# q-expansion of newform 2200.2.a.q, downloaded from the LMFDB on 23 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 = 2200
weight = 2
poly_data = [-1, -1, 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, 1], [0, 0], [-1, -1], [1, 0], [3, -4], [-2, 5], [-5, -2], [1, 1], [4, -7], [-5, 6], [-1, -2], [-7, 6], [2, -8], [-1, 2], [3, -1], [-7, 6], [-5, -5], [-12, 0], [2, -10], [11, -1], [-8, -1], [-8, 9], [-10, 7], [11, 1], [-13, 1], [-8, 7], [-7, 10], [-2, -9], [7, -2], [-4, -3], [-7, 9], [-4, 9], [-5, 10], [0, 12], [11, -8], [-2, 4], [-7, -1], [3, -6], [15, 6], [-1, -5], [5, 1], [-4, 13], [12, 4], [3, -13], [-14, 1], [-1, 0], [-17, 14], [12, -3], [12, -1], [-11, 9], [-23, -1], [2, 3], [8, -17], [10, 8], [-13, 4], [-2, 1], [9, -6], [-15, 18], [3, -2], [12, -25], [-14, -4], [-22, 3], [1, -8], [-2, 14], [4, 1], [1, -20], [26, -16], [3, -1], [3, 18], [7, -20], [22, -16], [23, -21], [7, -12], [27, -4], [-14, 8], [28, -8], [-18, 11], [-2, -4], [-23, -4], [-15, -4], [9, 7], [-7, -2], [1, -16], [6, -29], [-28, -6], [17, 15], [21, 3], [-3, -16], [35, -2], [31, -6], [3, 12], [-19, 10], [1, 14], [35, -9], [-22, 10], [25, -3], [4, 2], [-10, 12], [25, -25], [-7, -7], [-5, 4], [7, 13], [-27, -11], [-15, 31], [18, -19], [14, 11], [-25, -14], [-14, 33], [9, 7], [-6, -24], [42, -7], [-12, 2], [37, -6], [23, -25], [-3, 12], [18, -12], [9, -12], [-48, -1], [-8, 19], [13, -22], [5, -22], [12, 22], [5, 10], [19, -35], [-40, -4], [-12, 20], [35, -24], [-10, 5], [-7, -18], [-1, 3], [-33, 23], [8, 13], [-5, -16], [8, 0], [-33, 14], [-27, 35], [-3, -2], [-30, 9], [-28, 36], [-9, 8], [-28, 40], [0, -24], [8, -2], [-4, 31], [-7, 3], [-6, 9], [3, 24], [-18, 6], [45, -12], [-36, 16], [-1, -25], [-11, 24], [17, 0], [4, -32], [-18, -8], [-40, 20], [-43, -10], [-8, -12], [-15, -10], [-3, -18], [-10, -16], [-12, -15], [-1, -1], [15, -30], [-36, 8], [-46, 5], [48, -13], [-45, 12], [24, -38], [13, -11], [-17, 35], [-32, 19], [25, -7], [37, -8], [-10, -20], [-12, 12], [15, 28], [-2, -26], [-52, 8], [-20, 10], [-33, 9], [-17, 10], [30, -24], [25, -38], [-19, 20], [-10, 10], [-5, 3], [7, -10], [-20, -20], [-8, 11], [28, 16], [11, -18], [-2, -9], [-49, -5], [-51, -6], [32, -22], [21, -30], [11, -11], [44, -15], [-23, 23], [27, -13], [-32, -3], [20, -9], [22, -32], [-3, 5], [12, -8], [-29, 8], [-30, 24], [-17, -1], [11, -16], [-37, -2], [23, -11], [52, -22], [-19, 38], [-38, 46], [-20, -4], [-50, -11], [63, -1], [-1, 33], [-40, -13], [-11, -8], [-12, -16], [54, -15], [-11, -2], [23, -16], [-5, -27], [12, 22], [16, 20], [-14, -31], [3, -3], [1, -36], [13, -16], [43, -32], [-2, 11], [13, 36], [22, -26], [-3, 21], [-28, 31], [-25, 17], [45, -12], [-38, 14], [-58, 22], [3, -48], [-22, 30], [-39, 7], [-39, -13], [29, 9], [20, 23], [8, 30], [20, -14], [-15, 26], [14, -42], [51, 0], [31, -15], [-14, 18], [41, -50], [-23, 48], [-23, 53], [6, 12], [0, 36], [2, -30], [-35, -14], [-52, 47], [21, 26], [-6, -18], [69, -8], [29, -37], [-13, -2], [31, -56], [48, 10], [49, -20], [-16, 2], [-29, 58], [66, -19], [7, -22], [4, 8], [28, 19], [-53, 1], [33, -32], [-6, -37], [-12, 8], [17, -19], [43, 24], [-11, 55], [44, -12], [36, -49], [-13, 11], [13, -3], [46, -37], [13, -40], [29, -34], [-16, -4], [-18, 21], [31, 12], [-27, 41], [15, 30], [45, -2], [26, -44], [-11, -12], [-9, 10], [42, -4], [-13, 1], [-27, 6], [-30, 31], [-35, 30], [0, 26], [-21, -15], [-12, 3], [-19, -8], [-18, -1], [22, -33], [-9, -17], [-37, 38], [-69, 26], [42, 30], [-18, -4], [-10, 46], [64, -34], [35, -63], [23, 15], [5, 13], [-11, 10], [-33, -16], [30, 19], [-22, 16], [-12, 47], [11, -47], [44, -6], [18, -48], [42, -30], [-11, 24], [32, -21], [-71, -2], [6, -42], [-50, 56], [-48, -6], [2, 30], [47, -43], [43, -27], [-54, 38], [-43, 20], [3, -2], [53, -55], [-23, 24], [-66, 23], [-38, -33], [41, -8], [22, 35], [11, -8], [29, -65], [59, -23], [25, -26], [-68, -5], [-16, 32], [35, 0], [-38, 36], [-57, 41], [15, 30], [-44, 0], [21, 13], [-23, 10], [21, -60], [8, 47], [11, -59], [-3, -22], [40, -6], [-37, 2], [4, 40], [44, 15], [-68, -8], [85, -7], [22, 1], [42, -57], [-57, 40], [-29, 51], [49, 21], [78, -24], [-1, -45], [-7, 31], [-36, -18], [-65, 10], [18, -22], [-86, 1], [-21, -12], [-8, 51], [-75, 15], [16, 20], [-9, 20], [-54, 16], [55, 9], [32, -32], [-13, -10], [-16, 58], [-23, 60], [19, 10], [36, -73], [-6, 42], [-3, -38], [73, 5], [-28, 32], [-21, 51], [61, -19], [-9, -12], [-51, 6], [25, -60], [54, -42], [-71, 4], [-28, 48], [-48, 42], [-46, 65], [15, -44], [41, -72], [-33, 25], [-70, -8], [5, -34], [39, 40], [-20, 23], [-37, -10], [20, -5], [-23, 54], [-39, 86], [75, -20], [51, -16], [14, 22], [28, -12], [34, -4], [-26, 27], [-90, 0], [-76, -5]]