# q-expansion of newform 45.8.a.e, downloaded from the LMFDB on 27 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 = 45
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 = [[13], [0], [125], [1380], [3304], [8506], [9994], [41236], [-84120], [-132802], [-55800], [228170], [139670], [-755492], [-836984], [-1641650], [989656], [-1658162], [-4523844], [389408], [5617330], [3901080], [9394116], [-2803746], [5099426], [-15172266], [4705268], [-2631204], [-4300594], [-3982334], [28017676], [8199192], [1666458], [-58745596], [19369742], [-53395208], [20452922], [-733588], [-16892520], [-11818554], [31374640], [-58355530], [-4061656], [-133220566], [-13077138], [-69850232], [32853500], [69519356], [230778860], [146157054], [-311907438], [-227310296], [-198482654], [132535968], [35864226], [479640488], [70876414], [407490256], [-79347702], [87006846], [277611876], [-246490482], [384964564], [-46443456], [210558002], [960971366], [399922812], [269184538], [821868004], [648353902], [-652666326], [877430856], [286988988], [-1770132542], [1463113060], [990456408], [796901166], [795584194], [2016318342], [-54802966], [-110924536], [-2120637986], [2485326480], [29995642], [-1048734680], [-204679116], [-2639623418], [1837383370], [-3576778602], [1280677668], [4199840036], [-2482019880], [-1080653044], [947786672], [-2076919244], [-2732654896], [-4524697154], [279532670], [1463762068], [-2721936690], [-673048220], [7199944134], [-5801136324], [1670688894], [5701386956], [2430634946], [3333782076], [-3422803438], [928661192], [-6569738374], [-3507897644], [3639701194], [-7825585182], [8351725844], [-21681968], [7853610750], [-8818195092], [8719969024], [665755238], [9995126680], [-2891603978], [-5832356070], [12252424782], [9692933868], [3332853076], [2731988702], [344619886], [-3357493728], [-13201695988], [4301789378], [-18241424636], [236847232], [-14432560904], [16493357962], [2968454046], [8154002498], [2780590398], [-8209818596], [54791606], [-2049730330], [-15947121708], [-9659231274], [3903976844], [13222440964], [12048232350], [-31060309000], [25215704442], [-22531444814], [4078626820], [11064318272], [36059458650], [26295355606], [2788729388], [18296966616], [-1212451716], [20918089408], [-18154477040], [33922505414], [36402256938], [-50711742490], [-23099415492], [4163677114], [-52046779324], [364572720], [18178540474], [18577791360], [18252005552], [-1245342558]]