
# q-expansion of newform 7623.2.a.m, downloaded from the LMFDB on 25 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 = 7623
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], [1], [0], [5], [7], [-6], [4], [-9], [-2], [9], [7], [-6], [-2], [-3], [-2], [6], [8], [-6], [-10], [-14], [0], [9], [17], [14], [-16], [18], [-1], [15], [-8], [12], [-2], [20], [-13], [-2], [2], [10], [12], [6], [-6], [-5], [10], [5], [7], [6], [-24], [-8], [-6], [15], [9], [12], [2], [-4], [17], [8], [11], [-8], [11], [18], [14], [9], [28], [18], [-19], [14], [34], [13], [10], [-15], [33], [28], [-14], [26], [-32], [16], [9], [-29], [-17], [-9], [-32], [1], [-30], [-11], [-4], [-4], [19], [33], [21], [-26], [-24], [14], [-22], [28], [-16], [16], [-6], [-30], [-20], [2], [-2], [-38], [22], [42], [-32], [-15], [2], [-37], [-10], [-5], [28], [31], [-45], [14], [46], [-9], [-4], [-32], [-46], [22], [-17], [-14], [-51], [-44], [-34], [35], [26], [-6], [-6], [-3], [-2], [10], [46], [17], [-45], [19], [-30], [-40], [-2], [18], [2], [10], [12], [-34], [-29], [26], [25], [38], [0], [30], [27], [49], [22], [8], [-30], [36], [14], [33], [-29], [-3], [-42], [37], [50], [28], [-57], [30], [-14], [-47], [1], [-6], [20], [58], [44], [1], [44], [1], [-20], [-47], [-50], [-7], [44], [62], [30], [-9], [-54], [-6], [27], [12], [49], [-40], [-34], [10], [-6], [27], [-42], [-39], [-35], [-62], [-33], [14], [30], [-18], [-21], [-1], [-14], [-22], [-12], [-36], [-39], [-14], [33], [-21], [30], [-28], [-36], [41], [58], [-18], [36], [-51], [-49], [-36], [39], [44], [-66], [43], [69], [-18], [8], [10], [-59], [-52], [38], [5], [-38], [22], [30], [-21], [68], [50], [-16], [-8], [42], [-39], [-26], [6], [-4], [46], [24], [34], [14], [23], [0], [-19], [-63], [-62], [25], [-42], [-57], [62], [32], [-54], [-71], [54], [21], [34], [-42], [-30], [40], [-6], [-13], [36], [37], [4], [46], [14], [60], [-37], [74], [18], [46], [-58], [44], [-35], [20], [-54], [30], [-78], [-56], [-17], [-35], [-38], [9], [36], [-78], [-34], [-86], [11], [-30], [-68], [-54], [15], [-64], [50], [-22], [-50], [42], [-9], [18], [-45], [6], [-43], [9], [56], [18], [-5], [-2], [-68], [10], [5], [68], [7], [-55], [-40], [-71], [2], [18], [8], [36], [79], [-25], [-66], [10], [-14], [36], [-42], [49], [22], [25], [82], [-31], [54], [-51], [20], [70], [-2], [7], [-52], [-6], [39], [-14], [54], [-39], [-54], [-1], [51], [-48], [2], [29], [-56], [10], [-54], [-12], [4], [-12], [-53], [57], [4], [33], [-18], [12], [-38], [-5], [38], [1], [-32], [2], [55], [87], [54], [-46], [-69], [-8], [2], [86], [12], [-18], [-37], [64], [-38], [38], [-10], [4], [-8], [-12], [2], [-2], [-15], [-56], [-25], [-23], [83], [48], [42], [-74], [-90], [-27], [13], [-16], [-14], [39], [31], [-16], [10], [13], [87], [-40], [0], [58], [-90], [-23], [-58], [-6], [70], [13], [87], [-42], [-26], [28], [34]]
