
# q-expansion of newform 405.3.l.m, downloaded from the LMFDB on 13 March 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, 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 = 405
weight = 3
poly_data = [28561, 0, 0, 0, 235936, 0, 0, 0, 15630, 0, 0, 0, 256, 0, 0, 0, 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], [[51693551, 67695680, -141865857, 0, 22110205, 0, -13061075, 0, 583505, 0, -250015, 0, 2691, 0, -1037, 0], 135391360], [[0, 0, 634533683, 0, 0, 0, 38949635, 0, 0, 0, 630165, 0, 0, 0, 2453, 0], 220010960], [[-1029994667, -3394588431, 2952657455, 0, -55782675, -280768605, 194748175, 0, -609245, -4332705, 3150825, 0, -2197, -16211, 12265, 0], 880043840], [[1029994667, -3394588431, -2952657455, 0, 55782675, -280768605, -194748175, 0, 609245, -4332705, -3150825, 0, 2197, -16211, -12265, 0], 880043840], [[-1470016587, -6250453144, -4221724821, 0, -55782675, -222539980, -272647445, 0, -609245, -2301000, -4411155, 0, -2197, -7124, -17171, 0], 880043840], [[672016163, 0, 1844256141, -2538134732, 287432665, 0, 169793975, -155798540, 7585565, 0, 3250195, -2520660, 34983, 0, 13481, -9812], 1760087680], [[-672016163, 0, -1844256141, -2538134732, -287432665, 0, -169793975, -155798540, -7585565, 0, -3250195, -2520660, -34983, 0, -13481, -9812], 1760087680], [[-3447962505, 0, -9966373679, 2538134732, 64301965, 0, -609198725, 155798540, 5148585, 0, -9353105, 2520660, 26195, 0, -35579, 9812], 1760087680], [[11440569920, -23975329833, 0, -68431243099, 0, -2207321675, 0, -4682276425, 0, -42252535, 0, -77850245, 0, -175253, 0, -305879], 22881139840], [[0, -23975329833, 32995751516, 68431243099, 0, -2207321675, 2025381020, 4682276425, 0, -42252535, 32768580, 77850245, 0, -175253, 127556, 305879], 22881139840], [[-78929397833, -79545549356, -114298469987, 233419796594, -11028929015, -5857313540, -9077325725, 15033968390, -209494935, -98577700, -157214525, 243107710, -854633, -385996, -625027, 945194], 22881139840], [[-78929397833, 79545549356, -114298469987, -233419796594, -11028929015, 5857313540, -9077325725, -15033968390, -209494935, 98577700, -157214525, -243107710, -854633, 385996, -625027, -945194], 22881139840], [[40708966571, -81255890872, -224063315333, -227521685930, 9578579465, -2893019740, -16166159295, -15284306650, 193654565, -29913000, -271904555, -251349590, 797511, -92612, -1071473, -982950], 22881139840], [[-13389930671, 0, -38384546915, 126094691315, -725174775, 0, -2531726275, 8576649205, -7920185, 0, -40960725, 140730765, -28561, 0, -159445, 549515], 11440569920]]

hecke_ring_character_values = [[326, [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0]], [82, [0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]]]
aps_data = [[0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 1, 0, 0, 0, 1, 1, 1, 1, 1, -1, 0, 0, -1, -1], [-4, 0, 0, 0, 0, 0, 0, 2, 4, 1, 4, -4, 0, 0, 0, -1], [-2, -1, 0, 4, 1, 0, 0, 1, 2, 1, 2, -2, -2, 0, -2, -1], [2, 3, -1, -1, 0, -1, 2, -1, -3, -2, 1, 3, 0, 0, 2, 1], [6, -4, 0, 6, 2, 2, 2, 0, -1, -1, -10, -10, 0, 0, 0, 0], [1, 3, -6, -7, -5, 5, -1, 2, 0, 4, -1, 1, 1, 1, 2, 1], [-8, 0, 0, -4, 0, 0, 0, 5, 10, 10, 4, -4, 0, 0, 0, -5], [15, -6, 5, 0, -2, 3, 0, 7, 1, -1, 15, -1, 1, 0, -1, -2], [0, -5, 1, 0, 0, 3, 0, 5, 4, 0, -14, 0, 0, 0, 0, -3], [-1, 0, 0, 3, 2, -2, 0, 2, 1, 3, 0, 0, -4, -4, 0, 0], [0, -2, -3, 7, -1, 3, -5, -2, -4, 1, 0, 17, 0, 5, 5, 2], [-9, 10, 0, -4, -10, 0, -8, -2, 0, 2, 5, 9, 0, 0, 4, 2], [-8, 3, -13, 10, -10, 20, 0, 3, 6, 3, -2, -8, 0, 0, -6, -3], [-14, -4, 0, 10, 2, 2, 0, 17, 18, 1, 28, -20, -4, 4, 0, -2], [0, 3, -13, 0, -13, 10, 7, -3, -3, -13, 7, 0, 0, 7, -7, 3], [-5, -6, -12, 0, -9, 3, 6, 3, 3, -12, 11, 3, -3, 6, -3, 9], [6, 9, 0, 2, -6, -3, 0, 29, 9, 0, -10, -8, 12, -6, 0, 11], [-4, -21, 0, -32, 6, 6, -4, -17, -19, -2, 4, 60, -4, 4, 0, 8], [25, -12, 24, -25, 9, -9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [4, 5, -11, 8, -12, -4, 8, 1, -2, 8, -4, -32, 4, -8, -4, -12], [9, 7, 0, 4, -7, 0, 0, 11, 0, -3, -5, -9, 14, 0, 0, -3], [17, 25, 0, 0, -2, -2, -3, -22, -21, 1, -37, -3, 3, -3, 0, 1], [-16, -9, 0, -6, 3, 6, 0, -11, -22, -29, 22, -4, -6, 12, 0, 29], [4, -8, 3, -24, -9, 14, 0, -12, -4, 5, -4, 12, 4, 0, 4, 9], [-2, 6, 5, 54, 0, -16, -2, 1, 3, 2, -54, -56, 0, 0, -2, -1], [19, 18, 0, 19, -3, -3, 14, 0, -7, -7, -24, -24, 0, 0, 0, 0], [1, -1, 2, 116, 19, -19, -1, -2, -20, -20, -1, 1, 1, 1, 2, 1], [-84, 9, 0, -33, 0, -9, 0, 16, 52, 32, 33, -51, 0, -18, 0, -16], [66, 0, 0, 50, -8, 8, 0, 9, -6, 3, 0, 0, 16, 16, 0, 0], [0, 15, -21, 19, -11, -4, 19, 15, 5, 11, 0, -19, 0, -19, -19, -15], [-32, -4, 20, 34, 16, -32, 0, 15, 30, 15, -2, -32, 0, 0, -30, -15], [34, 22, 9, -13, -17, -23, 17, -12, -20, -17, -34, 13, 34, -17, 17, -6], [218, 5, -10, 0, 1, -6, -11, -5, 6, 1, -120, 0, 0, -11, 11, 5], [4, -26, -23, 0, 3, 0, 0, 26, 52, 3, -4, 0, 0, 0, 0, -3], [0, 0, 0, -91, 0, 0, 0, 34, 19, 0, -91, 91, 0, 0, 0, -4], [-44, -10, 20, 44, -6, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [70, 15, 2, -34, -30, 15, 2, 1, 1, 0, -34, -36, 0, 0, -2, -1], [-37, 1, 0, -76, -1, 0, 0, -16, 0, 9, -39, 37, 2, 0, 0, 9], [0, -23, 0, 0, 13, 13, 21, 2, -6, -8, 21, 21, -21, 21, 0, -5], [-100, 9, -18, 0, -19, 19, 8, -1, 6, -11, 8, -8, 8, 8, -16, -8], [-8, -39, 45, 52, 14, -20, 0, -31, 8, -6, 8, -26, -8, 0, -8, -14], [16, 29, 4, 57, 0, -37, 16, -8, -24, -16, -57, -41, 0, 0, 16, 8], [27, 21, 0, 27, -33, -33, -22, 0, 11, 11, -76, -76, 0, 0, 0, 0], [4, -17, 34, -124, 19, -19, -4, -21, -10, -23, -4, 4, 4, 4, 8, 4], [0, 20, 5, 0, -7, -23, -7, -6, -4, 7, 129, -14, 14, -7, -7, 16], [70, 13, -18, -26, -31, 0, -52, -13, 0, 13, -96, -70, 0, 0, 26, 13], [-3, 4, 14, 42, 18, -36, 0, -18, -36, -18, -39, -3, 0, 0, 36, 18], [-6, -16, -23, -90, 3, 59, -3, -10, -30, 3, 6, 90, -6, 3, -3, 56], [7, 2, 0, -5, -1, -1, 0, 16, 6, -10, -14, 10, 2, -2, 0, 20], [52, 32, -52, 0, -14, -18, -50, -32, 6, -14, -76, 0, 0, -50, 50, 32], [205, 35, 17, 0, -11, -7, -14, -28, -49, -4, -219, -7, 7, -14, 7, 11], [-31, 13, 0, -25, 7, 7, -31, 44, 68, 24, 31, 19, -31, 31, 0, -17], [-158, -17, -10, 83, 34, -17, 8, 4, 4, 0, 83, 75, 0, 0, -8, -4], [-78, -17, 0, -122, 17, 0, 0, 77, 0, 0, -44, 78, -34, 0, 0, 0], [73, 48, 0, 0, -33, -33, 3, -51, -87, -36, -143, 3, -3, 3, 0, 69], [2, 38, -76, 0, 21, -21, -36, -74, -13, -15, -36, 36, -36, -36, 72, 36], [-88, 0, 0, 0, 0, 0, 0, -30, -60, 31, 88, -88, 0, 0, 0, -31], [-8, -7, 42, -174, 43, -78, 0, 1, 8, -35, 8, 87, -8, 0, -8, -43], [-22, -91, 52, -43, 0, -13, -22, 11, 33, 22, 43, 21, 0, 0, -22, -11], [-210, -6, 0, -111, 0, 6, 0, 40, 151, 80, 111, -99, 0, 12, 0, -40], [21, 0, 0, 23, 1, -1, 0, 46, -85, -39, 0, 0, -2, -2, 0, 0], [0, 22, 24, -58, -16, -6, 28, 22, 62, 16, 0, -114, 0, -28, -28, -22], [-110, 65, 10, 28, -55, 0, 56, 14, 0, -14, 138, 110, 0, 0, -28, -14], [29, 8, -55, -16, -47, 94, 0, -21, -42, -21, -13, 29, 0, 0, 42, 21], [118, 11, -16, 0, 5, -32, -64, 21, 74, 37, -182, -32, 32, -64, 32, -5], [-28, -42, 0, 87, 28, 14, 0, -34, -48, 0, 143, -59, -56, 28, 0, 62], [202, 9, 44, -112, -18, 9, -22, -11, -11, 0, -112, -90, 0, 0, 22, 11], [-14, 42, 86, -28, 30, 14, -28, 56, 126, -16, 14, -38, -14, 28, 14, 30], [-87, -26, 0, -122, 26, 0, 0, -32, 0, -20, -35, 87, -52, 0, 0, -20], [10, 22, 0, 0, 7, 7, -18, -4, 21, 25, -38, -18, 18, -18, 0, -32], [-56, 24, 0, 16, -8, -16, 0, -16, -32, 34, 40, -88, 16, -32, 0, -34], [46, 9, -34, -60, 0, 59, 46, -23, -69, -46, 60, 106, 0, 0, 46, 23], [5, -33, 66, -71, -16, 16, -5, -38, 39, 11, -5, 5, 5, 5, 10, 5], [188, 18, 0, 112, 0, -18, 0, -11, -38, -22, -112, 76, 0, -36, 0, 11], [22, 50, -49, 0, 3, -4, 0, -52, -2, 1, 22, 2, -2, 0, 2, 3], [165, 0, 0, 191, 13, -13, 0, -4, 22, 18, 0, 0, -26, -26, 0, 0], [0, 44, -168, -77, -84, 40, 4, 44, -40, 84, 0, -85, 0, -4, -4, -44], [-68, -84, 14, -31, 34, 22, -34, -16, 32, 34, 68, 31, -68, 34, -34, -12], [-8, -22, -52, 0, 4, 18, 40, 22, 78, 4, 44, 0, 0, 40, -40, -22], [97, -70, 38, 0, 59, 49, 98, 21, -7, 10, 1, 49, -49, 98, -49, -59], [-34, 117, 0, 14, 3, 3, -34, 151, 182, 31, 34, -62, -34, 34, 0, -28], [-67, -112, 224, 67, 108, -108, 46, 46, 69, 23, 46, -46, 0, 0, -92, -46], [0, -48, -19, 0, 29, 0, 0, -48, -96, -29, 0, 408, 0, 0, 0, 29], [-4, 70, 0, -148, -70, 0, 0, 44, 0, 24, -144, 4, 140, 0, 0, 24], [-54, 17, 0, 0, -19, -19, 3, -20, -42, -22, 111, 3, -3, 3, 0, 41], [90, -87, 0, -58, 29, 58, 0, -148, -296, -134, -32, 206, -58, 116, 0, 134], [-32, -14, -30, 296, -12, 56, 0, 18, 32, 44, 32, -148, -32, 0, -32, 12], [-66, -34, 37, -290, 0, -40, -66, 33, 99, 66, 290, 224, 0, 0, -66, -33], [-28, -126, 0, -28, 12, 12, -184, 0, 92, 92, -128, -128, 0, 0, 0, 0], [4, -143, 137, 0, 8, -2, 0, 129, -14, -6, 4, 14, -14, 0, 14, 8], [80, 0, 0, 162, 41, -41, 0, -27, 87, 60, 0, 0, -82, -82, 0, 0], [0, -27, 129, -127, 43, -16, -11, -27, 59, -43, 0, -105, 0, 11, 11, 27], [26, -29, 15, 619, -13, 12, 13, -55, -53, -13, -26, -619, 26, -13, 13, 25], [-58, -64, 0, -6, 32, 32, 0, -52, -114, -62, 116, 12, -64, 64, 0, 124], [-48, 45, -61, 0, -130, 85, 40, -45, -114, -130, 64, 0, 0, 40, -40, 45], [-43, -41, 0, -13, 10, 10, -43, 2, 35, 33, 43, -17, -43, 43, 0, -23], [-377, 44, -88, 377, 30, -30, -24, -24, -36, -12, -24, 24, 0, 0, 48, 24], [-55, 25, -50, 0, 32, -32, 11, -14, 79, 43, 11, -11, 11, 11, -22, -11], [-399, 21, 0, 14, -7, -14, 0, 131, 262, 91, 385, -427, 14, -28, 0, -91], [-153, -51, 0, -153, -36, -36, 8, 0, -4, -4, 314, 314, 0, 0, 0, 0], [216, 18, 0, 126, 0, -18, 0, -84, -2, -168, -126, 90, 0, -36, 0, 84], [247, -50, 50, 0, -40, 40, 0, 90, 40, 0, 247, -40, 40, 0, -40, -40], [0, 68, -4, 0, 14, -74, 14, -96, -106, -14, 240, 28, -28, 14, 14, 88], [271, 0, 0, 271, 0, 0, 0, -96, 21, -75, 0, 0, 0, 0, 0, 0], [-271, 152, -172, 458, -20, 40, 0, 42, 84, 42, -187, -271, 0, 0, -84, -42], [204, 134, 0, -70, -67, -67, 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