
# q-expansion of newform 8100.2.a.m, downloaded from the LMFDB on 12 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
    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 = 8100
weight = 2
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 = [[0], [0], [0], [4], [-3], [4], [0], [5], [-6], [9], [5], [-2], [9], [10], [-6], [-12], [-9], [-10], [-2], [-3], [4], [-4], [6], [9], [-2], [15], [10], [6], [11], [12], [-20], [15], [12], [-13], [18], [17], [-14], [4], [0], [-6], [-9], [-13], [-3], [-14], [-12], [-16], [5], [-8], [-24], [14], [-30], [24], [29], [-12], [-6], [6], [-21], [32], [-26], [18], [22], [30], [-2], [3], [-20], [18], [5], [-8], [24], [17], [-6], [-15], [-14], [28], [-4], [36], [18], [-2], [18], [-10], [0], [17], [-33], [34], [17], [6], [39], [10], [3], [34], [0], [-3], [16], [3], [41], [-36], [-18], [-18], [16], [-1], [-20], [-24], [-18], [27], [11], [-20], [18], [42], [3], [-19], [22], [4], [-48], [-28], [5], [-3], [-14], [-30], [48], [12], [-19], [22], [6], [-24], [-28], [9], [38], [15], [-44], [22], [11], [-48], [-40], [52], [3], [5], [6], [-32], [-12], [9], [29], [-15], [-14], [12], [-1], [-15], [4], [0], [-31], [-24], [16], [33], [40], [-42], [22], [51], [-25], [15], [-26], [-6], [-18], [-24], [-44], [-3], [-12], [-12], [-43], [28], [-7], [48], [-39], [-13], [-51], [-14], [-40], [-3], [-28], [27], [-20], [-1], [-2], [0], [-32], [-30], [-30], [-27], [10], [-20], [-37], [-36], [-50], [-18], [-28], [-39], [-18], [-36], [-7], [-14], [48], [36], [21], [-13], [16], [65], [-36], [36], [-31], [24], [18], [44], [58], [-39], [-68], [72], [21], [-7], [-50], [27], [-42], [-30], [-55], [-49], [-39], [16], [0], [35], [42], [24], [-2], [12], [16], [-61], [-40], [-30], [28], [-54], [-25], [60], [-12], [-9], [24], [8], [-50], [62], [12], [27], [22], [-12], [41], [-30], [22], [-45], [18], [-13], [66], [0], [-25], [46], [24], [-20], [-2], [-30], [17], [-44], [18], [-37], [6], [-33], [-32], [-42], [-25], [16], [28], [11], [-14], [-74], [-6], [-19], [2], [-84], [-66], [32], [-18], [-58], [-20], [45], [-2], [-42], [65], [-30], [45], [-72], [24], [-39], [-8], [39], [-7], [-36], [15], [22], [34], [-72], [47], [-42], [77], [88], [-12], [41], [51], [-8], [30], [69], [3], [10], [12], [-79], [-39], [45], [-62], [15], [-52], [-68], [-63], [-20], [-24], [-79], [65], [46], [-72], [-6], [-67], [30], [-61], [-60], [47], [-30], [47], [-6], [5], [-56], [-2], [54], [-9], [-76], [-12], [-60], [-7], [-26], [75], [-24], [47], [-44], [54], [-26], [-7], [-30], [-9], [3], [-18], [-24], [16], [51], [36], [-33], [-98], [4], [48], [46], [-79], [-51], [44], [18], [-69], [-7], [-68], [33], [33], [34], [15], [-44], [42], [54], [70], [-30], [11], [12], [-64], [-86], [28], [78], [-10], [-60], [84], [34], [27], [-50], [-31], [-30], [68], [-3], [-49], [-48], [-14], [-72], [-9], [56], [16], [78], [70], [-45], [28], [-42], [12], [-13], [-92], [-30], [9], [-38], [36], [-18], [90], [34], [36], [21], [-2], [30], [-6], [-6], [11], [0], [-25], [69], [-103], [-24], [70], [33], [65], [-7], [-8], [80], [24], [-27], [-46], [-48], [11], [-12], [94], [-18], [89], [50], [4], [-96], [78], [-6], [-26], [-66], [-37]]
