# q-expansion of newform 6.18.a.b, downloaded from the LMFDB on 26 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 = 6
weight = 18
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 = [[-256], [6561], [-72186], [-8640184], [1159304460], [2801062862], [32979662226], [5778498836], [169116994200], [3631735478814], [6880978560608], [-35464500749338], [-8923766734806], [-129966457018324], [129499777218480], [218262107088054], [-1783401246652740], [1469145893932670], [5051560974054596], [-793480696785720], [6343500933237962], [-8292883305185392], [-24031501915598508], [-15466463339248422], [79745962551777122], [175677759352523238], [-100291893446960344], [-265518533305019412], [-22785042253550290], [56621972939186802], [-87046639270509760], [-291549572371166460], [1046112600962512266], [-2658520752132073204], [3059411321900707734], [2175984693454883288], [-5885411044465864738], [-1486877489574133084], [-8412524946090259800], [-13436931431786925522], [7868635363999246644], [20960871047274814454], [18859502676328149696], [-38628547028020225342], [-46052640653977067706], [8822744580056914568], [-52364196104193750700], [-29483848842204730720], [16685347507559524068], [16785975053486579750], [166092898884473294442], [-197773581854213504016], [120673658743619366642], [9476950003076922876], [144667851468351535362], [-165479848326316437240], [765963371346168916494], [-520542578911414482736], [-1093216137330405440938], [-424847879779312339302], [-122463743781547150948], [727954982157066933414], [1005252255579613687220], [-2323496172215958857160], [-916081290063323328838], [-1094589144734135273922], [90641184430944416972], [-5804036064165082220974], [-6712270729702768655268], [3891194297556060517790], [1496165779888419977826], [7578067497970434639336], [-11544706195082833515088], [2921174309179230785846], [-2670869639554197582724], [-7545015537822469484160], [17096102155549991668614], [3540924939937737165518], [-9136317226324741085166], [-16623235093642838161894], [-25753026774646988221020], [26935175637243579928550], [22898569761114481715760], [14834490498960419366066], [5898076413240070097528], [21367273232477178640572], [30597580461238927669314], [-33893127118235006170294], [42778615043407360580430], [-15989253070608240248944], [32804710655527500604884], [-7313033842947786124704], [64809324197450548801832], [87014090888929710587244], [103055915750727562537076], [55744202591084471414904], [240566486195582525694], [-83170139497192930995702], [-183385509783811343771380], [144977709796663614372830], [-32668696610552341744348], [58956545319738540601518], [-104476746159552982468812], [-322207346977641161094342], [-125037617924645435674180], [27454530336896530797122], [431074307935051140904908], [-647623958884219774438830], [-119466018560868739226664], [614541426771929995050458], [-52288189589396804879584], [188812148152033675632422], [779200774819457286821610], [91343352240180037475564], [127853076182494639729592], [146204680345535161124610], [-518713761080833627607164], [-716385630090760443816696], [1122660500000277316159758], [807222245993307114101268], [-1821656753669091603155626], [1676361700586355316427426], [-299561092153995832559706], [162279764183211465107628], [2061486871205378691697844], [-2138379862334493741342786], [1185970673377252179878918], [-2440862326938609699868080], [1827756687299437501779992], [1103707018721031076220318], [-2282473756287798089956252], [-566594224789948421045784], [3958732004428428078405680], [-4718598724584336272019658], [1212124850996846216027898], [850447281223390027672322], [3880221735560377351184262], [-6887451136550419635942124], [-2019515132783836390389090], [3683950789246829784528426], [6753567402353299544792876], [4186543693878082012827318], [1809393110303015884340600], [-99875108128514922146628], [-12036368138481116026868482], [-4295125253076801118577976], [-288158716892683723056106], [-3738250920613818354313638], [5445797481615106531498076], [13586444487638829106030944], [-4393607264492487695329618], [1211307377304836119637106], [544602199544639505212276], [-3395962869304708425749640], [-5671386255258848720457076], [-11611076260199317796692848], [-21673195618598503081176616], [-25657513393522986950291166], [-9789974149090113301259734], [-9974022852869782968295122], [20530475177014849429965876], [-5550898323145055496184134], [7038755234804881209794696], [-13292103964980636327056436], [23815099371702341923936338], [14070631223040808870268376], [23537006469258488811140384], [29221271103261101903787686]]