
# q-expansion of newform 2151.2.a.f, downloaded from the LMFDB on 25 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 = 2151
weight = 2
poly_data = [2, -7, -5, 22, 8, -10, -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, 0], 1], [[3, -5, -7, 1, 1, 0, 0], 1], [[-3, 2, 21, 7, -10, -1, 1], 1], [[3, 0, -21, -7, 10, 1, -1], 1], [[-6, 1, 22, 7, -10, -1, 1], 1], [[-6, -4, 22, 8, -10, -1, 1], 1], [[-12, 11, 51, 6, -21, -1, 2], 1]]

hecke_ring_character_values = None
aps_data = [[0, 0, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -1], [1, 1, 0, 0, 1, 0, 0], [-2, -1, -1, 0, 0, 0, 0], [-1, 0, 0, 1, 0, 0, 0], [-2, -1, -2, -1, -1, 1, 0], [-1, -1, 0, 0, 0, -1, -1], [-3, 0, 0, -1, 0, 0, -1], [-2, 1, 0, 1, -1, -1, 3], [-1, -1, 0, 0, -1, 1, 0], [2, 0, 0, -2, -1, 1, 0], [-4, 1, 2, 1, 2, -1, 2], [1, 0, 0, -1, -3, 2, 0], [-2, 2, 0, 2, 0, 0, 3], [0, -1, 0, -1, -1, 2, -2], [1, 2, 0, -1, 1, -1, -1], [0, -1, -1, -2, 0, 2, -3], [2, -2, 0, 0, 0, -2, 2], [4, 2, -1, 1, 1, -1, 1], [-2, 1, 2, 1, 2, -2, 1], [1, 0, 0, 1, -1, 0, 3], [-5, 1, 1, -1, 0, 0, 1], [-5, -2, -2, 1, -1, 0, -1], [-2, -3, 2, -1, -3, 1, -3], [-3, 0, 2, -1, 3, 1, 0], [-4, -5, -2, -3, -5, 4, -3], [-3, 1, -4, 2, 1, -1, 1], [3, 2, -1, 2, 2, -2, 0], [-3, 1, 2, 2, 1, 1, 3], [1, -4, -4, -1, 0, -2, -3], [0, 0, 2, 0, 0, 4, -1], [6, 0, -2, -2, 0, -1, 0], [0, 5, 6, 3, 4, 1, 2], [-1, -1, 2, -2, -1, 0, -1], [-4, 0, -2, 0, -3, -2, 2], [-6, 0, 2, 2, 3, -1, 3], [5, 3, 2, 2, 5, 1, 1], [-6, -2, 2, 2, -2, 0, -2], [4, -1, 2, -1, -2, 1, 4], [-4, -1, 2, 1, -1, 3, -3], [-3, -1, -4, -2, 1, -1, -3], [-4, -5, 0, -1, -5, 5, -3], [1, -2, 0, -1, -2, -2, -3], [-3, -4, 2, 1, 3, -3, 1], [-6, 5, 6, 1, 1, 0, 5], [-3, -2, 0, -5, -3, 3, -7], [-1, -2, 0, 5, 3, -2, -1], [2, 0, 0, 2, 1, -3, 1], [-5, -1, -4, 0, 1, -1, 7], [0, 4, 2, 0, 0, -1, 0], [1, 0, 0, 0, 0, 0, 0], [-10, -2, 2, 2, 1, 1, 0], [-4, 4, 4, 0, -1, -1, 1], [-3, 5, 4, 6, 4, -4, 1], [4, -5, 5, -2, -1, -1, -4], [6, 6, 2, 2, -3, -1, 4], [3, 1, -6, 4, 4, -4, -3], [4, 3, 2, -3, 1, -5, 2], [-10, 2, -2, 0, 2, -1, -2], [6, 2, 0, -4, -3, -1, -1], [9, 2, 0, -1, 1, 1, -8], [-1, 0, 0, 3, 1, 1, -5], [10, -1, 3, -2, 2, -2, -3], [-8, -2, -4, -2, -2, 6, -1], [-6, 3, -4, 1, 6, -1, 3], [4, 2, -2, 2, 6, -3, -3], [8, -5, 1, 2, -6, 0, 2], [-3, -5, 2, -2, -7, -1, -1], [-17, 3, -1, 1, 3, -3, -2], [5, 2, 2, -5, -2, 1, 4], [-4, -4, 0, -6, -9, 7, -8], [-1, 2, 2, -1, -5, 1, -1], [-9, 2, 0, -1, -2, 6, 0], [12, -3, -6, 5, 1, -4, 0], [12, 9, 4, 5, 10, -4, 4], [-4, 2, 4, 4, 5, 1, 5], [1, -1, 0, -2, 6, -2, -4], [-5, -1, 4, 0, -1, -5, -3], [-8, 1, -1, 4, -2, 0, -5], [-3, -2, -6, -9, -4, 4, -3], [-1, -1, -1, -1, -1, -3, -3], [-1, -6, -2, -3, -4, 4, -3], [-10, -5, -8, -1, 0, 2, 0], [-10, -4, -2, -2, 0, 0, 1], [-14, -1, 4, 1, 2, -8, 2], [1, 2, 0, 3, 6, -5, -1], [12, 5, 6, 1, 10, 0, 4], [7, -4, -6, -1, -6, 7, -6], [21, 1, 4, 2, 7, -6, -1], [7, -1, -6, -4, -4, -2, -4], [3, -1, -7, -1, 1, 1, 5], [6, -3, -4, 3, -3, -5, 1], [-4, -1, 4, -1, -3, -1, 9], [1, -6, -8, -9, -3, 6, -5], [3, 3, 2, 2, 6, 4, 1], [16, -3, -4, -3, -2, 2, -1], [2, 0, -4, 2, 4, -3, -7], [15, 1, -8, -4, -5, 3, -3], [-2, 4, -8, 0, -7, 5, 1], [-3, -1, 0, 4, 6, 1, 0], [-1, 4, 4, -1, -7, 2, 6], [7, -4, 1, -6, -9, 5, 0], [13, 1, 0, -4, -2, 2, -8], [1, 1, 2, 6, 3, -7, 2], [-9, 11, 3, 9, 5, -5, 8], [-10, 6, -7, 5, 6, 0, 4], [-10, -9, -8, 1, -3, 2, -4], [-4, 4, 7, 5, -1, 1, 2], [-9, -1, -4, -2, 0, 2, -3], [-12, 0, -6, 2, -1, 2, 1], [-1, -4, -5, -2, -7, 5, 0], [-7, -3, 2, 4, 8, -1, -3], [9, 9, 8, 0, -1, -2, 2], [-6, -4, 6, -6, -2, 8, -9], [10, 9, 6, 1, 1, 1, 0], [3, -8, 0, -7, -7, 3, -2], [9, 1, -7, 7, 3, -5, 7], [-1, 0, -4, 5, 7, 0, 7], [0, 5, 0, -1, 3, 1, -2], [-4, 1, 10, 5, 4, 2, 2], [-9, 4, 0, -3, 3, 3, -1], [5, 9, -2, 6, 13, -8, 7], [-11, -5, 0, 6, 4, 0, -9], [5, 9, 0, 4, 5, -7, 10], [-4, -1, -2, -7, -9, 0, 0], [-12, -8, -4, -2, -1, 3, 3], [4, -2, -15, -3, -3, -1, -6], [-19, 3, 14, 4, 2, -2, 7], [11, 1, 10, 0, 0, -2, 0], [18, 13, -4, 1, 0, 0, 11], [3, -4, -14, -1, 1, -3, 4], [-16, -4, 4, 0, -1, -5, 1], [-14, 4, 5, 3, 5, -3, 0], [2, 4, 2, -2, 4, -6, -6], [-8, -1, 8, -3, -4, -6, -4], [-1, 1, -8, 2, 1, -8, 1], [19, -8, 0, 1, -9, 2, -6], [0, 0, -12, -2, 1, -3, -3], [6, -7, -4, -11, -10, 9, -12], [1, 5, 8, 4, 3, -2, 8], [0, -2, 8, 4, 0, -7, -2], [13, 0, 0, -1, 4, -5, 5], [-2, -3, 0, -9, 1, 7, -1], [-21, 1, 4, 0, 0, 2, 5], [-5, 2, 2, 3, 1, -11, 1], [-15, 0, 9, 0, -2, -6, 4], [-6, -6, -10, -6, -10, 1, -9], [-8, -1, 4, 1, -5, 3, 8], [-13, 1, 0, -8, -11, 5, 5], [-11, -6, 3, -4, -9, 3, -3], [-10, 6, 10, 8, 4, -9, 4], [18, 6, 6, 2, 1, 5, 7], [7, -4, 3, -6, 0, 10, -14], [23, 9, -4, -2, -6, 1, -5], [-20, 3, 8, 3, 2, -2, 0], [8, -2, -10, -10, 0, 0, -17], [-1, 3, 0, 2, 0, -5, -3], [-5, -6, -5, -4, 3, -1, -6], [-4, -6, -6, -12, -5, 10, 2], [5, -4, 6, 1, 7, -1, -6], [2, 1, 12, 3, 8, -1, 5], [18, 5, 0, 1, 4, 4, -1], [4, 2, 4, -2, 0, 4, 0], [2, -9, 0, 1, -9, 4, -12], [-18, -18, -8, -4, -19, 5, -5], [-9, -9, -4, -6, -4, 7, -8], [-2, -3, -12, 3, 6, -10, -11], [-7, 3, -6, 2, -2, 6, 3], [-9, 12, 12, 9, 5, -6, 5], [10, 9, 10, 5, 7, 3, 15], [-14, -3, -4, -3, -15, -3, 3], [21, -9, 6, -8, 0, -2, -3], [1, 10, 2, 1, 7, -3, 6], [1, -9, 0, -6, 1, 5, 2], [1, -3, -4, 0, -7, 1, 9], [-2, 9, 6, 9, 10, 1, 10], [-1, -14, -8, -7, -7, -2, -10], [9, -2, 4, 11, 5, -8, 3], [3, -3, -17, -5, 0, 6, -5], [15, 10, -4, -1, 13, 2, -6], [-8, -4, 5, 3, -11, 3, 12], [2, 14, 10, 6, 16, -8, 11], [-1, 0, -14, 1, 3, -2, 4], [7, 10, 10, 3, 11, -1, 5], [-1, 13, -2, 6, 12, 0, 13], [10, -2, 4, 8, -4, -4, 7], [21, 1, 0, -2, 4, 9, 4], [-23, -1, -10, -2, -8, -4, 3], [-19, -2, 8, 7, 0, -2, 2], [-9, 5, -13, 9, -2, -4, 12], [-29, 0, 4, 3, -2, 2, 10], [30, 10, 6, 4, 8, -1, 3], [-18, -4, 10, 10, 4, -4, 12], [11, 5, 2, -10, 9, 1, -9], [-6, 4, 14, -2, -4, 11, 8], [27, 0, 6, 1, 2, 2, -15], [-25, 5, -8, 8, 3, 1, 8], [-7, 5, 6, -2, 9, 7, -9], [19, 3, 10, 2, 13, 1, 6], [-3, -5, -4, 2, -4, 4, -8], [-18, 5, 10, 5, 4, 6, 4], [15, 12, 4, 3, -3, 1, 6], [-6, 12, 5, 5, 7, -1, 7], [-19, 4, -3, 4, 5, -9, -2], [-9, -3, 2, 4, 0, -9, 6], [-13, 15, 0, 4, 9, 2, 2], [-22, 2, 0, -6, 10, 2, -3], [-2, -3, -20, -11, -2, 1, -6], [-5, 13, 14, 4, 14, 0, 9], [9, -4, -13, -14, -12, 0, -10], [-7, -5, -10, 2, 1, 2, 16], [3, 6, -12, 3, 9, -5, -4], [-7, 3, -2, 4, -5, -1, -6], [14, 18, 0, -2, 13, 1, 0], [-17, -16, -4, -5, -5, 7, 0], [2, 5, -2, 1, 9, 1, 9], [1, -14, -6, -9, -5, 7, -19], [14, 6, 10, 6, 8, -6, 8], [-5, -2, -2, -3, 10, 5, -4], [-11, 0, 24, 9, 3, -1, 8], [-19, 8, 8, -5, -7, 3, 12], [9, -7, -4, -8, 1, 2, -9], [11, -8, 20, -3, -12, -3, -5], [25, 6, -9, 2, -3, -1, -1], [2, -2, 2, -6, 7, -3, 2], [-12, 2, 2, 8, 1, -14, 4], [-8, 1, 16, 9, 5, 3, 3], [-1, 13, 4, 6, 6, -11, 13], [-3, -4, -13, 4, 3, 1, 7], [5, -2, -8, -5, 2, -2, 3], [11, 4, -14, 5, 2, -2, 2], [-10, 7, 2, 5, -5, 0, 2], [8, 0, -12, 0, 3, -4, 2], [-6, 5, -18, 1, 3, -5, 9], [22, 1, 10, 3, 0, 4, -13], [-6, -7, 3, -6, 2, 4, -17], [-11, -4, -4, -5, -10, 1, 2], [12, 6, 18, 10, 3, 3, 12], [11, 6, 14, 1, 16, -4, 7], [-24, -14, -6, -2, -13, -1, -14], [-13, 7, 16, 10, 14, -5, 16], [-7, -1, 10, -4, -10, 0, -11], [8, -3, 6, 9, 15, -5, -1], [21, -3, -6, -2, -3, 4, -7], [11, 5, -10, 2, 8, -6, -18], [-25, -1, -12, 2, -1, 1, 9], [8, -1, 14, 5, 6, 2, -4], [6, -12, -4, 0, -7, 5, -4], [13, 9, 0, -4, -6, 0, -5], [9, -1, -7, 5, 4, -10, -5], [1, -6, 2, 1, 1, 8, -8], [-5, -3, -10, -6, 6, -2, -8], [14, -4, 12, 6, -5, 1, -16], [11, -1, -10, 0, 3, -12, 6], [-18, 10, 8, 12, 13, -17, 15], [11, 2, 3, 10, -1, -5, -7], [32, 3, -2, 11, 6, -12, 7], [24, 8, 4, 0, 4, -7, -5], [16, 5, -14, 3, 4, -14, 9], [15, 6, 0, -3, 5, -2, 11], [-48, 8, 2, 4, 4, 4, 1], [-3, -8, -6, 3, -4, 0, 15], [-19, -23, -1, -9, -9, 9, -15], [-28, -3, 14, 1, 0, -2, -4], [10, -6, -10, 2, 8, -3, -14], [2, 3, -6, -5, 8, 0, 9], [-26, -7, 0, -3, 8, 4, -2], [19, -1, 0, -4, 0, 4, -7], [2, -7, 6, -11, -8, 4, -15], [4, 0, 16, 2, -1, -9, -5], [25, 0, 0, -3, 12, 1, -20], [-6, -3, -6, -1, 8, -6, 4], [9, 21, 16, 2, 11, -2, 7], [10, 13, 12, 3, 7, 7, 12], [1, -1, -6, -10, -14, 8, -16], [8, -11, -18, -3, -8, 2, -15], [15, 2, 7, 6, -9, 3, 11], [-25, 19, 9, 7, 14, -4, 11], [12, -4, -4, -10, -17, 5, -6], [-3, -15, 2, -4, 0, -10, -10], [-4, -4, -6, 14, -2, -3, 8], [30, -4, 5, -7, -16, 2, 3], [-7, 9, -6, 0, -10, 6, 9], [2, 6, -2, 6, 12, -1, 14], [16, -5, -4, -1, 9, 11, -1], [34, 1, -12, 1, -1, 1, -5], [7, 19, 0, 8, 5, -3, 3], [19, 1, 8, 2, 5, 14, -8], [-12, 4, -2, 6, -4, 2, -15], [-13, -7, 0, 4, 1, 6, -2], [-43, 2, 2, -1, 4, -2, -1], [-23, 9, -2, 14, 11, 1, 7], [11, 5, -8, 0, 1, -9, 9], [18, 6, -10, -8, -4, 6, 4], [7, 8, 14, 5, -2, 1, -1], [-8, 2, -2, -8, -14, -5, 4], [6, -3, 6, 7, 10, -2, 2], [14, -6, -19, 3, 4, -6, -9], [-5, 5, -2, -2, 4, -8, 6], [-5, 9, -13, -5, 12, -2, 5], [21, 1, 2, 2, 8, 12, -18], [2, 3, 6, 5, 1, -9, -2], [22, -12, 3, -5, -13, 5, -4], [7, 1, 6, -12, 9, 5, -6], [14, -5, -4, 3, 11, -1, -7], [11, -2, 6, -7, -1, 5, -6], [-13, 2, 0, 3, 17, -11, 9], [28, -3, 0, -5, -4, 6, 7], [-7, -4, 14, -1, -2, 4, -7], [9, 4, -4, 3, 10, -14, 13], [-13, 4, -8, -5, -5, -1, 2], [0, -7, 16, -7, 4, 4, 1], [6, -10, -2, -8, -6, 9, -9], [-11, 19, 2, 16, 21, -11, 28], [-5, -16, 4, -3, -10, 0, 10], [-5, -14, -2, -9, -18, 6, -2], [21, -7, -18, -10, -4, 0, -5], [-15, 14, 15, -2, 0, -4, 7], [7, 2, -22, -5, -3, 10, -2], [27, 7, 12, 4, -3, -10, 2], [4, -1, 11, 0, 8, -2, -2], [-24, -15, 14, 5, 3, -6, -3], [5, -7, 0, -4, 5, -4, -10], [0, 6, 8, 14, 12, -16, 9], [4, 7, 12, -9, 7, 1, 0], [-3, -9, 24, -4, -11, 2, -9], [-3, -9, 16, -2, -9, 0, -12], [7, -15, 16, -6, -15, 5, -2], [10, -17, -12, -11, -6, 14, -15], [2, -1, 8, -7, 0, 6, -3], [-29, -9, -12, -8, -11, 0, 1], [26, 7, 26, 9, 7, -3, -2], [6, 3, 18, 9, 16, -2, 11], [22, -16, -4, -12, -5, 15, -14], [-29, -2, 2, 5, -2, 6, 11], [27, -4, -12, 7, 10, -4, -11], [16, 5, 6, 1, -4, -4, -6], [2, -11, 2, -5, -3, 13, -19], [-14, -8, 0, 0, -13, 7, 18], [-10, -11, 11, -2, -3, 11, -8], [-18, -5, -10, -3, -12, -7, 18], [-4, 9, 14, 5, 12, 9, 6], [-31, 3, 2, 4, 4, -10, 9], [4, 1, 2, -9, -12, 2, -17], [4, 1, 0, 9, 6, -4, -6], [-10, -4, 22, -8, -6, 6, -1], [26, 11, -4, 7, 0, -10, 7], [-10, -4, 9, 1, 19, -3, 6], [-18, 6, -6, 4, -7, -10, 7], [-21, -16, -4, -7, -14, 14, -5], [38, -5, 5, -12, -8, -2, -8], [21, -7, 4, -2, -13, 4, 2], [-11, 21, 16, -4, 7, -1, 12], [21, 1, -16, 4, 7, -5, -5], [-14, 10, 4, 0, -1, 7, 6], [15, 8, 11, 4, 0, -6, -1], [29, -18, -2, -13, -12, 4, -15], [-2, 7, -6, -5, -8, 4, -13], [-10, -7, 5, 12, 4, 0, 16], [9, -5, -8, -2, -7, 5, 3], [9, -2, -16, 3, 5, -7, 2], [1, -21, -12, -10, -8, 6, 5], [13, 11, -10, 10, 17, -13, 11], [-20, -19, 6, -11, 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