# q-expansion of newform 2.16.a.a, downloaded from the LMFDB on 23 October 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 = 2
weight = 16
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 = [[-128], [6252], [90510], [56], [-95889948], [-59782138], [-1355814414], [3783593180], [-11608845528], [-28959105930], [253685353952], [817641294446], [-682333284198], [366945604292], [695741581776], [12993372468702], [9209035340340], [-42338641200298], [30029787950636], [115328696975352], [43787346432122], [79603813043120], [-3417068864868], [-377306179184790], [-166982186657374], [60680236307502], [587676402004952], [656968133121156], [-2090227783678810], [-1632955081904238], [6513337595398016], [-11263608694217748], [5103277652953146], [2766818784713060], [26846924611329150], [-36547941539093848], [29968513790601206], [-42949655950023988], [7100640246563736], [-56930880794733978], [-2656549234371780], [51764838979787102], [53675288757279552], [114449562301688642], [-207129734322721074], [-35983759623512200], [90269907274561052], [-322894137103599328], [784855840799205516], [-784091561182107730], [-717047533944033318], [693110354447654160], [965565808917411602], [-846640700229572748], [1401559966131752706], [1388044612099161912], [-2854722622192901370], [-1761582601077887248], [396897243417223166], [336756164587013802], [-4565289799382321068], [851075979146367342], [3969990685049876156], [3186658737303622152], [1598656477025729162], [-4922618687492902314], [-6428968905776435548], [14491233801832096946], [21149973039491539476], [-10478334124056258250], [13944183624623216802], [-10302717043689278760], [-11790047285366852464], [-33255145150314778978], [-6272369177076237580], [32791907326704162432], [26907402185918135310], [6983264957016571526], [12257125471824703602], [41328507748505589290], [-65831242648134829620], [-6815718740136590098], [-24344421555688828848], [37320935465926720082], [88815328885211578760], [-32446267771575487308], [140615166466843114050], [-110893306450658169094], [29612440593010015302], [-133020853136210791888], [-7073926250320974564], [13096370593308018720], [88223174902963614296], [-232523484978530205948], [191679747549886408700], [264371805300770023752], [135341978803894506390], [116387056808493897402], [237253140768304358372], [-201502494205089813898], [-212933722628900575924], [273615733273571378406], [-223410733057050565188], [-524953154866077681270], [-154330587745396275148], [719412993597562352066], [-431990037395095194204], [-419124315111945415758], [318808163037472852200], [-985640986046284507798], [1082967215047827402656], [1474879968902539772462], [-186047580614032187814], [544498671360295182980], [-134412385356022376248], [-215011191928405903998], [-1367048309326701260308], [114998262473166478776], [-1233911394764611979898], [1630818763318769448540], [-1387400997879304820098], [-1339365665163582571678], [1994084993241420893166], [-1251802004294577191868], [-1464310331335620951748], [-297987165417885862698], [1218811167980636150990], [-3018556524927520743120], [897709687558340647016], [-852856783886536991818], [2191956683366898194060], [1899287834890543771992], [-4401280949941167736048], [3326002274080363141406], [6663618118618587085002], [1777254644139518262530], [-4171454431854670481778], [4968355486347358187996], [4550556629628882583926], [11203979375955922171290], [1564306061231781619652], [5315337445585456827102], [2483800092476862910472], [-6014203217716988743884], [-4336525918553531989930], [7922236969592446631160], [4779756045324840685502], [-13850590188241015167894], [11338903675489670737940], [13450603597907554238112], [8712642599626390432166], [18428777183204026694802], [6386963409192241412732], [-12676116339862959720504], [1296697253709090336356], [22628014577804573066352], [-4370465969452095455320], [-24342274882313732208030], [16477551313221255640346], [-20938348648574896646298], [-33603580255297160054724], [34529196504570780833802], [1345370761206985481336], [1503340877222658441252], [-43613260394881822774734], [3219859061745904400232], [-48501045826223818529248], [53554291372107703319726]]