
# q-expansion of newform 2001.2.a.m, downloaded from the LMFDB on 19 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 = 2001
weight = 2
poly_data = [-64, -128, 352, 604, -717, -939, 750, 647, -420, -216, 124, 34, -18, -2, 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, 0, 0, 0, 0, 0, 0, 0], 1], [[0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], 1], [[-3, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], 1], [[-128, 256, 604, -701, -939, 750, 647, -420, -216, 124, 34, -18, -2, 1], 16], [[112, -336, -604, 717, 939, -750, -647, 420, 216, -124, -34, 18, 2, -1], 16], [[8496, 10560, -37648, -30667, 63135, 25206, -46019, -7004, 15640, 108, -2450, 210, 142, -21], 368], [[-2124, -800, 12241, 3544, -21091, -1989, 15225, -664, -5152, 686, 808, -156, -47, 11], 92], [[-1604, -2620, 7661, 9877, -9660, -12843, 4986, 7444, -1196, -2090, 130, 278, -5, -14], 92], [[48, 3144, 5168, -16273, -19251, 25826, 19119, -17452, -7728, 5636, 1362, -858, -86, 49], 184], [[-6544, -3040, 49860, 2763, -97451, 15014, 74967, -19396, -26680, 8412, 4386, -1522, -266, 97], 368], [[-1266, -31, 7283, -1282, -12121, 3486, 8530, -2912, -2852, 1046, 446, -169, -26, 10], 46], [[-544, 961, 4020, -4510, -8027, 6727, 6257, -4419, -2231, 1407, 365, -212, -22, 12], 46], [[5088, 18072, -17164, -77057, 4761, 106310, 12435, -65388, -8832, 19748, 2002, -2858, -146, 157], 368], [[-6144, -18608, 8992, 85233, 21091, -119906, -33175, 74340, 15640, -22484, -2986, 3242, 198, -177], 368]]

hecke_ring_character_values = None
aps_data = [[0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0], [0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, -1, 0, -1], [-1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0], [-1, 1, 0, -1, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1], [0, 1, -1, 1, 0, 0, 0, 1, 1, 0, -1, -1, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-3, 0, 0, -1, 0, 0, 0, -1, 0, -1, 1, 0, 0, 0], [-1, -1, -1, 0, -2, 0, -1, 1, 0, -1, 0, 0, 0, 0], [-3, 0, -1, 0, -1, 0, 0, 0, 0, 0, 1, -1, -1, 0], [1, 0, 1, 0, -1, -1, -1, 1, 1, 0, -1, -1, 0, -1], [-2, -1, 1, 0, 0, 1, -1, 0, 0, 0, 1, -1, 1, 0], [-2, 2, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, -1, 1], [-3, 1, 1, 0, 0, 1, 1, -1, 0, 1, 1, -2, 0, 0], [-2, 0, 1, -1, 1, 2, 1, -1, 0, 0, 0, 0, 0, 0], [0, 1, -1, -1, 0, -1, 1, -2, -1, 0, -1, 1, 0, 0], [-4, -1, 2, -1, 1, 1, 1, 1, 0, 0, 1, 0, 0, -1], [0, 3, -1, -1, 0, 1, 0, -1, 1, 0, -1, 1, -1, 1], [-5, 1, 0, 1, -1, 0, 0, 0, 1, 0, -1, -1, -2, -1], [0, 3, -1, 0, 2, -1, 1, 1, 0, 1, -1, 1, 1, 1], [2, -3, 1, 0, -2, -2, -1, 3, 0, 0, -2, 2, 1, 0], [2, 3, 1, 0, 0, 1, 0, 1, 1, 1, -2, 0, 0, 2], [0, 0, 0, 0, 2, 0, 1, -1, 1, -1, -1, 1, -1, 0], [0, -2, 3, -1, -1, 2, 0, 1, 3, 0, 1, -2, -1, -1], [-2, 1, -3, 2, 0, -1, -2, 0, -1, -1, -1, -1, -1, -1], [-3, 1, 2, -2, 1, 1, 1, -4, -1, 1, 1, 0, 0, 1], [-3, 1, 0, 0, 1, 1, 0, -2, 1, 0, -1, 2, -1, 1], [0, -1, -1, 1, 0, -1, 0, 1, -1, 0, -1, -1, 1, 1], [-3, -5, 4, -2, -1, -1, 0, 1, -1, 2, 2, 0, 2, 0], [-1, 3, 1, 0, 2, -2, 0, 0, -1, 1, 0, 3, 2, 4], [1, -3, -1, 1, 0, -3, -1, -3, -4, 1, 2, -1, 1, 0], [1, -3, 1, -3, 2, -1, 2, -1, -2, -1, 3, 2, 3, 2], [-3, 2, -2, 0, 0, 2, 0, 0, 0, 0, 0, 1, 1, 1], [2, -2, 3, -2, -1, -1, -2, 0, -2, 0, 2, 0, 2, 1], [-1, 1, 0, 0, 3, 1, 1, -1, 0, -1, 2, -2, -2, 1], [-5, -4, 4, 1, -2, 2, -2, 0, 0, -1, 1, -4, -2, -4], [0, -5, 0, 3, -1, -1, -3, 0, -1, -1, 1, -3, -1, -1], [-4, -2, -2, 0, 0, 0, 2, -2, 0, 0, 1, -1, -1, -3], [4, -2, 1, 1, -1, 2, -2, -1, 1, -1, 0, 0, -2, -1], [-1, 1, -1, 1, 2, -2, 1, 1, 0, 0, 1, 1, 1, 3], [0, 0, -3, 0, 1, 0, -2, -2, 0, 0, 0, 3, -1, 4], [1, 0, 0, 2, 0, 0, 0, 1, 0, 0, 1, 0, 2, -2], [-5, 2, -4, 1, 0, 0, 0, 1, -2, 0, 0, 2, 1, 2], [-6, -2, -3, -1, -1, 0, 2, 0, 0, 1, 1, 0, 2, -1], [-1, -4, 4, 0, 2, 0, -1, -2, -1, -1, 2, 3, 1, 0], [4, -1, -2, -1, 1, -1, 2, -3, -2, 1, -2, 2, 0, 2], [2, -1, -4, -3, -1, -2, 3, -1, 0, 0, -2, 2, -2, 1], [-1, 1, 3, -2, 2, 1, 2, -3, -2, 2, 1, 3, 3, 2], [1, -6, -1, 0, -3, -2, -1, 3, -2, 1, -1, 2, 2, 2], [2, -4, -2, 2, -4, 0, -1, 2, 1, -3, 0, 0, -2, -3], [-1, 0, -2, 2, 2, -3, 0, 2, -1, 0, 1, -2, 2, 0], [3, 1, -4, 1, -3, -3, -1, 4, 0, 0, -3, -1, -1, -2], [-6, -2, -1, -1, -1, 2, 3, 0, 1, 1, -2, -1, -1, 0], [-6, 1, -1, 3, -2, 3, -1, -1, 3, 0, -2, 1, -3, -1], [7, 1, 0, 2, 3, -1, 2, 0, -1, 3, -2, 0, 2, -1], [6, 3, 2, 2, 1, -1, -1, 0, 2, 0, 2, 1, 1, 1], [-1, 8, -7, 2, 5, 0, 4, -2, 2, 0, -3, 0, -1, 4], [1, 0, 2, -3, 0, -2, 1, -1, -3, 1, 2, -2, 1, -3], [0, -4, -1, 6, 1, -1, -3, 0, -1, -1, 0, -1, 0, 2], [1, 5, 2, -3, 3, 3, 3, -1, 2, 2, 1, -1, -1, 4], [-11, 2, -2, -2, 2, 2, 2, -8, -2, 1, 0, 1, -1, 0], [-1, -2, -4, 5, -2, -2, -1, 1, -1, 2, -3, 0, 0, 1], [6, 0, -6, 2, 0, -2, 0, 4, -2, 1, -1, 3, 3, 2], [-4, -3, 4, -1, 1, 1, -1, 0, 1, -1, 0, 2, -2, 0], [-3, -2, -2, 1, 0, 1, -2, 3, -1, 2, 2, 1, 1, 2], [2, 0, 2, -2, 2, 0, 0, 2, 0, 0, 1, 1, 1, 1], [8, -1, -2, 2, -3, -3, -4, 5, 0, -2, -2, 0, -2, -2], [4, 4, -3, 0, -1, -2, 0, 1, 1, 1, -2, 0, 0, 2], [-6, 5, -4, 0, 3, -2, 2, -3, -4, 2, -1, 4, 2, 2], [10, -2, -5, 0, -3, 0, -3, 4, 1, 0, -5, 3, 2, -1], [0, 1, 2, -3, 1, 1, 2, 1, 1, -2, 1, 4, 0, -1], [-7, -4, -2, -3, 0, 1, 2, -3, 2, 1, 5, 2, 1, 3], [-2, -5, 1, -3, 0, -5, -2, -3, -8, 1, 6, 1, 3, 0], [-4, 5, 1, -3, 2, 2, 3, 0, 4, -3, 0, 2, -1, 2], [-2, 1, 0, 2, -1, 3, 2, -2, 2, 0, 0, -4, -6, -4], [0, -9, 0, 2, -3, -1, 1, 1, -2, 1, -1, 3, -2, -3], [-3, 1, 0, 0, 1, 3, 0, -2, 2, 0, -3, 0, -4, 0], [-4, 3, 4, -2, 1, -1, -1, 2, 0, 1, 0, 1, 3, 6], [-1, 0, 2, 5, -4, 1, -3, 3, 1, 1, -2, -6, 0, -6], [-7, 0, 4, -5, 0, 0, 1, 0, 0, -2, 0, 6, 0, -1], [0, -4, 2, -1, 0, 0, -2, -1, 2, -1, 4, -4, -2, -2], [-1, 2, -2, 2, -2, -4, -2, 1, 0, -1, -2, 1, -1, 2], [-3, 2, 4, -1, 4, 0, 3, -2, 2, 1, 1, 1, 1, 1], [-6, 2, 2, 1, -2, -2, -1, 4, 4, 1, 0, -3, 1, 1], [1, 0, 6, -4, -2, 2, -1, 3, 2, -2, 1, -3, 1, -3], [3, 1, 1, 0, 4, -3, 3, -3, -6, 1, 2, -1, 5, 1], [1, -3, 0, 1, 1, 3, 0, 0, 2, -4, 4, 3, -1, -2], [-2, 2, -2, -4, 0, 3, 3, 3, 4, 1, 1, -1, -1, 0], [-3, -1, 9, -5, -2, -3, 0, -1, -1, -2, 2, 3, -1, -3], [-5, 6, -3, -3, 3, 0, 6, -2, 0, 1, 2, -4, 0, -1], [-5, 17, -3, -1, 4, 1, 4, 3, 6, 1, -3, 3, -1, 6], [3, -2, -5, 0, -3, -2, 1, 4, 3, -3, -6, -2, -4, 0], [1, 6, 0, 0, -2, 3, -1, 1, -2, -2, 3, -2, 0, 0], [-3, 7, 1, 1, 2, 5, 3, -6, 0, 0, 1, -4, -2, -2], [4, -1, 4, 0, -1, -3, -3, 4, -2, 1, 0, 0, 3, 2], [-4, -1, 0, 1, 1, 1, -1, -6, 0, 1, -2, 2, -1, 3], [7, -4, -2, 6, 0, 2, -2, -3, -2, 0, -4, -3, -3, -3], [-3, 8, -5, 3, -1, 2, 0, -3, 1, -3, -2, 5, -1, 0], [2, -2, -3, -6, -1, 0, 4, 2, 2, 0, 2, -4, 1, -2], [1, 0, -2, -3, -2, -3, -2, 4, 2, 0, -1, 5, 3, 2], [-6, -4, -1, 3, 1, -2, -5, 2, 1, 0, -1, -1, 2, -2], [-2, -7, 3, -1, -4, 1, -5, 3, -4, 2, -2, -1, 0, -2], [-1, -1, 1, -6, -6, 1, -4, 2, 2, -2, -1, -5, -1, 0], [-1, -6, 6, -1, 0, 0, 2, -2, 0, 1, 1, -4, 2, -4], [-6, 4, -1, -4, 3, 0, 0, -4, -2, -2, 1, 6, 0, 1], [-12, 7, -6, 2, -1, 5, -1, 0, 4, -2, -3, -4, -4, -2], [4, 0, -2, 4, 0, -2, -3, 3, -1, -2, -6, -2, 2, -6], [-1, -2, -5, 2, 1, 2, 0, -1, 1, 3, -5, 4, 0, 2], [2, 4, 2, 0, 4, 6, 6, -3, 2, 3, 2, -2, 0, 3], [-2, 8, -4, 3, 0, 0, -1, 1, -1, -1, -1, 4, 1, 1], [2, 8, -3, 0, 5, 1, 1, -6, -3, 0, 1, 2, 1, 2], [-12, -2, -2, 1, -8, 2, -4, 4, 1, -1, 1, -7, -2, -2], [-6, -7, -2, 2, 1, -2, -2, 1, -4, -1, 4, 0, 1, -6], [-6, 2, -4, 4, 2, 4, 1, -5, -1, 1, 1, 0, 1, 2], [-3, 3, 1, 1, 0, -3, -3, -4, -10, 0, 5, 0, 4, 0], [-6, -5, 1, 2, -6, 5, -3, 2, 0, 0, -1, -7, -1, -4], [-13, 9, 0, 1, -1, 2, 0, 4, 6, 0, -3, -1, -1, 0], [-1, -1, -1, 0, -2, -1, -3, 6, 3, -1, -1, -3, 3, -3], [1, 1, -4, -6, 3, -3, -1, -4, -1, -1, 2, 7, 1, 2], [-10, 6, -2, -4, 0, 4, 5, -6, 0, 0, 1, 5, -2, 0], [0, -7, 7, -3, -4, -3, -2, 3, -4, 3, 2, -1, 3, -4], [6, -4, -2, 1, 4, 0, 4, -2, 0, 0, -2, 0, -2, 1], [14, 5, -1, 3, 2, 1, 2, -1, -4, 1, -2, 0, -1, 2], [-4, 0, 0, 0, 0, 2, 3, 3, 4, 1, -1, 0, 2, 3], [-13, -1, 8, 2, 7, 1, 0, -4, -1, -1, 9, -1, 1, -4], [-8, -4, 3, -8, -1, -2, 4, -4, -7, 3, 3, 6, 3, 3], [3, -5, 5, -4, 4, 3, 1, -5, 0, 3, 0, 5, -1, 3], [6, 1, 3, 6, -6, -1, -3, 10, 2, -1, -8, -2, -2, -4], [-9, 4, 5, -2, -1, 3, 3, -4, -1, 1, -3, 0, 0, -2], [-14, -4, 4, 2, 0, 1, -2, 0, 1, -3, 3, -2, -1, -4], [-3, -1, 1, 2, 2, 2, 2, -2, 3, 3, 0, 2, 0, 2], [3, -4, -10, 4, 2, 2, 0, -2, -3, 1, 1, -1, 3, -4], [-1, 11, -2, 6, 3, 0, 0, 2, -1, 1, -6, 3, 1, 0], [-15, -1, 2, 1, -3, 3, -3, 5, 2, -2, 1, -8, -3, -6], [5, -4, -6, 2, -8, -4, -2, 6, 0, -2, 1, 1, 2, -6], [2, 6, -5, 2, 7, 2, 7, -7, -4, -1, 3, -4, 3, -1], [-4, 3, 0, 4, 3, 1, -3, 1, 0, 0, -2, -3, 1, -1], [3, -5, 2, -3, -1, -3, -3, 5, 0, -1, 2, 6, 4, 4], [0, -1, 1, 3, 4, 3, 5, -2, 1, -1, 3, 2, -4, -1], [7, -2, -4, 4, -4, -2, -4, 7, 3, -1, -5, 1, -3, -2], [-1, 3, 7, -2, -4, 4, -4, 3, 1, -1, 1, -1, 1, -3], [-1, -2, 0, 6, -2, 2, -6, -1, -2, 0, 2, 1, -3, -3], [3, 11, -5, 7, 10, 1, 8, -1, 2, 2, -2, -4, -2, 0], [13, 1, 6, 1, -3, -3, 1, 1, 3, 1, -2, -1, -1, 5], [10, -4, -4, 1, -8, -4, -3, 3, -2, -4, -4, -3, -3, -4], [4, -3, 2, -2, -1, -1, 2, -3, -4, 0, 5, 1, 1, -1], [-5, 0, -5, -1, -3, 2, -4, -1, -1, 2, -4, 1, -1, 3], [1, 1, 2, 0, 1, 5, 3, -3, 3, -2, 4, -1, -5, -7], [-3, 5, -5, -2, -2, -2, -1, -5, 3, -1, 0, 4, -1, 3], [2, -15, 6, 3, -5, -1, -3, 0, -6, 0, 2, -1, -3, -8], [3, -7, 3, -3, 2, 1, -3, -1, -1, -2, 5, 1, 1, -3], [-9, 3, -7, 1, 0, -2, -1, -1, 0, -2, -3, 4, -1, 3], [-23, 0, -2, 3, -6, 1, -4, 6, 2, -2, 1, -5, -1, -6], [-1, -1, 0, 3, -1, -5, -3, -1, -11, 0, 3, -3, 4, -1], [-4, -7, 0, -6, 5, -3, 5, -9, -5, 4, 1, 1, 1, 3], [-2, 8, -3, -2, 1, 0, 7, 2, 5, -1, -2, 5, 1, 5], [1, 6, -6, -3, 4, 2, 5, 5, 4, 0, -3, 4, 3, 8], [4, -3, -5, 9, 4, 3, 1, -3, 3, 2, 0, -5, -3, -1], [-8, 6, 3, -5, 7, 2, 8, -4, -3, 2, 4, 1, 2, 0], [18, 3, -4, 7, -5, -1, 1, 10, 4, -4, -3, -4, -4, -1], [5, -3, -6, 2, -1, 3, 2, 1, -2, 6, -6, -5, -1, 3], [-5, -2, -3, 6, 3, 4, 2, -1, 2, -2, 4, -7, -6, -7], [-7, -8, 2, -1, 4, 1, 5, 1, 2, 4, 1, -6, 2, -3], [-16, -1, 11, -3, 4, 1, 1, -3, -3, -2, 2, 7, 1, -1], [16, -1, 3, -3, -4, -5, -4, 5, -2, -3, -2, 1, 1, -4], [-10, 4, 6, 0, 0, 4, 2, -1, 4, -1, 2, -6, -4, 3], [-4, -2, -7, 3, -5, 0, -1, 5, 4, -3, -5, 1, 1, -6], [18, 0, -1, 3, -5, -6, -8, 5, -1, -2, -5, 3, 5, 1], [22, -7, 3, -3, 6, -1, 1, 0, -7, 2, 0, 3, 5, 1], [2, 7, -3, -3, 2, 5, 2, -2, 8, -1, 5, 10, 2, 7], [10, 0, 5, -1, 7, -3, 3, 0, -1, -1, 0, 4, 0, 3], [12, 11, -8, 9, 1, 1, -1, 3, 8, -2, -8, -1, -5, 0], [-3, 2, 7, 0, 1, 0, 1, -3, 1, -2, 3, -8, 1, -4], [-11, 3, 3, 0, 4, 1, -1, -4, -1, 3, 2, 8, 2, 8], [10, 8, -5, 2, 7, -4, 4, 2, -4, 3, -1, 0, 2, 0], [2, -3, 0, 9, -1, 3, -3, -5, -3, 1, -1, -2, 1, 1], [7, -1, -3, 5, -4, 3, -6, 3, 0, 0, -4, -6, 0, -8], [3, 2, 9, -9, -3, -2, 1, 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