
# q-expansion of newform 4014.2.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 = 4014
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], [0], [1], [-4], [2], [-6], [8], [-6], [2], [0], [-4], [-1], [2], [6], [-3], [-14], [-10], [-6], [-13], [-6], [-11], [-5], [9], [12], [18], [-10], [-7], [-2], [1], [-2], [-8], [-7], [17], [2], [10], [-23], [-14], [-11], [8], [13], [-15], [-7], [22], [2], [-12], [20], [-20], [1], [15], [4], [-26], [27], [23], [-5], [22], [24], [15], [0], [-8], [0], [-18], [30], [-23], [0], [-8], [-2], [4], [6], [-12], [5], [26], [-12], [-36], [18], [2], [0], [-24], [-28], [-30], [14], [36], [8], [-4], [31], [-1], [3], [-22], [12], [0], [-30], [0], [-24], [-26], [-30], [-16], [-26], [-8], [-3], [12], [-10], [8], [9], [18], [13], [37], [33], [38], [9], [32], [-32], [-25], [-2], [-2], [43], [3], [-41], [-28], [-24], [42], [-33], [-2], [6], [-2], [23], [-35], [4], [10], [-3], [22], [34], [3], [41], [-2], [-14], [11], [6], [26], [23], [-18], [-49], [-28], [42], [13], [0], [32], [0], [-18], [54], [-13], [12], [18], [40], [47], [9], [18], [-21], [-56], [-36], [28], [30], [-12], [-47], [-16], [24], [17], [52], [-44], [-37], [-30], [58], [40], [-46], [33], [44], [-52], [-34], [-1], [48], [10], [-14], [47], [-1], [-29], [9], [-17], [-46], [-33], [-46], [-31], [-9], [-2], [26], [-4], [4], [-63], [-56], [-33], [51], [-25], [-30], [-35], [44], [22], [-8], [-12], [28], [28], [-30], [41], [-67], [58], [9], [-4], [-36], [20], [-18], [30], [-27], [39], [-62], [-23], [-8], [-61], [8], [-17], [20], [9], [40], [-68], [56], [14], [-26], [56], [-22], [1], [-20], [74], [33], [-60], [-50], [68], [-45], [-57], [-47], [-38], [-4], [-8], [-46], [-50], [-34], [31], [11], [0], [-16], [-7], [-25], [-39], [-12], [42], [8], [19], [-39], [-38], [34], [-26], [36], [-52], [55], [14], [-18], [-40], [9], [24], [-4], [0], [-79], [-6], [-76], [65], [40], [-29], [-32], [84], [68], [-68], [-72], [-31], [-69], [-21], [-66], [-77], [-72], [-11], [32], [34], [-42], [-80], [-7], [58], [-2], [58], [-60], [25], [48], [-78], [32], [10], [-30], [41], [44], [39], [8], [-22], [-36], [-12], [77], [-66], [58], [-14], [55], [-38], [1], [9], [-56], [-78], [70], [32], [56], [21], [-3], [-4], [38], [-38], [46], [40], [-59], [2], [30], [58], [-37], [5], [-75], [79], [2], [13], [17], [22], [-17], [13], [-2], [-35], [-64], [-93], [14], [12], [6], [15], [27], [-19], [-56], [72], [50], [-12], [34], [-70], [-64], [39], [-74], [-55], [-15], [42], [56], [51], [76], [16], [-84], [-51], [0], [90], [-59], [-35], [54], [55], [81], [-5], [15], [98], [22], [-10], [-60], [-86], [6], [2], [-68], [-38], [29], [20], [-72], [-74], [-18], [31], [-21], [-24], [-10], [6], [-20], [2], [-30], [-100], [-31], [0], [25], [88], [28], [23], [70], [97], [28], [-19], [31], [74], [69], [35], [58], [-64], [-6], [-11], [102], [-74], [-64]]
