0, 0, 10], [0, -18, 27, 0], [-9, 0, 0, 21], [-15, 15, -17, 17], [-11, 11, 12, -12], [0, 37, -15, 0], [0, 33, 1, 0], [-24, 0, 0, -8], [12, -12, 11, -11], [0, -11, 0, 0], [-47, 47, 9, -9], [0, 1, 12, 0], [-6, 0, 0, -6], [0, 0, 0, -4], [-31, 0, 0, -4], [24, -24, -20, 20], [0, -36, 18, 0], [-20, 20, 8, -8], [-4, 0, 0, -22], [48, -48, 1, -1], [0, 16, 24, 0], [0, 0, 12, 0], [-12, 12, -5, 5], [0, 1, -30, 0], [16, -16, 22, -22], [0, 36, 13, 0], [12, 0, 0, 26], [-12, 12, -18, 18], [-5, 5, 14, -14], [25, -25, 24, -24], [42, -42, -11, 11], [-40, 0, 0, -12], [9, -9, 3, -3], [0, 25, 26, 0], [0, 18, -12, 0], [6, 0, 0, -6], [0, -23, 13, 0], [0, 18, 10, 0], [31, -31, -5, 5], [35, 0, 0, 12], [-26, 26, 28, -28], [6, -6, 32, -32], [0, 16, 12, 0], [0, -18, 16, 0], [13, -13, 18, -18], [3, 0, 0, -21], [-12, 12, 24, -24], [-18, 0, 0, 23], [0, 9, -31, 0], [-4, 0, 0, -6], [7, -7, 21, -21], [-40, 0, 0, 0], [36, -36, -20, 20], [0, 24, 3, 0], [-31, 0, 0, -19], [0, 0, 40, -40], [-23, 23, 20, -20], [27, 0, 0, -23], [-59, 59, -9, 9], [-6, 0, 0, 25], [21, -21, 1, -1], [0, 1, -30, 0], [0, -9, -23, 0], [-54, 0, 0, -10], [17, 0, 0, -12], [0, 40, 18, 0], [-21, 0, 0, -25], [41, 0, 0, -9], [0, 13, 13, 0], [0, -3, 15, 0], [19, -19, 12, -12], [-10, 0, 0, -2], [21, -21, 17, -17], [0, 67, -6, 0], [12, 0, 0, 20], [0, 30, 9, 0], [16, -16, 12, -12], [36, -36, -6, 6], [-23, 23, -26, 26], [-22, 0, 0, 30], [0, 34, 6, 0], [-47, 47, -6, 6], [-56, 56, 14, -14], [5, 0, 0, -27], [24, -24, 14, -14], [0, -23, -20, 0], [38, 0, 0, -16], [0, 18, 14, 0], [-33, 33, 11, -11], [-2, 2, 12, -12], [0, -30, 8, 0], [0, -38, -6, 0], [19, -19, -9, 9], [-6, 0, 0, 5], [-43, 0, 0, 1], [15, -15, 7, -7], [0, 7, 0, 0], [-18, 0, 0, 28], [0, -12, 9, 0], [0, 0, 0, 20], [-45, 45, -11, 11], [-18, 18, 35, -35], [0, 7, 33, 0], [30, -30, 11, -11], [0, -59, -2, 0], [0, 21, -15, 0], [-18, 0, 0, -3], [0, 34, -24, 0], [28, -28, -18, 18], [21, 0, 0, 1], [29, 0, 0, 4]]
