
# q-expansion of newform 4008.2.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 = 4008
weight = 2
poly_data = [12144, -72184, 112850, -14532, -72240, 26911, 16968, -7582, -1879, 901, 99, -49, -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], 1], [[0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], 1], [[1211669325949820, -4373968597429142, 1650853947740980, 1724063123712248, -1037037612348531, -196205928405368, 201417980697138, 5016057310831, -16627326024925, 295838273673, 576861363065, -12640530894, -6207348777], 111451400959900], [[3605634089719788, -3707736754746094, 598916455112988, -993417294727940, -714691137288825, 1012871433621172, -90263735212158, -193085396549923, 48247617829913, 14015645726951, -4532098501229, -351643713690, 126516435801], 557257004799500], [[-261781210799648, 719430531278666, -61985566741568, -552823809736903, 129168526434186, 149114772131121, -42939671149010, -18714943276923, 5918945839452, 1102737562371, -363301449706, -24439858116, 8155241826], 27862850239975], [[-9603799562310, 1829362384230014, -2780474603245950, -967840084997036, 1772258179746447, 236986143576626, -406570644479846, -30214877144177, 43420560752365, 1857640717639, -2192557864275, -42904580362, 42253210649], 55725700479950], [[-455409403401910, 1829362384230014, -2724748902766000, -967840084997036, 1772258179746447, 236986143576626, -406570644479846, -30214877144177, 43420560752365, 1857640717639, -2192557864275, -42904580362, 42253210649], 55725700479950], [[-2901404480931816, 10417867358098218, -6707972583182866, -5762733210647085, 4728856566083805, 1217996115089036, -1131795821929059, -121899916781369, 127066606088584, 5771355044103, -6791649579897, -105435570025, 138385219703], 139314251199875], [[-629228580752360, 6347787481933558, -5320985228785100, -4408101894281592, 3686070397711199, 1109495790415652, -912294162144422, -129559753690019, 104000386651945, 7117312024823, -5533191476745, -148746957474, 111023509273], 111451400959900], [[6802024064231936, -25352491816897518, 14675109344372636, 16707098673252820, -11231871461142475, -4297098906328266, 2761108370208174, 524566352898419, -309203328646339, -30174775561853, 16238231918487, 655210507370, -323942240303], 278628502399750], [[-1313465392184157, 7530986189108326, -7596810398586632, -3276596851987855, 5522030332256130, 348943217873482, -1435122721494978, 10485608743617, 173349897433318, -3015289235129, -9810853937919, 101272023955, 208422301596], 139314251199875], [[-20950363454455596, 67210663867287138, -31856351450553896, -44471075457654780, 26745397139049795, 11473935236033836, -6930554296860074, -1397243777337779, 807810238699029, 79678278406623, -43690158347557, -1710596413190, 889692540773], 557257004799500], [[-2219186123392628, 10312125439960266, -10980877487070868, -4406202874244528, 7475418034130291, 558179494985656, -1812887391077730, -13254732421323, 205661421513157, -1496522255749, -11028950583521, 63390661414, 224267911141], 111451400959900]]

hecke_ring_character_values = None
aps_data = [[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, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0], [1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0], [2, 0, 1, 0, 1, 1, -1, -1, -1, 1, -1, 1, 1], [0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0], [1, 0, 0, 0, 0, -1, 0, 1, 0, 0, -1, -1, 0], [0, 1, 0, 0, -1, 0, 0, 0, 0, 0, 1, 0, -1], [0, 0, 0, -1, -1, -1, 0, 1, 1, -1, 0, -1, -1], [-1, 0, 0, 0, -1, 0, 1, 1, 0, 0, 0, -1, -1], [-1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0], [2, -1, 0, 0, 1, 0, 0, 0, -1, 1, 0, 0, 0], [3, 0, 1, 0, 0, 0, 0, -1, -1, 1, 0, 1, 1], [2, -2, 0, 0, 1, 1, 1, -2, -1, 0, 1, 1, 0], [1, -1, 0, 0, 0, -1, 0, 0, 1, -1, -1, -1, 0], [4, -1, 1, 0, 1, 0, 0, -1, 0, 1, 0, 0, 1], [1, -1, 0, 0, 1, -1, 0, 1, 0, 0, -1, -1, 0], [4, -2, 0, 0, 1, -1, 0, -1, 0, 1, 0, 1, 0], [3, 0, 1, 0, 0, 0, 1, -1, -1, 0, 1, 0, 0], [3, -1, -1, 0, -1, 1, 0, -1, 0, -2, 2, -1, 0], [0, 0, -1, 0, -1, -1, 0, 1, 0, 0, -1, -1, 0], [1, 0, 1, 0, 0, 1, -1, -1, 0, 1, 0, 1, 1], [1, -1, 0, 0, 0, 1, 0, -1, -1, 1, 1, 0, 1], [4, 0, 1, -1, -1, 0, 0, -1, -1, -1, 0, 0, 0], [2, 1, 0, 2, -1, 1, -1, 0, -2, 1, 1, 1, 0], [-1, 0, -1, 0, -1, -2, 1, 2, 1, 0, 1, 0, -2], [3, -2, 1, 1, 1, -1, 0, 0, -1, 1, 0, -1, -1], [0, 1, -1, 0, -1, -1, 1, 2, 2, -1, 0, -1, -2], [-1, 0, -2, 0, 1, 0, 0, 3, 1, 2, 0, 0, -1], [1, 0, -1, 0, 0, 1, 0, -2, 0, -1, 0, 1, 2], [3, 0, 1, 0, 1, -2, -1, 1, 0, 0, -1, 0, 0], [2, 0, 0, -2, 0, -1, 0, 0, 2, -1, -1, -1, 1], [3, -2, 1, 0, 1, 1, 0, 0, 0, 1, -2, -1, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -2, -1, 0, 2, 0, -1, -1, -1, 1], [2, -4, 2, 2, 0, 2, -1, -3, -2, 0, -1, 0, 0], [0, -1, 0, -2, 1, -2, 0, 1, 3, 1, -1, -1, 0], [-3, 0, -2, 0, -2, 0, 0, 0, 0, 0, 1, 1, 1], [3, 0, 0, 0, -1, 0, 0, 2, 2, -1, 1, -2, 0], [-3, 0, 0, 1, 0, 4, -1, 0, -1, 1, 1, 2, 0], [-1, 1, 0, 0, 0, 3, 1, -1, -1, 1, 0, 1, 0], [3, -3, -1, 2, 1, 0, 0, -1, -2, 0, 1, 0, 1], [-1, 1, 1, -2, 1, 1, 1, -3, 0, 2, 1, 1, 0], [0, -2, 1, 0, -1, 0, -2, 0, -1, -2, 0, 0, -1], [5, 0, -1, 0, 0, -1, 0, 3, 1, 0, -1, 0, 0], [3, -1, 2, 0, 1, -1, 1, -4, -1, 0, 2, 1, 2], [-4, 0, 1, 0, 0, 1, 1, -2, 1, -1, -1, -2, 0], [4, -1, 0, 0, 1, 1, -1, 1, -1, 2, -3, 1, 0], [-1, 0, -1, 0, -2, -2, -2, 2, 1, -1, -2, -2, 0], [0, 0, 0, -2, 2, 2, 0, -3, 1, -1, 0, 0, 2], [0, 0, 0, 0, -1, -1, -1, 0, -2, 0, -2, 1, 1], [-1, 0, 1, -2, -1, -2, -2, 1, 0, -2, -3, 0, 1], [-1, 3, 0, -2, -1, 0, 1, 0, 0, 1, -1, 0, 3], [3, 0, 0, -2, -3, 3, 0, 0, 0, 0, -1, -1, 0], [-1, 0, -1, 2, 1, 2, -1, 0, 1, 2, -1, 2, 0], [0, 0, 1, 0, -2, 0, -2, 0, 1, -2, -1, -2, 0], [-1, -2, -1, 2, 2, 0, 2, -1, 1, -1, 2, 0, -2], [2, 0, 1, 0, 1, -2, 2, -1, 0, -5, 0, 0, -1], [-7, 1, -1, -2, 1, -4, 1, 2, 2, -1, -2, -2, 1], [6, -1, 0, 0, 1, 2, -1, 1, 1, 1, 2, 0, -3], [-6, -3, -1, 0, 4, -1, -1, 1, 0, 0, 2, 1, -2], [-2, 0, -1, -1, 1, 1, 0, -3, 0, -1, 3, 3, -1], [3, 0, 0, 0, -2, -1, 1, 0, 0, 1, -1, -1, -1], [-5, 2, -2, 0, -1, -1, -1, -1, 0, -2, 2, 1, 0], [6, 2, 0, -2, 2, 0, 0, 0, 2, 0, 1, 0, 1], [3, -1, 2, 0, 4, 2, 0, -4, -3, 3, 2, 2, 1], [0, -1, 0, 0, 3, 1, 1, -2, -3, -2, 1, 1, 2], [-5, 2, 0, 0, 2, -2, 0, 1, 2, 1, -1, 0, 0], [4, 2, -1, 0, 0, 0, 2, 1, 1, 2, 0, 0, -1], [-3, 1, -1, 4, -2, -1, -2, 4, 1, 0, -1, -1, -3], [-6, 3, 1, 0, -1, 1, -2, 0, -2, 2, -1, 0, 1], [-5, -2, -1, 3, -3, 3, 1, 2, 0, -2, 1, -1, -3], [5, -1, -1, -2, 0, 1, 0, -2, 0, -1, 0, 0, 1], [1, 3, 0, 0, -1, -2, -1, 1, 0, -3, 1, 0, -1], [4, 1, 1, 0, -1, -3, 0, 0, 2, 4, -3, -2, -1], [5, -1, 1, 2, 0, 2, -1, -3, -4, 0, -2, 3, 2], [5, -1, -1, 0, 3, -4, 2, 0, 2, 0, 3, -2, -1], [-14, 3, -2, 2, -2, -1, 3, 1, 1, -1, 5, -1, -2], [3, 0, 3, 0, -3, -1, -1, -2, 1, -3, 0, 0, 2], [-3, 1, -1, 2, 0, 0, -1, 3, 1, -1, 0, -2, -1], [5, 0, 1, -2, 2, -1, -1, -3, -1, 2, 1, 3, 2], [0, 1, 0, 0, 2, 1, -1, 0, 1, 1, -1, 1, -2], [10, 2, -1, 0, 0, 0, 1, 2, -1, 2, -2, 0, 2], [-3, 1, -2, -2, -3, -2, 1, 2, 3, -4, -1, 0, -1], [0, -1, -2, 0, 3, -4, 2, 2, 0, 6, -3, 0, 1], [0, -1, 3, -2, -1, 3, -1, -4, 0, -2, -2, 3, 2], [1, -5, 1, 4, 1, 2, -1, -3, -4, 4, -2, 2, -1], [-9, 4, 3, -2, 1, -3, -1, 2, 2, 1, -4, -1, 1], [-2, 2, 1, 2, -2, -3, 0, 2, -1, 1, 0, 0, -2], [-4, -1, -3, 3, -2, 1, 2, 1, 1, -2, 4, -1, -2], [4, -3, 0, -2, 5, 1, -1, -2, -1, 2, -3, 2, 1], [1, -1, -1, 0, 0, -1, 0, 1, 0, 3, -4, 1, 2], [-4, 1, -1, -2, 0, -3, 1, 5, 3, -1, 0, -1, -2], [7, 0, 1, 2, -1, 3, -2, 2, -1, 2, -3, -1, 1], [-3, 0, -1, 0, -2, 0, -2, 2, -3, -3, -1, 0, -1], [2, 0, 0, -2, 1, -2, 1, 2, 1, -1, -4, 0, -1], [-5, -1, -1, 0, 1, 5, -3, 0, -2, 3, -1, 1, 2], [-3, -3, -2, 0, 2, -3, 4, 2, 1, 3, -1, -1, 0], [-4, -1, -1, -2, -1, 1, -1, -1, 1, -3, 2, 1, 0], [3, 0, 2, -2, 4, -4, 2, 0, 1, 2, 2, 0, 1], [-3, 1, -2, 0, -3, -5, 3, 5, 1, 0, -2, -4, -3], [4, -1, 0, 0, 0, -2, 2, 1, 1, -1, 2, 0, 0], [0, 2, -2, -2, -4, -3, 2, 5, 3, 0, -3, -3, -3], [-8, -2, 0, 2, 1, -2, -3, -2, -2, -3, 1, 2, 2], [2, 1, 1, -2, 1, -3, 0, 6, 2, 1, -1, -1, 0], [-7, 1, -2, 2, -3, 3, -3, 1, 0, -2, 0, 1, -2], [0, 2, -1, -2, -1, 1, -4, 1, 1, -1, -4, 0, 0], [-8, 1, -2, 0, -6, 1, -1, 4, 3, 0, -1, -1, -2], [6, 0, 1, 1, -1, 2, -2, -1, -2, 2, -4, 4, 2], [-15, -3, 0, 2, -1, -1, 1, 2, -1, -4, 0, -1, -2], [4, -3, 1, 4, 1, 5, -2, -6, -5, 1, 1, 1, 4], [3, -1, -1, 1, 2, 1, 2, 0, 1, 1, 1, 1, 0], [-2, -5, -1, -2, 1, 5, -1, -2, -2, -4, 0, 1, 2], [-2, -1, 2, -2, 3, -2, 2, 0, 3, 5, -1, -2, 1], [-9, 1, 2, -2, -1, -5, 1, 4, 3, -1, 1, -3, -1], [4, -1, 2, -3, 3, -3, 0, -1, -3, -1, -1, -1, 2], [3, 2, -4, -4, 0, -1, 1, 0, 2, 0, 1, 1, -1], [3, -5, 0, -2, 4, 2, -1, -2, -1, 0, -3, 2, 3], [8, -3, 1, -1, 3, -5, 1, -3, -2, 1, -1, -1, 4], [-2, 3, 1, -1, -2, -4, 0, 1, 1, -3, 3, 0, -1], [-9, -4, 4, 0, -1, 4, -1, -1, 0, 1, -2, -2, -2], [6, 2, -1, 2, -5, 4, -3, -3, 1, 0, 1, 2, 3], [-5, 0, 0, 2, 1, -1, 1, -3, 0, -4, 3, 1, 1], [-7, -5, -2, 2, -1, -4, -2, 2, 2, -1, 0, -2, -3], [5, 0, 0, 2, 0, 3, -4, 4, -2, 4, -4, 2, 0], [-7, 2, -5, -2, -2, -6, 2, 8, 4, -4, -1, -4, -3], [-6, 3, 0, 0, 2, -1, 1, 0, 1, 5, 5, -1, -2], [1, -6, 0, -4, 0, 0, 2, -4, -1, -4, -3, 0, 4], [-1, 3, -1, 2, -4, 2, -3, 4, 2, 0, -4, 2, -1], [-11, -1, 0, 2, 2, 1, 0, -2, -3, -2, -1, 3, 2], [6, 3, -2, 0, -5, -3, 3, 3, 3, -4, 1, -5, -2], [-4, -3, -2, 0, -5, -2, -2, 5, 2, 0, -4, 0, -2], [7, -5, 2, 5, 0, 2, -2, -4, -4, 0, -3, 0, -1], [-4, 2, 0, 1, -6, 1, 0, -1, -3, 3, 2, 3, -1], [-3, 3, -3, -4, 1, -6, 3, 5, 6, -1, 0, -2, -1], [-6, 6, 0, -2, -1, 1, -4, 0, -1, 0, 0, 3, 4], [-8, -4, 1, 1, -3, -3, -2, 1, 2, -1, -5, 1, -1], [0, -1, 3, 0, 1, 1, 2, -1, 1, 5, -1, 1, -2], [3, 1, -2, 1, 6, -2, 1, 2, -1, 1, 0, 0, -1], [5, -5, 1, 0, 1, 1, 2, -4, 0, -3, 3, 3, -1], [-13, -1, 2, -2, -4, 2, -1, 2, 3, 1, -7, -3, 0], [15, 0, -1, 2, -1, 6, 1, -1, -1, 1, 3, 3, -1], [-2, -3, 3, 2, -1, 5, 0, -7, -4, -1, 3, 1, 2], [6, -4, 2, -2, 3, -3, 1, 1, -2, 1, -2, 1, -3], [-3, 0, 2, 2, 1, 0, 0, 3, -2, 3, 1, -4, 0], [3, 3, 1, 0, 4, 0, -1, -4, -2, 4, -2, 4, 3], [-5, 2, -3, -2, -5, 1, 1, 3, 0, -1, -2, -3, -1], [1, -4, -3, 0, 0, 0, 3, -2, -3, -7, 3, 0, 2], [1, 0, 2, 0, 4, 0, 0, -3, -4, 3, -1, 2, 4], [4, -1, -1, 4, -1, 2, -1, -3, -2, 1, 5, 4, 3], [-12, 1, 0, 4, 2, 3, -1, -1, -1, 1, 1, 1, 2], [-12, 3, -4, -4, -4, 0, 3, 6, 7, -2, 4, -4, -7], [-3, 7, 1, 0, -4, 2, 1, 2, 2, -5, 6, 0, -3], [-5, -6, 2, 0, -4, 1, 1, -2, 0, -6, 3, 1, -3], [-9, 4, 2, 5, 3, -4, 2, 2, -3, 1, 4, 0, -2], [-6, -4, -1, -4, -1, 2, -1, 2, 5, -6, 2, -5, -1], [-5, 0, 0, 0, 2, 2, -4, 2, 1, 6, 1, 0, 0], [-5, -2, 1, 2, -3, 1, 0, -3, -1, -2, 3, -1, 1], [-9, 3, -1, 2, -3, 0, 2, 2, 0, 6, -4, -2, -4], [-13, 7, 2, -4, 1, -6, 1, 2, 2, 1, 1, 0, -3], [-2, -7, -2, -2, -2, -1, -1, 4, 5, -6, -1, -5, -2], [-9, -6, 5, 0, 1, 2, -3, -2, 0, 0, -5, -2, 0], [5, 1, 3, 0, 4, -6, 4, -2, 2, 6, -1, -1, 0], [0, -4, -1, 0, 2, -5, 4, -2, -1, -4, 6, -2, 0], [-16, 2, -4, -2, -3, -2, 3, 4, 6, 2, 3, -4, -6], [7, 5, 2, 2, 2, 3, 0, -2, 1, 1, 6, 1, -3], [-4, 0, 2, 2, 1, 4, -1, -4, -2, -4, 3, 2, 0], [-12, 2, 0, -4, -4, -1, 0, 6, 4, 4, -1, -3, -3], [1, 6, -1, 4, 1, -2, -1, 2, -1, -2, 1, 1, -1], [-12, -3, 1, 2, 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1, 4, -3, 0, -8, 6, 1, 2], [-18, 2, -7, 4, -8, 0, -2, 4, 3, -2, -1, 4, -4], [25, -9, -1, 0, 7, -1, 0, -2, -4, 7, -2, 3, 0]]
