# q-expansion of newform 60.8.a.d, downloaded from the LMFDB on 28 July 2026.

# 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 = 60
weight = 8
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], [27], [125], [-1408], [-4044], [-5890], [31002], [-40300], [-78912], [-157194], [114824], [-471994], [-404310], [-253852], [437688], [334926], [562596], [3246662], [3895148], [-2345160], [5726954], [-5222008], [-2928132], [-3160230], [-1898686], [-9829842], [5891072], [-22537164], [3291782], [23374554], [16564232], [34005612], [1408626], [-29949556], [-39372738], [-12628240], [31610654], [-1746580], [74141616], [-32805018], [-83747844], [93211070], [-48070656], [17222642], [-30728802], [-181385728], [-108304060], [-225721480], [271001100], [-276697186], [-215356110], [-412011648], [-42490222], [57304788], [176317386], [237052608], [100666374], [28976024], [-338054746], [-336322374], [33132404], [-502960386], [-619829044], [327307704], [75755162], [-889914618], [-475587028], [778710626], [299149476], [1281922694], [-1065894246], [-778277544], [766621832], [-765360922], [-808440628], [-242686296], [-1223989266], [-114697954], [2388216786], [2734265834], [2138728908], [1462254110], [-2297834352], [2511707522], [885268592], [2547652212], [-1478846238], [-2493746470], [-2780243034], [2439501944], [3092021196], [-4537517664], [-1009249936], [-2812986156], [-2572288924], [-3112573488], [-3299845962], [4623183642], [2141781044], [222350678], [3099215564], [-5673748026], [-2920893780], [-8372187126], [5150749340], [-1049474686], [5340260532], [1571402778], [6493273704], [6405325418], [7677300056], [7850222534], [218100018], [9023037356], [5587035872], [3298461714], [-1924387348], [-572440464], [-1547632602], [3707828652], [-2935244722], [-7090598350], [369688782], [-14101472748], [-8612158492], [-10074350634], [18191391326], [-630763008], [-12094384096], [-17893066882], [-12375535660], [13074243792], [5290320632], [-8944797082], [22577543082], [-786783262], [2857662846], [10917722924], [14416991094], [10953612714], [-16187629780], [3296331006], [-5473508656], [-2648041836], [-9412933642], [645230616], [-22445197834], [-13743960366], [-3199095220], [-8086868424], [12536703758], [-5176443870], [-28571851204], [-14675621568], [-2659211884], [35046691968], [23954123792], [25592910882], [-4512553894], [5615805990], [-35215191444], [-23657169918], [12786344240], [-8301430812], [-48233805606], [19164355104], [54456853208], [22388058998]]