
# q-expansion of newform 240.4.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
    R = PolynomialRing(QQ, "x")
    f = R(poly_data)
    K = NumberField(f, "a")
    betas = [K.gens()[0]**i for i in range(len(poly_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 = 240
weight = 4
poly_data = [0, 1]

# The basis for the coefficient ring is just the power basis
# in the root of the defining polynomial above.
hecke_ring_character_values = None
aps_data = [[0], [3], [-5], [-20], [24], [74], [54], [124], [120], [-78], [-200], [-70], [330], [-92], [24], [450], [-24], [-322], [196], [288], [-430], [520], [-156], [1026], [-286], [-1734], [-452], [1404], [-1474], [1086], [-1244], [-2328], [2118], [-2324], [258], [808], [2378], [52], [3720], [426], [1440], [-3130], [-3576], [2666], [-2718], [3832], [-1100], [-1964], [-660], [-1906], [-1458], [-1176], [866], [-432], [2526], [-5448], [-2574], [3184], [3962], [-8286], [2716], [6018], [-9236], [-1536], [-7342], [-3894], [-3692], [-8998], [-5244], [6302], [3414], [-4824], [3508], [10802], [-1460], [4872], [-14046], [-2734], [-15942], [8714], [-11976], [11054], [-720], [-15622], [9880], [16116], [9018], [-3670], [17562], [-1172], [-6876], [-2280], [3076], [18912], [-9956], [10656], [-2766], [10530], [-12692], [18110], [-3620], [-14166], [13404], [-18654], [7684], [-1726], [-10596], [2862], [23592], [-9574], [-17444], [-2374], [-12162], [-8804], [12688], [-9150], [-25292], [2736], [22218], [-14520], [-10618], [1370], [-13758], [-11988], [-32996], [-25902], [-27394], [-34848], [-28028], [18002], [-15284], [18768], [-8696], [-38662], [23874], [23618], [11538], [14884], [-11334], [44730], [42748], [-31686], [-11036], [-25884], [15950], [-13800], [-27862], [-7314], [28780], [32688], [36650], [-2646], [-10892], [43464], [14884], [1248], [6640], [29946], [45002], [6090], [-56388], [10854], [42316], [-24480], [-6906], [-6960], [-47792], [9938]]
