# q-expansion of newform 18.26.a.d, downloaded from the LMFDB on 25 August 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 = 18
weight = 26
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 = [[4096], [0], [292754850], [3580644032], [-15111573238212], [1221071681246], [-2518250853863682], [-7992693407413060], [99645642629247624], [2080672742244316890], [-4937672075835729208], [19829154107621718182], [-224696060863159376442], [-72221008334482349884], [-189872435947262116992], [2645676034335389555874], [16454608826354674865340], [-35546954389065591688738], [106703750402023286661692], [-73672004836753334994312], [-262402855870448192600374], [-1002642123108883497568840], [-1558588706101147601425596], [-2181670205644666928498490], [-440165308375605500117758], [5025816466960161774796338], [-15231132413116222331069104], [24723232336147920033402828], [-28577088988318093284377890], [17323993971537134780614254], [323348791716576759188607032], [363273194835322968636254148], [656741995091887721031840198], [672509382550429897533038900], [798626851788367611214144530], [-1036330071584944559083620688], [-3363461783241608608462639138], [800443041447900266135244236], [5179823242791965026125053928], [3408798534214068994276285914], [12361024688698930704034878900], [3385164534828342405435586982], [-38113957824947567554261304832], [63884180703822232991912580866], [87368006040135342497540658738], [27676620242123935779517509200], [-20847903083577826132074696148], [290505533386818563228819293496], [-323765484141151392384515736012], [-511098914973490127945336061130], [120130126304444209044165968694], [-334916172666816135840861227760], [-353238065443818531917291264878], [-1076032866769126355578367407572], [-6562583967503843506745344002], [-2038182223470016481986689918456], [-1302160522888136340250317085590], [2733839991949733137863539874632], [-483846314350783888045698250618], [7578181846561495517766147813798], [-1304434491439386969086272188124], [404020502709184111649020217394], [6956357800191461593641789968972], [-2749129675327262552220956664552], [9393826054165747022559281525306], [-18654308209875405963563764351782], [-47163781801817945578817539796908], 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