# q-expansion of newform 1445.4.a.c, downloaded from the LMFDB on 25 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 = [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 = [[-3], [2], [5], [5], [17], [1], [0], [-76], [-7], [86], [178], [-6], [266], [-138], [305], [-197], [213], [-112], [-794], [1190], [982], [-162], [-718], [-107], [714], [-586], [-768], [756], [-1076], [-582], [1256], [1157], [-894], [1876], [2608], [-1172], [2117], [736], [-1389], [3763], [-771], [3080], [2938], [-3324], [1885], [-4916], [-749], [4597], [1294], [-258], [-516], [2190], [-1193], [4169], [6024], [-4219], [162], [6210], [1331], [2269], [3922], [8353], [7700], [3242], [6652], [8650], [3247], [-826], [-6928], [12094], [-5578], [-3528], [6979], [9533], [-6820], [-712], [10746], [6675], [12897], [6349], [-11805], [6798], [1376], [12772], [5538], [-7236], [-17010], [-5822], [2180], [17929], [-4716], [4316], [-5812], [-19677], [15721], [19907], [-21610], [-4662], [-21326], [-14064], [-4180], [-22919], [-3030], [6162], [-8273], [8088], [3806], [-19254], [-13416], [5673], [9559], [-7013], [-13428], [-27211], [-14002], [26679], [21928], [-6805], [29793], [-1660], [-26548], [8968], [-30435], [30378], [3763], [12312], [8968], [-10400], [13475], [-2838], [14985], [-83], [-10566], [35099], [-6711], [-33625], [26826], [516], [-8645], [26598], [695], [-25824], [-3388], [-24670], [-3754], [16818], [-17053], [-20530], [18244], [-47935], [965], [-25695], [34148], [-14736], [-24056], [23930], [41842], [-3618], [35798], [-11444], [30548], [29556], [36392], [1468], [-45354], [17912], [-53560], [33529], [-51309], [37794], [-40089], [39128], [38790], [-52382], [42708], [-2655], [-16417], [-68298], [3809], [18240], [30952], [-8479], [30387], [-6910], [66705], [53496], [-13439], [13178], [39666], [70854], [26102], [4882], [36167], [-2086], [-20236], [40398], [52433], [-25567], [-54348], [-76561], [35028], [-81096], [-41851], [-45359], [3691], [-25231], [53796], [12076], [-43206], [42735], [50654], [-68286], [-61848], [14484], [-49486], [-38466], [43861], [-24279], [-27079], [-36930], [-56576], [10514], [-91321], [-46336], [-72028], [34810], [-36188], [-6822], [14820], [3359], [53654], [33019], [-9044], [-61263], [-2322], [7317], [49797], [-11886], [79588], [9690], [-90074], [-33637], [17936], [49966], [81390], [23226], [-55339], [-67233], [-117125], [-26825], [-50141], [64535], [66008], [54095], [88287], [104343], [55600], [110054], [43219], [-22078], [56039], [-83942], [-31744], [25327], [-86524], [7361], [-53070], [-122997], [77306], [-80202], [-6288], [-99388], [-128576], [-103510], [42420], [80416], [44352], [-53608], [128858], [-31481], [-24340], [-105878], [-141189], [-118688], [21090], [7762], [-81476], [111397], [20362], [42023], [21982], [-55650], [-10092], [-75551], [-34742], [62926], [48140], [12685], [89040], [111424], [-59922], [-59063], [159154], [116252], [-39868], [81098], [-20998], [-39592], [-1154], [4657], [18528], [-143860], [-54559], [107878], [44880], [33077], [-4851], [-115798], [-119306], [-26241], [74765], [-90964], [22870], [73517], [125418], [-89665], [-167149], [162322], [-98936], [-61065], [77656], [14209], [23774], [56926], [112300], [-42050], [94248], [-120480], [-76963], [72039], [-193166], [-102490], [46674], [131730], [-107225], [-133095], [-91580], [-144646], [140068], [104614], [-19824], [-121186], [30774], [-212768], [19468], [128086], [50688], [145485], [149552], [-27581], [191302], [123803], [67960], [-180765], [76286], [-24048], [17494], [73008], [-185610], [-23052], [122492], [-108797], [-217948], [-208074], [155762], [-99908], [160776], [-32894], [46921], [-245468], [157470], [-247966], [98659], [225696], [-159049], [-49511], [-83324], [-15881], [-245340], [-147535], [-179230], [-25811], [265644], [24364], [229338], [-2572], [-254006], [-50009], [-95177], [-163740], [-196748], [6698], [-85743], [-235786], [92754], [119994], [151258], [219617], [-72834], [136155], [131538], [36698], [294414], [-141532], [-124286], [265414], [89472], [29347], [270988], [192192], [-54842], [118510], [69651], [147888], [33978], [55897], [76380], [-130162], [277200], [-80192]]