
# q-expansion of newform 585.2.i.f, downloaded from the LMFDB on 19 April 2024.

# 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, ZZ
    R = PolynomialRing(QQ, "x")
    f = R(poly_data)
    K = NumberField(f, "a")
    betas = [K([c/ZZ(den) for c in num]) for num, den in basis_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 = 585
weight = 2
poly_data = [81, 63, 391, -14, 1245, 133, 1221, -9, 784, -4, 289, -18, 74, -4, 11, -1, 1]

# The entries in the following list give a basis for the
# coefficient ring in terms of a root of the defining polynomial above.
# Each line consists of the coefficients of the numerator, and a denominator.
basis_data  = [[[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], 1], [[0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], 1], [[-15267025981692153, -8427487924594911, -85814814008979952, -21804108527677176, -88736525406721413, -16621010179418725, -61053441921082007, -6835814771962126, -22854640199651823, -655850999572883, -6008483088362072, 29888665074896, -899262551344468, 51742196981879, -82999536459994, -482086147342], 9139478381478873], [[12833642664201015, 82246147650769, 28007951530950219, 17630686105303827, 18517847455679833, 13241752875048308, 5162807733352603, 6528933517925721, 543181503433067, 1498667977954772, 16556478346708, 189377660393089, -2697108934580, -8290755936386, 7812059681008, -3469225716849], 9139478381478873], [[188168453461782, 47347580732583266, -29817520384644188, 36951118425472407, -33863989763810635, 36230288966505302, -26605135287110714, 13576539103281376, -11225648306363066, 5043192392654460, -3319089373556393, 939963055735038, -554550721812852, 157853267854272, -59652178722282, 6246753964615], 9139478381478873], [[10022852458122111, -26296054994954299, 1711574395907992, -43280949748322271, 346318987982825, -28231418467650946, -3020771374915802, -10896063663200009, -2287548272381948, -2850156002058894, -634932225031991, -568340771522781, -118768769872326, -83292291525414, 69644913861, -10150138176959], 9139478381478873], [[3465152003612358, -14181352848452430, -45244610063730803, -21257885787172131, -42465191302472856, -7293677405066531, -29202452252119336, 3410762945368948, -9946106370147720, 2842579273044968, -2740194677046448, 1026293635905532, -394886776051364, 176286820995922, -45341940014915, 16744116205687], 9139478381478873], [[7952721863900868, -15105298015423202, 66520663704511511, 8345422433200593, 86744344986601537, 3455848364026450, 58933039323116513, 8193067138457009, 24965334872155991, 2174607198860307, 6370987760378243, 899822933969919, 1032516115451622, 112052967333795, 73952194191366, 26595513341993], 9139478381478873], [[845713476880083, 993128556611358, 2099036328412212, 12190149830375740, 4368639176864571, 10645673810520968, 2179877825131728, 7278933995715497, 914336412587371, 2653504647339595, 80541385038920, 701826108448870, 1974780746302, 103630645608798, -6203109654385, 10058702437115], 3046492793826291], [[1006054223780547, -5950075068841603, 37008967590250226, -25890870569987892, 27771099489894347, -31705409312404125, 22118706479439852, -21814004313126223, 8085804827482501, -8810642319636721, 2669287309933332, -2311957038839840, 436182822290110, -363159263988374, 61498828677274, -31322721365929], 9139478381478873], [[-18971878017965067, -11838012848643870, -39267507171071219, 8592989216359623, -28360469660381838, 15550848768311545, -17754813181134811, 18436346491318150, -5381830987437618, 7939379548056260, -1738189450969768, 2346977874531172, -261408818119538, 360549986338030, -42906025379669, 35353445168842], 9139478381478873], [[-18277046259484959, -10729277200479152, -33052810323229990, 10463030136046236, -19746766936098476, 24635102803156015, -11054480958405496, 19687039264476329, -1319639334852235, 8674336787121810, -864033044630458, 2518026227560113, -112029436571388, 436243278002799, -55102847496393, 47477396037032], 9139478381478873], [[27855550205743248, 5618911102245064, 91314609178178160, 52649488556200071, 109067758838777617, 53253791588998406, 59485031448387424, 36094915770719934, 22166122749372422, 13772024529051905, 4030094128429255, 3837661637756542, 489098601157807, 620697587330554, -20896415688563, 67174987872150], 9139478381478873], [[-7147318694846175, -23673309380329397, -36447991581067544, -110918644261366557, -62266292327076857, -98955484082891763, -38750539349362944, -62381927939342093, -15810686435053003, -22404008242064108, -2688129593090100, -5885557766462179, -287803296009742, -891308516903932, 39343350153959, -88820209793387], 9139478381478873], [[1006054223780547, -5950075068841603, 33962474796423935, -25890870569987892, 27771099489894347, -31705409312404125, 22118706479439852, -21814004313126223, 8085804827482501, -8810642319636721, 2669287309933332, -2311957038839840, 436182822290110, -363159263988374, 61498828677274, -31322721365929], 3046492793826291], [[-1957620602448711, 47651869820136828, -43331342630160394, 262801827503435319, -17879932341420474, 280378078725409877, -30755186597641787, 187862907036005150, -12238829708683665, 70587911675040145, -7008331821372974, 18188750599418894, -1384847430059254, 2707842518071070, -269626419785329, 241239132189497], 9139478381478873]]

hecke_ring_character_values = [[326, [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0]], [352, [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]], [496, [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]]]
aps_data = [[0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -1, 0, -1, -1, -1, -1, 1, -1, 0, 0, -1, 0, 1, -1, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0], [1, 2, 1, 0, 1, 1, 0, -1, 1, 1, 0, 1, 0, -1, 1, 0], [1, -1, 0, 0, 1, 0, 0, 2, 0, 1, 0, -1, 0, 1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0], [1, 2, 3, 0, 1, 1, 0, 0, 1, 0, -2, 1, 0, 1, 2, 0], [-1, 1, 0, -2, 0, 2, -2, -2, 0, 0, 1, 0, 0, 0, 1, 0], [0, 0, 2, -2, -1, 0, 1, 1, 0, 0, -1, 0, 0, 0, 0, -1], [2, 2, 1, 1, 4, 3, 0, 0, 1, 2, 3, 2, -1, -2, 1, 0], [0, 2, -2, 2, 1, 0, 1, -1, 2, -3, 3, 0, 0, -2, 1, -1], [-2, 0, -1, 4, 1, -2, 2, 0, 0, 0, 2, 1, -3, -3, 0, -3], [0, 0, 3, -1, 0, 0, -1, 1, 0, 2, -3, -1, 0, 3, 1, 3], [3, -1, -1, 2, 1, 0, -1, 0, -2, 3, 3, 1, -2, -1, -2, 0], [1, 0, 1, 2, 2, 0, 2, 0, 3, 1, -3, 1, -2, -1, 3, 0], [1, -2, -2, -3, -1, -3, -1, 1, -1, 0, -1, -1, -1, 3, -2, -1], [0, 0, 1, -5, 0, 0, -5, 5, 0, -1, -1, -5, 0, 1, -1, 1], [3, 1, 2, -3, -3, -1, -3, -2, -1, 3, -2, -1, 3, 1, -1, 0], [0, 0, -2, 1, 0, 0, 0, -1, 0, -2, -1, 3, 0, -1, 1, -1], [-5, -2, -3, -1, 0, -1, 0, 1, -3, 0, 0, 0, 0, 2, -2, 0], [-2, 2, -3, 4, 2, 5, 1, -5, -1, 0, 6, 2, -2, -7, 2, -2], [6, -5, -1, -1, -1, 0, -3, 1, -4, 6, 4, -1, 1, 1, -4, 0], [0, 1, 1, -3, -3, 4, 3, -5, 4, 0, -2, 3, 3, -3, 4, 0], [2, 2, 4, 2, 1, 0, 2, 1, 4, 0, -3, 1, 0, 0, 2, 0], [1, 3, -1, 3, 3, -2, 3, 1, 2, 1, -1, 0, -3, 0, 2, 0], [0, 4, 3, 3, 1, 4, 2, -1, 5, 0, -6, 4, -3, -4, 5, 0], [0, 0, -3, 3, 0, 0, 2, -3, 0, -5, 4, 1, 0, -2, -4, -2], [1, -1, -6, 3, 0, 3, 3, -4, 0, 0, 5, 0, -1, -7, -1, -1], [3, 2, 6, 0, 0, 2, -1, 2, 4, 0, -3, 0, 4, 2, 2, 4], [0, 6, 1, 3, 3, 0, 1, 0, 6, -2, -1, 1, 0, 0, 2, 3], [-4, -2, 0, 1, -1, -4, 0, 3, 0, 0, 0, -1, 0, 1, -2, 0], [0, -3, 2, -6, -4, 0, -3, 2, -3, 1, -3, -2, 0, 3, -1, -1], [0, -2, 1, -3, -1, 0, -9, -1, -8, 0, 8, -1, 3, 1, -8, 0], [0, -1, -4, 5, 6, 0, 0, 1, -1, -4, 8, -4, 0, -5, -2, 1], [0, -3, -5, 1, -1, 0, 2, -2, -3, -2, 7, 0, 0, -5, 0, -6], [-1, 2, -1, -4, -7, -1, -4, -1, -5, -1, 1, -2, 4, 2, -5, 0], [0, 4, 4, 3, 1, 0, 5, -2, 4, -3, -5, 6, 0, 1, 4, 2], [-1, -4, -6, 2, -1, -2, 0, 1, -5, 0, 5, -1, 0, -2, -4, 0], [0, -3, -7, 7, 1, 0, 6, -6, -3, 0, 9, 4, 0, -7, -3, -6], [1, 0, 3, -6, -2, -2, -1, 0, 2, 1, -5, -4, 6, 4, 2, 0], [-6, -3, -12, 3, 0, 0, 2, -6, 0, 0, 10, 0, -6, -9, -3, -6], [4, 0, 2, -8, -3, 0, -12, -1, -7, 0, 5, -3, 2, 4, 0, 2], [4, 5, 0, 5, 5, 1, 3, -2, 3, 4, 1, 4, -5, -4, 3, 0], [0, 6, -1, 5, 4, 0, 1, -1, 6, 3, 3, -1, 0, -1, -2, 3], [4, 3, 5, -3, 0, 5, -7, -1, -2, 0, 2, 0, 4, 2, 3, 4], [-10, -2, -7, 9, 4, 3, 5, 1, -1, 0, 5, 4, 0, -4, -2, 0], [0, -3, 0, -9, -7, 0, -5, 2, -3, -11, -4, -1, 0, 3, -3, -4], [1, -15, -5, 0, -7, -4, -1, 4, -6, 1, 0, -4, 0, 4, -6, 0], [1, -7, -6, -2, -15, -7, -4, -3, -10, 1, 0, -3, 2, 3, -10, 0], [0, -3, -9, 7, 1, 0, 1, -6, -3, -3, 11, -1, 0, -4, -6, -3], [4, 7, 10, 0, 2, 2, -2, -2, 0, 0, -3, 2, -2, 0, 7, -2], [0, 6, 2, -1, -1, 0, 1, 0, 6, -6, -3, 2, 0, 1, 8, 0], [-8, -2, -2, 1, 1, -4, 5, 3, 3, -8, -4, -3, -1, 3, 3, 0], [6, -5, 0, -2, -4, -9, -9, 6, -8, 0, 6, -4, 1, 4, -5, 1], [0, 1, -6, 11, 7, 0, 5, -4, 1, 5, 8, 3, 0, -7, -1, 0], [-5, 5, 3, 0, 4, 3, 2, 5, 5, -5, -5, -1, 0, 1, 5, 0], [-3, 2, -4, 13, 8, 3, 9, 0, 8, 0, 2, 8, -5, -5, 2, -5], [-4, 1, -6, 10, 6, 6, -3, -2, -2, 0, 14, 6, -2, -6, 1, -2], [6, 9, 9, 4, 16, 6, -5, 6, 4, 6, 1, 2, -4, -2, 4, 0], [4, -6, -5, 3, 3, -7, 3, 0, -2, 4, 5, -2, -3, 2, -2, 0], [0, 4, -5, 10, 1, 0, 7, -9, 4, -9, 2, 10, 0, -3, 0, -2], [0, -6, -13, 7, 5, 0, 0, -2, -6, 4, 19, -6, 0, -11, -4, -6], [3, -1, -6, 3, 2, 3, -8, -2, 0, 0, 14, 2, -1, -3, -1, -1], [0, 8, 3, 4, 4, 0, 4, 0, 8, -1, -8, 9, 0, -1, 5, 3], [-4, -13, -12, -2, -13, -12, 3, 0, -9, -4, 3, -7, 2, 7, -9, 0], [-12, -7, -2, 1, 2, -7, -5, 4, -7, -12, 5, -5, -1, 5, -7, 0], [11, -7, -6, 0, -9, -4, -2, 0, -8, 11, 3, -2, 0, 2, -8, 0], [0, -3, 3, 3, 0, 0, 4, -3, -3, -7, 0, 1, 0, 2, -1, 2], [0, -9, 5, -10, -5, 0, -10, 5, -9, 3, -14, -1, 0, 10, -5, 5], [-1, 0, 2, 1, 4, 2, 7, 1, 9, -1, -7, 1, -1, -1, 9, 0], [-7, -3, -8, 7, 1, -3, 7, 0, -1, -7, 5, 2, -7, -2, -1, 0], [-9, -5, -2, 2, -1, -15, 6, 7, 1, 0, -6, -1, -7, 4, -5, -7], [12, 19, 7, -3, 1, 5, -1, -10, 6, 12, -3, 6, 3, -6, 6, 0], [0, 5, -12, 8, 2, 0, 3, -6, 5, -3, 11, 4, 0, -9, -5, -7], [0, -7, -5, -2, 0, -7, 3, 9, -2, 0, -7, 0, 2, 11, -7, 2], [0, 3, 8, 0, -2, 0, 4, -2, 3, 6, -7, 3, 0, 6, 5, 4], [13, 11, 8, 5, 10, 9, -1, -4, 7, 13, -1, 9, -5, -9, 7, 0], [1, -4, -3, -2, -2, -5, 0, 3, 3, 0, -1, -2, 0, 3, -4, 0], [0, -14, 5, -10, -9, 0, -9, 1, -14, -13, -9, -5, 0, 13, 1, 4], [0, 1, 0, -5, -6, 0, 1, -1, 1, -3, 0, 1, 0, 0, 4, -6], [0, 7, 14, 0, 1, 0, 8, 1, 7, -1, -18, 12, 0, 5, 4, 6], [-5, 17, 3, -1, 6, 8, 9, -7, 12, -5, -1, 7, 1, -7, 12, 0], [-5, -10, -4, -16, -14, -12, -9, 2, -7, 0, 1, -14, 4, 4, -10, 4], [4, 4, -2, 13, 9, 9, 9, -1, 10, 0, 1, 9, -1, -5, 4, -1], [3, -14, -13, -5, -14, -7, 6, -4, -7, 3, 7, -4, 5, 4, -7, 0], [0, 20, 2, 5, 5, 1, 8, -6, 10, 0, -6, 6, -5, -6, 10, 0], [-1, -3, 5, -14, -9, -5, -9, 2, 4, 0, -4, -9, 6, 7, -3, 6], [-2, -12, -8, 10, 3, -1, -4, 11, -12, -2, 12, -1, -10, 1, -12, 0], [-2, -11, -5, -7, -11, -12, -7, 5, -12, -2, 1, -12, 7, 12, -12, 0], [0, -7, -9, 4, -7, 0, 0, -11, -7, -9, 7, 2, 0, 2, -6, -5], [0, -11, -14, -4, -6, -12, -2, 2, -8, 0, 7, -6, -4, 0, -11, -4], [4, 20, 6, -3, 9, 1, 2, -4, 8, 4, -1, 1, 3, -1, 8, 0], [-10, -6, -8, -2, -4, -7, 2, 0, -3, 0, 2, -4, -3, -2, -6, -3], [0, 2, 2, 0, 5, 0, 0, 5, 2, -10, -4, 2, 0, -3, -2, 2], [0, -7, 8, -18, -14, 0, -8, 4, -7, -7, -16, 0, 0, 12, -1, -2], [-1, 3, 9, -10, -5, -1, -3, -2, 3, 0, -8, -5, 2, 3, 3, 2], [0, -12, 2, -14, -10, 0, -12, 4, -12, 3, -7, -7, 0, 10, -5, 0], [-3, 4, 11, 0, -2, -1, -2, 1, -1, 0, -3, -2, 2, -1, 4, 2], [-4, -9, -20, 10, -1, -1, 5, -3, -9, 0, 17, -1, -3, -14, -9, -3], [-5, -7, -21, 22, 7, -2, 19, 0, 9, 0, 10, 7, -9, -15, -7, -9], [5, 7, 2, -6, -4, 1, 4, -11, 6, 5, -3, 3, 6, -3, 6, 0], [5, -1, -2, 2, 1, 4, 1, 1, 11, 0, 1, 1, 4, 0, -1, 4], [0, -9, -1, -8, -1, 0, -10, 7, -9, -7, 6, -15, 0, 2, 4, 1], [20, -13, -4, -7, -11, 2, -10, 5, -14, 20, 9, -5, 7, 5, -14, 0], [0, 10, -3, 7, 6, 0, -4, -1, 10, -12, -6, 5, 0, 2, 1, 8], [5, 8, 12, -13, -1, 2, -11, -5, 5, 0, -5, -1, -2, 7, 8, -2], [-9, -17, -5, -5, -6, -2, -7, 13, -12, -9, 6, -10, 5, 10, -12, 0], [1, -6, -14, 8, 1, -4, 3, -1, -1, 0, 12, 1, -6, -8, -6, -6], [0, 4, 4, -2, -2, 0, 1, 0, 4, 5, -8, 5, 0, 3, 14, 1], [12, 14, 11, 4, 16, 16, 4, 4, 15, 12, -6, 8, -4, -8, 15, 0], [0, -10, -4, -5, -4, 0, -11, 1, -10, -12, -2, -5, 0, 6, 8, 2], [1, 0, -3, 2, 4, 5, -5, 1, -10, 0, 6, 4, 2, 3, 0, 2], [0, -7, -6, 10, 2, 0, 8, -8, -7, -2, 10, 4, 0, -6, -2, -4], [-2, -4, -8, -2, -7, -8, 2, -18, -6, -2, 13, 4, 2, -4, -6, 0], [-8, -5, 9, -9, -6, -15, 1, 13, 2, 0, -18, -6, 4, 16, -5, 4], [-12, 2, 0, 4, 1, -10, -7, 0, -7, -12, 1, -3, -4, 3, -7, 0], [0, 4, 14, -2, 9, 0, 0, 11, 4, -7, -5, -9, 0, 3, 9, 12], [7, 7, 17, 7, 5, 0, 12, 8, 19, 0, -20, 5, 3, 6, 7, 3], [0, -13, -10, -5, -10, 0, -10, -5, -13, 1, 7, -7, 0, 5, -6, -5], [5, 11, 0, 14, 13, -1, 6, 5, 6, 5, -3, 4, -14, -4, 6, 0], [0, 1, 9, -12, -1, 0, -4, 11, 1, -6, -6, -7, 0, 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