# q-expansion of newform 1472.2.a.l, downloaded from the LMFDB on 30 April 2026.

# 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 = 1472
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 = [[0], [1], [4], [2], [4], [5], [-2], [-6], [1], [-1], [-9], [4], [3], [-8], [-5], [-6], [4], [10], [4], [-5], [-15], [-6], [-6], [-8], [10], [10], [10], [-6], [0], [6], [-7], [21], [4], [-5], [18], [11], [12], [-25], [12], [2], [-1], [12], [12], [1], [-21], [16], [16], [-16], [14], [-28], [-9], [-1], [-16], [-30], [-21], [22], [-21], [-8], [7], [-6], [16], [-20], [4], [-21], [20], [18], [23], [-16], [-28], [-1], [-3], [-18], [18], [8], [28], [-12], [-20], [25], [12], [17], [-4], [2], [-36], [4], [25], [5], [-10], [10], [7], [-8], [-22], [18], [-11], [11], [-11], [-22], [-33], [0], [-6], [3], [-9], [0], [-12], [-4], [26], [-39], [13], [-2], [40], [29], [32], [26], [6], [44], [16], [-48], [8], [11], [9], [26], [-22], [43], [-26], [31], [28], [-18], [14], [16], [0], [26], [7], [-18], [12], [-24], [25], [8], [48], [0], [-2], [-22], [-7], [30], [-5], [-12], [-38], [-6], [46], [-1], [-35], [21], [14], [-54], [-12], [15], [-14], [-42], [-2], [-49], [-32], [6], [-19], [34], [37], [18], [36], [-22], [48], [58], [14], [29], [-16], [35], [-30], [-24], [5], [-26], [-37], [5], [2], [-22], [37], [-36], [39], [18], [-32], [14], [30], [-6], [29], [32], [-17], [32], [32], [-18], [-52], [-16], [-44], [4], [12], [29], [60], [24], [-61], [20], [0], [15], [-64], [69], [-61], [-37], [-33], [45], [-36], [42], [56], [24], [-53], [21], [8], [-27], [5], [20], [-39], [-18], [33], [21], [64], [-39], [-46], [-60], [-50], [-74], [4], [66], [-12], [-10], [28], [44], [73], [5], [-48], [-43], [55], [-55], [-66], [-25], [-17], [10], [70], [16], [-6], [28], [2], [62], [-66], [-17], [-32], [4], [-57], [77], [16], [4], [-26], [10], [-18], [6], [-44], [-68], [-14], [3], [35], [-58], [48], [42], [-15], [39], [32], [38], [-72], [60], [-63], [52], [-8], [70], [84], [-12], [34], [-54], [-20], [18], [-26], [46], [66], [-26], [-5], [2], [38], [-27], [-29], [3], [-6], [6], [46], [63], [-40], [-37], [36], [-88], [-56], [38], [15], [-46], [-22], [19], [74], [78], [-9], [3], [16], [-1], [-44], [-2], [-71], [36], [72], [-36], [-52], [-73], [16], [-6], [23], [37], [41], [31], [14], [-17], [-8], [82], [-37], [32], [29], [-80], [30], [-50], [-16], [-56], [-35], [28], [-50], [12], [45], [41], [-6], [86], [-14], [-46], [14], [-20], [1], [-96], [40], [-83], [-35], [-12], [79], [93], [-91], [16], [-94], [63], [-52], [79], [70], [-4], [-27], [87], [16], [2], [-36], [39], [48], [-14], [4], [-65], [82], [5], [-55], [-14], [-56], [-2], [-82], [10], [29], [28], [58], [44], [8], [-78], [2], [62], [1], [1], [58], [78], [-83], [75], [46], [33], [-2], [12], [-1], [-27], [72], [58], [-42], [27], [0], [40], [-84], [0], [-6], [48], [-96], [36], [-43], [105], [102], [-54], [-75], [-33]]