# q-expansion of newform 11.19.b.a, downloaded from the LMFDB on 02 June 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 = 11
weight = 19
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 = [[2, [-1]]]
aps_data = [[0], [-20870], [-3063526], [0], [-2357947691], [0], [0], [0], [3592942977890], [0], [26636854831058], [-10400449085350], [0], [0], [-2113576298457550], [-1367378362647430], [-12373964663300278], [0], [37614783844431290], [91404547650956162], [0], [0], [0], [623295746335388018], [-1511803192851761470], [0], [2373115488942794690], [0], [0], [-5699180055764896990], [0], [0], [22986158593075686290], [0], [0], [0], [56049250986007652810], [-151048026623916596230], [0], [0], [324603134467754584538], [358776403363251390458], [420323003319338024978], [0], [0], [44253831881125468802], [0], [-2246832740431032682030], [0], [3422666504706625321562], [0], [0], [0], [-6980136967493176433398], [2572346996323815060290], [0], [5188654804501107708842], [0], [0], [0], [0], [0], [0], [-54402385869046911604318], [41061177557294111874290], [63131421556475453969930], [25732892023538661674858], [0], [0], [0], [59783438271590872141250], [0], [3919101574862500160690], [0], [-322643718526522189226038], [15501272195347811293970], [-301702306713166763919718], [-303102261975687551594710], [478476187857185098425698], [0], [-219821927639796787462342], [230786441232322324331738], [0], [-274946074955208315872350], [0], [1313844113131387625215370], [858422137941156075117698], [0], [0], [652603589928966505669490], [-665414092224812785646950], [0], [-1788973712574209618962750], [0], [-1687886701318657061879398], [0], [1119361263048067540678922], [-5471323677179321580566062], [0], [0], [0], [0], [0], [0], [0], [-11591951632579390033372030], [-12973763369512024377129430], [0], [13344730105707709222281698], [0], [0], [0], [-19536906388580967468888430], [-24956548990305196935486742], [25713727245884791477499042], [-24077960797496268813925822], [12240146148615680689943930], [31734221077934837149730690], [16888059639021741058652330], [0], [5369601473911421111031482], [0], [0], [43284748905122581838392490], [-67323501043006491018069478], [0], [-86925650608734421452690022], [83324630686264373074118258], [-111226097426843535852668830], [0], [0], [0], [-8974801809203621142027598], [-66015669936127300292554630], [0], [0], [-72387592517367430824535270], [0], [-149505556101660010670148790], [0], [0], [0], [-315801998083815145361435230], [0], [368805165983241020293990538], [393115247699508874734756482], [0], [0], [418692588985650860032999178], [-407782529113450235026372270], [0], [239393369004751223047542242], [646497658275327015666789530], [0], [611392784176031619606934250], [-345438237829752808929267982], [0], [1017118459322139558047376962], [0], [0], [953397329063626215179871770], [0], [0], [576019479350646970515116138], [298148396719684995215098850], [309555844129750044223454690], [-1648374493892974044963576622], [0]]