# q-expansion of newform 1445.4.a.i, downloaded from the LMFDB on 24 August 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 = 1445
weight = 4
poly_data = [-3, 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 = [[-2, 1], [1, -1], [-5, 0], [-1, 9], [19, -13], [-40, -28], [0, 0], [-90, 6], [21, -101], [108, -46], [-35, 141], [-176, -106], [-172, -142], [-44, 166], [-174, 8], [-82, 64], [-718, -66], [38, 268], [-22, -308], [755, 119], [-544, 18], [479, -437], [124, -238], [-102, -742], [-506, 436], [282, 410], [606, -8], [-359, -237], [-686, 176], [108, 562], [-1252, 462], [339, 151], [1708, -132], [373, -759], [1458, 560], [-234, -978], [1650, 932], [3269, -21], [109, 1927], [1620, 382], [954, -434], [1516, 1222], [-1708, 1320], [24, -902], [1900, -642], [765, -751], [2677, 569], [-1952, -530], [-2155, -1361], [2962, -1954], [-2854, 528], [4480, -740], [-420, -1406], [-2552, -1940], [-500, -1392], [-236, -318], [3422, -2124], [-2428, 468], [1530, -1820], [2610, -1908], [-2295, 3383], [2694, -180], [1442, 3668], [2427, -1745], [988, -2474], [3650, -4092], [-5486, -934], [3546, 2556], [-367, -629], [3574, -4132], [3214, -1956], [4082, 142], [-9485, 453], [-2144, -2924], [6455, 1023], [8052, -850], [-1114, 6510], [1346, 2392], [2442, -4812], [1546, 1600], [9993, 1369], [-1942, 8094], [-3505, -3153], [-2560, 1308], [1839, -4577], [-18, -6624], [4212, -3178], [9516, -568], [15262, -1956], [3162, -4532], [-12092, 1214], [-10035, 2609], [-6217, 1057], [-3298, -5898], [15113, -3111], [-21, -7199], [-7858, -1448], [4206, 3800], [8582, 6100], [12716, -4162], [-8447, -5085], [-15052, 1952], [-9968, -3114], [18238, -3660], [-15015, 1757], [-13440, 7032], [-24828, -774], [-20490, 1400], [1228, 10448], [1676, -1238], [-6693, -4291], [-15830, 2712], [6732, 7826], [-5329, -1365], [14414, -6622], [2316, -3170], [-14371, 579], [-6326, -12072], [-15400, -4574], [-10724, 4236], [-158, 1864], [8312, 8190], [14226, -3996], [30321, 2751], [-3989, -3605], [12546, -8122], [4716, -4758], [8851, 6355], [10962, -5332], [-16166, 8880], [-11618, -2722], [-9285, 4857], [-11557, -15649], [-232, 5248], [-17706, -12274], [5370, 12538], [-13632, 7128], [30435, 925], [-3938, -3396], [7474, 2208], [15005, 10653], [9346, 19252], [-32471, 7735], [5163, 17821], [10654, 2928], [21469, 5621], [-8908, 8562], [-14404, -15874], [5910, 12062], [16634, 8064], [-6922, 9380], [-16404, -14550], [-446, -15708], [-17201, 4465], [-9027, -2321], [-20803, -10919], [-22784, -5736], [-20740, 16474], [-33158, -1860], [-24412, -9406], [35467, 1437], [15552, -12800], [3584, 11662], [-158, -10298], [-26882, -6528], [6753, -26065], [56143, -2573], [7028, -6610], [-4746, -10372], [50644, -3990], [8364, -136], [-13262, -3574], [-9945, 1919], [47968, 8080], [-42938, 2274], [-26254, 2476], [33381, 13069], [-22344, -6566], [4768, -24666], [18662, 10470], [20670, 12876], [29745, 14577], [-16794, 3716], [31362, -7152], [-14124, -12494], [40966, 10518], [8710, 7772], [3690, 30092], [-464, 10450], [29029, 2877], [35622, 5040], [-31013, -6303], [-28006, 26714], [-42014, 6426], [7531, -13507], [-16396, 13590], [41612, 15430], [8974, 9712], [2640, -42078], [8214, -7052], [-29616, 6714], [28855, -11257], [-25042, -16484], [-37238, 6078], [10904, 14652], [-2304, -40808], [2524, -40748], [-284, -49074], [-14842, -16896], [60748, 12652], [15094, -11588], [-5350, 5122], [-23859, -13701], [-1596, -12714], [47091, 18639], [44024, -20154], [5178, 16616], [14378, -3352], [17313, 3219], [63772, 20640], [-36306, 5152], [57539, -1181], [870, -42352], [-39551, -18249], [14334, 19724], [-46418, 7330], [-82490, -7896], [-54739, 18565], [63052, 16562], [36313, 85], [17582, 44940], [23691, -6981], [74298, -2662], [-2942, -27182], [-202, -16500], [-36976, 26622], [1124, 31034], [-53874, -5436], [64223, -16905], [-34626, 38738], [26195, 28265], [93160, 104], [63050, -21972], [-10310, 12438], [-56626, 17900], [-22701, 19531], [-51045, 18253], [7065, 15245], [9590, -14922], [44320, 35350], [-50470, -11660], [13538, -41296], [-22836, -10846], [-11816, 57294], [-70656, 29336], [10316, -55952], [1924, -9366], [-199, 27395], [44160, 21850], [103696, 10052], [63017, 8719], [19786, -16032], [73040, 16878], [-45618, -21304], [39118, -14708], [9972, 28784], [48058, -50276], [-3802, -19728], [30605, -20245], [-40366, 9764], [-49816, 26846], [20766, -5664], [10426, -42172], [72746, 36974], [-25042, 34768], [-42500, -24814], [-20248, -22578], [-43418, 15942], [-16978, 49210], [878, -52754], [5770, 21664], [61707, 22887], [-75771, -18425], [64246, 14044], [-123711, -14381], [-60564, -4924], [8388, -35702], [8204, -5998], [-86638, 19526], [3174, 48678], [-12150, 23268], [-32383, -34525], [3772, -40608], [-11309, -7765], [46934, 55532], [-11362, -11684], [17572, 47668], [-63952, -42292], [-34697, -34525], [32252, -45730], [-48940, -7620], [-39348, -41708], [23855, 34715], [29151, -31835], [78335, -37349], [-3482, -33084], [-27914, 47816], [45608, 54978], [-4364, 32680], [-2706, -52576], [23105, -10553], [47520, -30014], [26242, -88848], [45508, 10206], [-50330, -73556], [140894, -12280], [84606, -50874], [78647, 57443], [128822, 13868], [-10646, -61616], [56851, -22317], [67518, -60704], [15426, -25000], [-59070, 29204], [64362, -49476], [87670, -20196], [9709, -45463], [-98339, 20059], [-18959, -27341], [70986, -3876], [-96902, -11828], [-130356, 9402], [-81189, -17157], [70170, -41740], [-11119, 101085], [62007, 22433], [36966, -44016], [-12668, 1026], [51980, -82074], [62076, -25254], [115262, 14196], [3006, 64332], [2918, 76940], [-74936, -2396], [86844, 54048], [-47375, -70399], [-99986, 54080], [41726, -63780], [10397, 3829], [31228, 18170], [100442, -16658], [32730, -75792], [24742, 14048], [33709, -4381], [28162, -12362], [12780, 6940], [-43278, -48018], [-150359, 9113], [-70476, 8262], [-81024, 19086], [-18436, -13998], [-29778, 4084], [19198, 79232], [177009, -17415], [34760, -41634], [-13258, -108556], [-1066, 44968], [-7466, 34252], [-225320, 12826], [-40122, 99126], [-55605, -14641], [-23583, -14497], [-35634, -61054], [31636, 43256], [-165296, 8490], [46929, 84565], [84537, -80875], [138368, 53388], [-18486, 54292], [-62110, 65216], [-61892, -65666], [213802, -8148], [-151209, -9663], [-50586, 66204], [94085, 70045], [152609, 59247], [-129534, 32662], [-61788, 36780], [-96953, 3965], [49538, -25012], [102224, 101362], [-31986, 126984], [76924, 14436], [-88482, 9292], [-49530, -62482], [-126504, -82950], [51224, -57948], [-39994, 112288], [-196983, 18285], [-138526, 49166], [-122342, -17928], [-19136, 124144], [-5738, 126072], [-224494, 14260], [62362, -109984], [208013, -24107], [20012, 62220], [9342, 89974], [102180, -46506], [77077, -34831], [142070, 46360], [147984, 18848], [19090, 13040], [14527, -59837], [50170, -26276], [-112848, 1050], [-67743, 28761], [-43897, -3639], [147154, -57304], [-58170, -72616], [-16754, -5736], [62814, 61232], [124193, -60869], [-242518, -10866], [-47186, -27320], [-192834, 9168], [123871, -17867], [206208, 12990], [18240, 45208], [78165, -55075]]