
# q-expansion of newform 4020.2.a.f, downloaded from the LMFDB on 18 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 = 4020
weight = 2
poly_data = [144, 360, 209, -12, -28, -1, 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, 0, 0], 1], [[0, 1, 0, 0, 0, 0], 1], [[348, 685, 180, -104, -17, 5], 24], [[-924, -1415, -204, 184, 19, -7], 48], [[492, 1055, 204, -160, -19, 7], 24], [[-708, -1103, -180, 160, 19, -7], 24]]

hecke_ring_character_values = None
aps_data = [[0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0], [-1, 0, 0, 0, 0, -1], [1, 0, 0, -1, 0, 0], [2, 0, 0, 1, 0, -1], [0, 0, -1, 0, 0, -1], [-1, 0, 1, 0, 1, 0], [1, 1, 1, 0, -1, 0], [0, 1, 0, -1, 0, 1], [3, 0, -1, 0, -1, 0], [1, 0, 0, 0, 1, 1], [1, -1, -1, 0, 0, 0], [0, 1, 0, 0, -1, 0], [2, -2, -1, 0, 0, -1], [1, 0, 0, 1, 0, 0], [2, 0, 1, 0, 0, 1], [1, 0, 0, 0, 0, 0], [-2, 0, 0, 1, 1, 1], [1, 1, 2, 0, -1, 1], [0, 2, 3, 1, 1, 2], [1, -3, -2, 0, -2, -1], [1, -1, 2, 0, 0, 1], [7, -1, -1, 1, 2, -1], [-1, -2, -1, 0, 1, -2], [1, -1, 1, 0, 2, 0], [0, 0, 0, 0, -1, 0], [3, -1, -4, 0, 0, -3], [2, 1, 2, -2, 0, 0], [5, -2, -1, 2, 0, -2], [-4, 0, 0, -1, 0, 1], [2, 0, 1, -2, 1, 3], [-3, 0, 2, -1, -1, 0], [4, 2, 1, -2, 2, 1], [4, -3, -5, 0, -1, -1], [5, -1, 1, 2, 1, 2], [7, 0, -1, 0, 0, 0], [0, 4, 2, 0, -1, 2], [5, -3, -1, -1, 1, -1], [-2, 1, 3, 0, 1, 1], [1, 3, 1, 0, 1, -2], [-3, 1, 2, 0, -3, -1], [10, 2, 1, 0, 1, 1], [1, 2, 2, 0, 2, 1], [-2, -1, 0, 0, -4, -2], [-1, 1, 4, 0, 2, 3], [9, -4, -2, 2, 1, -1], [-2, 3, 4, 0, 4, 2], [2, 2, 1, -2, -1, -1], [7, 1, 1, 2, 2, 0], [0, 2, 3, -2, -2, 5], [5, -2, -2, 4, 1, -1], [-6, 0, -2, 2, 1, 2], [6, -1, -3, 1, 0, 0], [5, 0, 2, -2, 3, 1], [5, -2, -4, -2, -1, -3], [2, -3, -3, -1, 2, 0], [6, -2, 0, 2, -1, -2], [2, -1, -1, 2, 3, 1], [8, 2, 0, -2, -1, 2], [0, 3, 4, 1, 0, 1], [3, -5, -1, 2, 2, -4], [-5, 1, 1, 4, 2, -4], [11, 3, 1, -1, -1, -1], [4, 0, -1, 3, -3, -2], [0, -1, -3, -3, -2, 4], [14, 0, -3, -1, 0, 2], [-7, 0, 1, -4, 1, 2], [5, 1, -2, 2, -4, -3], [4, 2, 4, 0, -3, 2], [5, -2, -5, 3, 0, 1], [5, 1, 4, 0, -2, 1], [15, 1, -4, -1, 0, 2], [0, -2, -6, -1, 0, -5], [5, -4, -4, 0, -1, 1], [5, -4, -2, 0, 0, -3], [1, 1, 3, -4, 2, 2], [-9, 5, 1, -2, -2, 2], [-1, 0, 3, -4, 1, 4], [-2, -6, 4, 1, 1, 1], [3, 0, 1, 2, 0, -6], [-9, 9, 1, -1, 0, 1], [7, -6, -5, 1, -2, -1], [4, 1, 6, 0, 0, 6], [0, -3, -3, -2, -1, 3], [1, 0, 1, 2, 0, -2], [13, 0, -1, -2, 0, 0], [-5, -4, 0, 0, 2, 1], [20, -2, 1, 2, 0, -1], [-8, -3, -4, -4, 2, 0], [0, -7, -2, -1, 3, -1], [-6, -5, -1, 2, -3, -1], [-19, -5, -4, -1, 0, -2], [2, -4, 1, -1, -4, -4], [-12, 7, 7, -2, -2, 3], [-4, 3, 5, 2, 3, -3], [-16, 7, 7, 0, 0, 5], [8, -5, 3, -2, 0, 3], [-5, 1, 2, -2, 7, 5], [-1, -4, -6, 2, -4, -1], [-8, 3, -5, 3, 0, -6], [-4, -3, -1, 4, -1, -7], [6, -2, 0, -2, -3, -2], [-4, -5, -3, 2, 4, -1], [2, 9, 2, -1, 1, 3], [-17, 2, 2, 0, 0, -1], [2, 3, 2, 0, 1, 2], [-8, -1, -2, -4, -5, -2], [-8, 1, -1, 0, 0, -5], [-8, -1, 2, -2, -1, 4], [2, -1, 1, 0, -2, 1], [-2, 5, -2, -3, 1, 1], [2, 7, 3, 2, -2, 7], [-2, -1, -3, 1, 1, -2], [0, 3, -1, -4, -4, 7], [11, 3, 0, 2, 0, -3], [-18, -2, 2, 2, 4, 0], [-7, 2, -4, -2, 1, -3], [-7, -5, 2, 3, -2, -4], [-4, -3, -2, 0, -6, -4], [2, 6, 3, 1, -2, -2], [-7, 2, 6, 0, -6, 1], [-5, 3, 7, 2, 1, 6], [2, 2, -3, 4, -1, 3], [-20, -5, -2, 0, 2, 4], [-1, -6, 1, 2, 4, 2], [-3, 1, 3, 3, 6, 5], [-7, -2, -2, 4, 6, 3], [9, 4, -1, 1, -4, 1], [-8, 7, 6, -1, 1, -1], [-15, -2, 2, 4, 2, 3], [7, 0, -4, 0, -3, -1], [-21, 1, 5, 5, 2, 1], [-7, -4, -1, -4, -3, -2], [-1, -3, -3, -2, -7, 0], [-19, 5, 3, -3, 2, 1], [-12, -1, -5, -2, -3, -3], [2, 7, 5, 1, 3, 4], [-6, -5, -3, -6, -1, -5], [-13, 3, 0, 3, 1, -4], [-10, -2, 4, 0, 2, 0], [8, 7, 6, -2, -2, 6], [-21, 1, -3, -2, -2, 2], [-11, 2, -4, 2, 0, -1], [-2, 3, 1, -7, -3, 6], [-2, 6, -2, -2, 4, 4], [14, -2, 1, 6, 2, 3], [-2, -8, -2, 0, -2, -8], [11, -6, 0, 2, -4, -7], [3, -2, -4, 4, 2, 1], [-2, 1, 2, -4, -3, 10], [9, 4, -4, 0, -5, -7], [-7, 5, -4, -6, 3, 7], [-10, -6, -6, -2, -2, -4], [6, 3, 5, 1, 2, -2], [-1, -3, 3, -1, 0, 7], [3, 4, 10, 4, 3, 7], [3, 1, -5, -3, -9, -1], [-11, -5, -2, 0, 1, 7], [2, -3, 1, 0, 3, -1], [-8, -1, 4, 3, 3, -1], [0, 7, 2, 2, 0, 6], [0, 0, -4, 3, 0, -3], [-5, -10, -5, -1, -2, -7], [8, -5, -6, 4, 5, -4], [4, -4, -1, -1, -1, -4], [5, -9, -10, 2, -5, -5], [-18, -3, 2, 0, 2, 6], [10, -5, -7, -2, -8, -5], [0, -6, -5, -1, 3, -4], [-3, -6, -3, 4, -3, -12], [9, 1, -2, 3, 0, 4], [-8, 2, 0, -1, -4, -9], [4, 1, -4, -3, 3, 7], [3, 1, -9, -2, -4, -6], [-24, -2, 4, 1, -3, -1], [3, -11, -6, 2, -1, -3], [14, -7, -8, 0, 3, -10], [4, -4, -9, 4, -2, -3], [7, 0, 5, -6, -2, -2], [15, -3, 1, 11, 0, -7], [3, 1, 3, 4, 2, -4], [6, -6, 0, 1, 1, 3], [-3, 1, -6, 2, -6, -3], [26, 5, 1, 2, 1, 3], [2, 0, -9, -1, 3, 4], [2, 0, -5, 4, -4, -7], [-14, -1, 0, -4, -2, 8], [-6, -5, 4, -2, 2, 0], [11, -6, 4, 0, 6, 1], [-5, -1, 5, -2, -3, 2], [13, 6, 2, -1, 9, 8], [5, -6, -3, 8, -3, -8], [-5, 8, 9, -4, -4, 10], [-12, 2, 6, 5, 6, -1], [3, -2, -2, 2, 6, -3], [12, -14, 1, 1, -1, -2], [19, 1, 4, 2, 5, 7], [15, -8, -9, 6, -1, -12], [-15, -2, 5, 0, -3, 0], [3, -2, 1, -4, -2, -4], [-8, -6, 2, -1, 5, 1], [-21, -2, 4, 0, -2, 1], [13, 5, -8, -2, 2, -1], [9, -1, -6, 7, -1, -6], [-8, 2, 6, -4, -2, 6], [-5, -5, 3, 0, 6, 6], [15, -6, 0, 0, 0, 9], [-10, -5, 8, 1, 7, 5], [5, -9, 3, 0, 0, -6], [-15, 8, 5, -6, -4, -4], [11, -10, 0, -2, 5, -3], [-6, 1, -8, -6, 3, 4], [3, 4, 5, 0, 5, 0], [7, -2, 2, 4, -2, 3], [2, -11, -7, 4, 1, 5], [-8, 7, 13, -4, 8, 7], [-3, 7, -3, -4, -3, -6], [-19, 2, 5, -5, -2, 13], [22, -7, -1, 4, -2, -1], [10, -1, 4, -12, 0, 6], [38, 2, 2, 4, 0, 2], [17, 2, -2, 4, 6, 9], [-30, -4, 1, 4, -1, 5], [-11, 1, -1, -2, -4, 2], [16, 9, -4, 1, -6, -3], [17, -4, 5, 0, -1, -6], [12, -1, -5, -12, -6, -1], [-2, -3, -2, -2, -6, 6], [-2, -5, 3, 5, 1, -2], [-15, 0, -5, -1, -4, -7], [9, -2, 0, 0, -4, 1], [-9, -2, -3, 2, 0, -8], [5, 6, 4, 2, 0, 7], [18, -11, -7, -2, -1, -3], [0, -2, -5, -4, 2, -7], [5, -1, -1, -1, 1, -1], [-5, -2, -7, 5, 2, -1], [22, -9, -5, 1, 0, 4], [-19, 7, 2, -2, -6, 5], [-2, 0, 7, 7, 4, -2], [21, -9, -3, -4, -2, -2], [10, 4, -1, 2, 5, 5], [8, -1, -7, -6, 2, -3], [7, 8, 11, -4, -1, 2], [18, 2, -10, 0, -8, -8], [-11, 2, -8, 4, 6, -3], [-2, 6, 0, -5, 3, -1], [4, 1, -5, 2, -5, 5], [2, 5, 4, -2, -8, -2], [12, -3, -11, 2, -1, -11], [-29, 10, 7, -2, 5, 6], [-20, -3, 5, -2, 9, 7], [-11, 7, 4, -6, -7, 9], [26, 4, 1, -2, 1, -3], [20, -2, -8, 2, -6, -4], [3, 8, 9, 0, 10, 6], [3, -11, -4, -3, -2, -8], [-6, 1, -4, 2, -5, -2], [8, 0, 8, -4, 5, 6], [-9, -5, 8, 4, 2, 3], [11, 0, -9, 6, -4, -2], [2, -11, 3, 7, -3, 0], [13, -13, -8, 5, 4, -2], [-11, 6, -1, -8, 0, 4], [-20, 10, 1, 0, 0, -3], [-1, -9, -9, 0, 1, -10], [-8, 5, 6, -8, -3, 2], [-20, 1, 1, -2, 6, 5], [22, 9, 4, -2, 4, 6], [-10, 2, -2, 0, -5, -12], [-22, 9, -1, -1, 3, 8], [40, -11, -4, 8, 5, 0], [-2, 6, 5, 2, 7, 7], [3, -4, 0, -4, -3, 7], [-20, 1, 5, 0, -2, 9], [-20, 2, 10, 6, 3, -6], [-20, 7, 12, -2, 3, 10], [9, 0, -1, 0, -2, -4], [5, -12, -4, 2, 0, -9], [-18, 4, 12, -2, -3, 8], [7, 4, -2, 0, 1, 13], [-9, 3, -2, 0, 0, -1], [-15, 7, 4, 2, -3, 7], [41, 5, -1, 3, -4, -3], [-10, -3, -3, 8, -2, 1], [-13, -1, 4, -1, -1, -2], [-20, -7, -8, 3, -4, -15], [51, -3, 7, -6, -6, 2], [-19, 4, 5, 6, 0, -2], [-11, -1, 5, 6, -1, -2], [-9, 11, 11, 1, 2, 3], [-6, -6, 9, 11, 1, 0], [-6, 10, -1, 3, -2, 4], [3, -2, -8, 2, -9, -1], [20, -9, 4, -1, -3, 5], [-12, 2, 3, -10, -3, 11], [-15, 1, 14, 3, -7, 4], [-21, 1, 10, -2, 3, -1], [-9, -14, -1, 3, 3, 5], [0, -3, -5, -7, -11, -4], [-11, 4, 8, 0, 9, 5], [-9, -5, -9, -12, -2, 2], [1, -18, -11, 0, -5, 2], [9, 1, -13, -7, -1, -3], [-23, -1, 5, -2, 0, -8], [14, -4, -6, -4, -2, -6], [20, 10, 6, 3, 1, 11], [-18, -6, 0, -7, 3, 11], [12, -13, -4, 0, 2, -4], [1, -7, 1, 0, 9, 4], [21, 7, 0, 2, -8, 1], [-7, -2, -2, 6, 0, -7], [-5, -6, 1, -2, 3, 10], [-1, 0, 5, 4, -4, 6], [27, 0, -3, 3, 0, -3], [-25, -10, -4, -2, 3, -7], [-34, 6, -5, -4, 2, 3], [-17, -5, -5, 4, -3, -4], [-7, -13, -3, 0, 2, -6], [25, -7, 0, 6, 4, 1], [-2, -5, -2, -2, 1, 16], [9, -4, 7, 0, -7, 4], [9, 7, -9, -1, 5, 7], [9, 7, 6, -4, 3, 9], [6, 17, 11, 0, 0, -3], [15, 1, -6, -8, -10, -5], [-1, -6, 5, 6, 2, -6], [17, 5, 8, 4, 9, 11], [25, -11, -6, 6, 10, -19], [12, 7, 4, 1, 1, 17], [30, -9, -1, 1, 4, 4], [30, -3, -16, -4, -1, -8], [-9, 1, -5, -4, -2, 10], [4, -3, -16, -4, -1, -12], [-13, -17, -11, -2, -1, 4], [6, 4, -4, -1, 0, 5], [-24, 10, 3, -10, -2, 7], [11, -7, 4, -2, 2, 1], [8, 2, -3, 2, -9, 1], [19, -11, -4, 3, -2, -2], [7, 9, 2, -8, 3, 3], [-10, 6, -12, -2, -1, -12], [-4, 9, 14, 1, 0, 7], [12, 4, -6, -4, 6, 6], [-6, 7, 10, -6, -7, 4], [24, 2, 5, 0, 6, -3], [-1, 2, 2, -5, 5, -2], [16, 3, 11, 0, -2, 9], [20, -3, 1, 2, 2, -9], [-11, 0, -3, 9, 6, 7], [-21, -9, -18, 2, 0, -19], [13, 4, 1, 8, 1, -18], [-1, 7, 12, 0, -7, 5], [7, -11, -2, 8, 0, -5], [27, -11, 2, 9, 6, 4], [-1, -5, 4, -6, -4, 1], [12, -3, 6, -8, 3, -4], [-16, 8, 13, -8, 0, 5], [-3, 0, 6, -14, 4, 3], [13, -5, -1, 14, 0, -8], [2, 9, 7, 6, 9, 3], [-10, 3, -6, -5, -5, 1], [32, 6, 0, -6, -4, 8], [16, 6, 1, -5, 3, -4], [30, -16, -15, 1, -1, -12], [19, -9, -12, 2, 3, -13], [8, 9, 0, 6, -2, 0], [-17, 4, -2, 0, -8, 13], [24, -10, -14, 4, -5, -2], [13, -9, -3, 4, -9, -6], [-20, 12, -5, 2, 5, 1], [-2, -6, 12, -1, 5, 19], [-10, -2, -6, 11, 4, -1], [11, -10, -2, -5, 1, 2], [18, -7, 0, 0, 2, 6], [-23, 17, 6, 2, 7, 7], [15, -18, -12, -2, -6, -9], [20, -2, -3, 4, 2, -1], [36, -1, -7, 0, 0, -9], [-61, 21, 13, -10, -3, 12], [-8, 3, 12, -4, 2, 12], [14, 15, 14, 8, -2, 2], [-31, -4, -7, -1, -2, 7], [-22, 4, 5, -4, 0, -5], [-5, -5, -5, -1, 7, -9], [-8, 4, 5, -2, -4, -3], [5, 5, 4, 0, 4, 1], [-7, -1, 7, 14, 4, 2], [-59, -5, -1, 0, -3, -8], [15, -18, -6, -6, 3, -7], [42, 6, 2, -3, 3, 7], [14, -1, -13, -8, 2, -3], [15, -4, 0, -6, 5, -3], [-51, -2, -6, -2, 3, -1], [33, -8, -8, 0, -4, -3], [26, -5, -1, 2, 4, -3], [-8, 4, 11, -11, 10, 14], [-7, 1, -9, 2, 0, 2], [16, -8, 0, 14, 0, -16], [-11, 3, -11, -3, 6, -1], [16, 8, -1, -4, -7, -1], [0, 18, 3, 6, -5, 11], [-19, -2, 11, 4, -9, 2], [17, -5, -5, 3, 5, 1], [-25, 3, 4, -1, 2, -6], [-28, -1, -4, 2, 3, -4], [11, 12, -10, -4, -1, -7], [0, 0, -12, 12, -3, -12], [-3, 8, -1, -10, -8, 4], [33, 2, -7, 3, 2, -7], [-9, -12, -12, -2, -4, -15], [10, -5, -1, 8, 1, 9], [-52, -9, -3, -3, 6, 6], [-11, 5, 1, 8, 4, 8], [34, 0, 2, -5, -2, 9], [1, 3, 10, -10, 6, 5], [7, -2, -8, -7, -8, 4], [5, 10, 1, -2, -14, 4], [16, 7, 0, 4, 8, 4], [6, 5, -5, -8, 10, 3], [-15, 3, -6, 10, -4, -7], [-7, 7, 13, -8, -6, 18], [-43, 0, 10, 5, 4, 4], [29, -3, 1, -5, 2, 1], [1, 0, 3, -8, -1, -4], [-5, -9, 1, -10, 0, -4], [-2, 14, -1, -4, -3, -3], [8, 5, 2, -8, 6, 2]]
