
# q-expansion of newform 4650.2.a.f, downloaded from the LMFDB on 24 April 2024.

# 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 = 4650
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 = [[-1], [-1], [0], [-1], [-5], [4], [0], [-5], [9], [-2], [1], [-8], [6], [-1], [12], [13], [10], [-14], [-14], [-9], [9], [5], [-6], [3], [18], [-7], [-8], [-11], [-2], [-13], [-20], [-6], [-2], [16], [7], [0], [-13], [-4], [9], [6], [-16], [17], [8], [22], [10], [-11], [-13], [-16], [-15], [-19], [1], [-6], [24], [-28], [7], [8], [-22], [-25], [2], [12], [-26], [14], [8], [-8], [-6], [-34], [-24], [-2], [18], [4], [4], [-9], [-18], [-1], [-11], [-12], [-16], [3], [-25], [-28], [30], [-26], [-32], [11], [20], [-13], [6], [22], [36], [-30], [-20], [-21], [-24], [15], [6], [-22], [-20], [-36], [-17], [0], [-16], [-7], [44], [-21], [-10], [2], [6], [10], [-39], [-12], [-13], [-4], [-9], [16], [31], [-2], [5], [-5], [-18], [30], [28], [-38], [-51], [-3], [27], [29], [-17], [-30], [-1], [14], [10], [43], [18], [20], [15], [-35], [27], [23], [2], [-5], [55], [-2], [2], [46], [29], [-21], [-21], [-10], [0], [31], [6], [6], [-23], [-8], [-4], [18], [-8], [-11], [54], [-48], [10], [52], [20], [2], [-22], [52], [-3], [18], [8], [-3], [12], [-29], [-8], [-58], [-4], [13], [44], [21], [-7], [-21], [24], [-23], [-6], [-8], [-18], [-24], [27], [8], [10], [9], [-19], [-45], [48], [-36], [-12], [10], [26], [39], [45], [30], [41], [-25], [43], [-35], [-6], [-2], [-66], [16], [-18], [4], [-57], [48], [12], [27], [24], [3], [-16], [30], [24], [54], [-29], [-70], [20], [23], [1], [9], [42], [-12], [-64], [42], [-30], [20], [54], [-45], [-44], [-33], [-3], [34], [21], [-58], [-28], [-2], [-58], [50], [-12], [61], [12], [-67], [6], [44], [-1], [-40], [3], [47], [-12], [24], [-38], [7], [12], [-62], [-35], [-10], [33], [35], [16], [-3], [5], [2], [18], [-18], [48], [-68], [-51], [-47], [28], [32], [-57], [46], [20], [-13], [-20], [-68], [84], [12], [52], [14], [-31], [-54], [-24], [18], [27], [45], [28], [60], [64], [-18], [56], [-24], [21], [-87], [22], [3], [-40], [-3], [-26], [-48], [-60], [-12], [-88], [-54], [-16], [-40], [-30], [19], [40], [-2], [-24], [72], [-16], [-55], [-58], [43], [-49], [79], [75], [-70], [-79], [-4], [52], [33], [-10], [82], [80], [0], [-81], [7], [-62], [41], [-19], [10], [20], [-64], [-86], [-7], [-28], [-42], [-34], [49], [80], [76], [-4], [-37], [89], [82], [-86], [-16], [0], [87], [-24], [46], [-47], [-69], [21], [-16], [90], [29], [-26], [-64], [-77], [-74], [-61], [-78], [-34], [4], [10], [37], [15], [16], [-8], [33], [-84], [-60], [-22], [-42], [-22], [10], [22], [50], [55], [-63], [8], [-55], [22], [-36], [85], [22], [-51], [-70], [-100], [98], [7], [-75], [-59], [-70], [-96], [26], [-10], [-38], [41], [45], [6], [0], [-78], [66], [-10], [-53], [56], [1], [-53], [16], [-76], [102], [35], [-16], [26], [-53], [69], [55], [-26], [30]]
