
# q-expansion of newform 1950.2.a.m, 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 = 1950
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], [-3], [-1], [1], [-8], [-4], [-7], [1], [-4], [-6], [12], [-3], [-5], [-9], [5], [11], [8], [0], [-8], [-7], [-8], [-6], [1], [14], [18], [2], [2], [-12], [-10], [-12], [-14], [10], [13], [15], [4], [-20], [5], [10], [1], [-10], [-6], [-6], [-26], [12], [-24], [3], [14], [-26], [3], [-10], [18], [7], [-30], [1], [1], [18], [-18], [14], [-6], [12], [18], [-3], [-20], [28], [-13], [14], [-28], [-14], [-15], [28], [-13], [11], [32], [34], [-26], [-28], [-2], [6], [-12], [12], [10], [-8], [12], [36], [6], [26], [11], [-24], [23], [25], [18], [25], [22], [34], [38], [2], [-8], [6], [6], [36], [3], [-36], [14], [39], [-34], [-26], [-35], [22], [34], [34], [-28], [-12], [-27], [-20], [-24], [21], [0], [-46], [43], [-22], [-27], [-43], [15], [38], [-26], [-14], [-28], [-47], [29], [28], [-43], [-36], [8], [46], [-7], [11], [-26], [-47], [-52], [4], [-3], [47], [16], [-8], [-10], [-24], [9], [0], [-21], [-48], [56], [-8], [28], [-14], [-34], [-21], [-60], [-47], [-57], [-7], [42], [-52], [11], [-8], [17], [-20], [33], [-12], [20], [46], [20], [46], [-15], [-55], [10], [6], [-9], [-32], [-56], [30], [-12], [49], [-42], [-43], [13], [36], [-5], [-2], [21], [-40], [-52], [26], [-13], [-28], [53], [6], [22], [8], [-32], [-38], [-39], [9], [58], [63], [34], [-44], [38], [-67], [-58], [-4], [51], [45], [28], [2], [9], [15], [-16], [-10], [48], [40], [-17], [0], [-17], [-10], [-46], [-10], [59], [46], [-60], [53], [18], [-48], [-32], [10], [0], [24], [-12], [-12], [0], [-50], [76], [-12], [62], [-68], [-56], [-29], [-74], [-32], [-56], [60], [-77], [30], [-45], [-38], [-1], [7], [22], [-22], [-48], [12], [22], [52], [-12], [-6], [36], [-77], [-67], [-38], [17], [-2], [54], [17], [5], [-4], [84], [80], [70], [58], [32], [-64], [2], [-5], [24], [-45], [-30], [63], [-73], [-21], [46], [78], [-75], [-9], [39], [-40], [-54], [-28], [23], [-19], [-66], [-10], [-12], [-50], [12], [48], [-46], [-24], [-11], [48], [-12], [47], [4], [-61], [-62], [1], [40], [-56], [-39], [-70], [58], [-6], [-43], [0], [35], [83], [1], [-38], [-51], [-6], [54], [26], [19], [55], [-51], [-2], [36], [-2], [-2], [-74], [-82], [-26], [80], [-4], [76], [-13], [11], [-45], [15], [-17], [48], [74], [-84], [18], [-1], [-15], [-19], [-13], [51], [80], [-38], [-84], [0], [-98], [-86], [70], [73], [-29], [-78], [-46], [56], [-69], [66], [37], [-33], [-30], [-8], [9], [46], [-38], [-12], [47], [-22], [-52], [-24], [-95], [83], [76], [-58], [-84], [-60], [59], [-62], [69], [83], [-7], [-51], [-22], [34], [32], [49], [-73], [42], [-42], [-42], [-34], [42], [71], [-12], [-31], [51], [30], [30], [82], [-29], [-92], [76], [-62], [26], [-63], [56], [-1], [64], [-24], [60], [-44], [-62], [-21], [6]]
