
# q-expansion of newform 3600.2.a.m, downloaded from the LMFDB on 13 September 2025.

# 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 = 3600
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], [-2], [1], [4], [-5], [-1], [-2], [8], [-10], [-6], [3], [-4], [4], [-6], [8], [10], [1], [-12], [3], [-6], [-13], [9], [-14], [-6], [-4], [-15], [14], [9], [6], [-12], [-13], [-9], [-8], [-2], [-14], [-7], [-12], [-8], [-19], [2], [-10], [-11], [2], [28], [1], [4], [-12], [4], [-6], [24], [-7], [15], [-30], [30], [16], [6], [8], [2], [1], [2], [-13], [14], [-22], [-12], [27], [19], [17], [-18], [10], [-18], [-20], [10], [5], [6], [-30], [4], [-21], [-11], [-3], [-12], [-34], [5], [-16], [-21], [39], [1], [-12], [36], [-28], [22], [38], [36], [4], [36], [-34], [35], [35], [-32], [19], [28], [-32], [-17], [20], [-13], [3], [-39], [30], [43], [0], [30], [18], [-44], [-10], [6], [12], [12], [34], [-21], [24], [-6], [32], [-3], [19], [28], [-28], [-14], [-8], [36], [-12], [30], [-24], [32], [3], [13], [0], [52], [-42], [-18], [-44], [38], [-28], [3], [-20], [-16], [-22], [3], [-41], [48], [4], [-30], [-5], [28], [-12], [28], [34], [2], [-5], [-4], [40], [39], [0], [-31], [-17], [6], [-38], [-12], [29], [16], [35], [10], [-34], [35], [16], [-39], [-7], [58], [56], [34], [28], [-27], [-58], [41], [-4], [-20], [26], [-41], [-7], [40], [47], [-9], [1], [-14], [-23], [22], [-45], [-54], [-11], [-10], [-18], [44], [32], [3], [-36], [-26], [-58], [-20], [-30], [-8], [-62], [-2], [-56], [-3], [10], [33], [-58], [-5], [-66], [-10], [-34], [20], [-25], [66], [-27], [-50], [1], [56], [-44], [35], [34], [45], [-50], [-7], [24], [-18], [45], [0], [55], [12], [37], [-19], [14], [4], [13], [-16], [-22], [68], [5], [-16], [-48], [5], [30], [-37], [-32], [75], [-40], [-75], [6], [69], [-46], [-73], [-48], [-28], [-53], [-37], [50], [-30], [51], [-34], [-4], [-43], [-33], [16], [2], [4], [13], [-40], [-34], [-3], [-60], [-30], [8], [4], [-55], [-8], [62], [24], [0], [30], [-34], [-12], [-51], [-81], [28], [18], [-2], [48], [4], [-3], [-39], [-54], [16], [65], [57], [73], [-12], [46], [-54], [38], [-26], [-66], [-9], [8], [-4], [-26], [49], [72], [-65], [29], [-44], [-67], [-36], [62], [38], [18], [65], [-20], [-70], [36], [44], [-82], [26], [1], [11], [24], [80], [-66], [55], [30], [8], [15], [2], [-20], [-70], [52], [28], [89], [-72], [42], [29], [18], [-70], [-68], [-46], [53], [-36], [-59], [-46], [44], [26], [53], [14], [-12], [21], [19], [96], [-56], [5], [-12], [55], [-4], [36], [-48], [-34], [-39], [74], [-37], [-57], [98], [70], [95], [48], [-87], [-48], [-62], [32], [0], [-96], [-14], [-5], [-84], [39], [-19], [50], [77], [-68], [-33], [-89], [30], [96], [75], [-52], [39], [-96], [48], [-62], [6], [-4], [-9], [-65], [-48], [59], [20], [31], [-30], [-26], [-10], [-81], [-42], [-24], [22], [58], [-77], [69], [24], [24], [-3], [-53], [38]]
