
# q-expansion of newform 4026.2.a.f, downloaded from the LMFDB on 19 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 = 4026
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], [-1], [2], [-1], [-3], [8], [4], [4], [-9], [-3], [6], [3], [4], [0], [8], [-3], [-1], [4], [6], [12], [-4], [4], [-17], [13], [-1], [12], [13], [3], [2], [0], [1], [14], [-19], [20], [4], [-7], [-5], [24], [11], [14], [1], [18], [-17], [-14], [14], [-25], [-12], [24], [-4], [-20], [-16], [-8], [-19], [26], [-6], [23], [20], [26], [20], [-32], [-8], [-11], [16], [-4], [3], [-4], [15], [-25], [36], [6], [-33], [-22], [-20], [-13], [14], [36], [-29], [25], [31], [-36], [-31], [18], [-4], [13], [-26], [-40], [-14], [-22], [6], [33], [0], [20], [-28], [-22], [38], [-4], [-18], [-13], [16], [11], [27], [-32], [-35], [-44], [8], [39], [6], [0], [-14], [1], [-33], [-19], [-31], [-8], [-6], [-34], [-10], [-12], [-39], [-21], [11], [-50], [-30], [33], [15], [21], [0], [-12], [14], [-31], [-5], [2], [-52], [-12], [13], [21], [-35], [42], [25], [44], [-38], [11], [-4], [-20], [9], [3], [17], [31], [17], [-36], [52], [4], [-24], [16], [15], [-49], [0], [-26], [27], [-48], [-42], [-23], [-36], [44], [6], [-10], [40], [55], [20], [-22], [-22], [-45], [-10], [-24], [4], [20], [0], [36], [14], [-53], [-12], [-24], [30], [-15], [-4], [26], [-45], [-54], [-11], [56], [39], [-51], [27], [38], [38], [36], [-31], [-42], [-42], [34], [-2], [43], [-39], [-12], [41], [-1], [46], [36], [50], [30], [-42], [10], [15], [-21], [-40], [-35], [10], [67], [-14], [55], [-23], [43], [58], [-58], [-24], [-41], [-24], [-14], [-60], [11], [28], [26], [58], [17], [-69], [50], [51], [-15], [-53], [-63], [68], [1], [2], [44], [-28], [12], [10], [20], [-4], [64], [63], [4], [18], [56], [12], [5], [16], [6], [-3], [-17], [45], [42], [51], [-30], [71], [31], [18], [-6], [-16], [14], [-39], [13], [-80], [-17], [-16], [-5], [-58], [-26], [29], [51], [-68], [24], [20], [9], [-69], [-8], [-42], [0], [-26], [-16], [-24], [39], [42], [-35], [20], [4], [2], [-84], [20], [14], [10], [-8], [84], [-38], [18], [-60], [20], [-24], [-16], [48], [-6], [-54], [-27], [22], [52], [-16], [-28], [-51], [6], [84], [-32], [-50], [19], [-20], [-76], [-24], [44], [-48], [34], [3], [18], [62], [-61], [9], [15], [-84], [-61], [-54], [46], [64], [-69], [40], [61], [-66], [-80], [-74], [26], [32], [49], [32], [8], [-78], [0], [90], [-86], [-55], [32], [72], [70], [86], [-38], [-44], [83], [-1], [62], [-37], [-24], [20], [-75], [30], [-12], [6], [0], [-16], [-7], [76], [11], [10], [-22], [2], [-81], [-76], [-43], [8], [-102], [-26], [-59], [-18], [75], [37], [54], [10], [-66], [-40], [-67], [-18], [-2], [-45], [-54], [-53], [87], [79], [10], [-23], [-25], [92], [-30], [7], [64], [-52], [46], [15], [18], [58], [-58], [38], [4], [47], [-48], [54], [-24], [-73], [60], [47], [-54], [39], [95], [-74], [-87]]
