
# q-expansion of newform 3720.2.a.f, downloaded from the LMFDB on 26 July 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 = 3720
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], [1], [-3], [3], [-2], [0], [1], [1], [-4], [-1], [4], [12], [11], [6], [11], [6], [-14], [-2], [-3], [-13], [1], [12], [-11], [10], [9], [16], [15], [-8], [19], [12], [-10], [14], [-14], [-13], [8], [7], [18], [-23], [-10], [-12], [15], [-8], [22], [-22], [25], [-7], [-2], [-21], [-9], [-11], [20], [22], [12], [-13], [16], [6], [11], [-28], [-4], [4], [-22], [12], [-16], [26], [20], [8], [-22], [0], [4], [-18], [-3], [-6], [19], [19], [-12], [-18], [31], [21], [-12], [12], [-14], [16], [-31], [10], [-39], [26], [22], [-36], [-20], [-24], [-15], [-28], [35], [8], [28], [0], [-22], [-1], [2], [2], [-33], [20], [13], [-22], [-44], [-12], [-6], [15], [-42], [-3], [10], [-9], [18], [31], [-30], [-7], [-33], [6], [20], [-2], [22], [43], [19], [29], [-3], [-3], [2], [-11], [-50], [-28], [15], [-42], [-2], [-23], [-23], [33], [15], [-30], [29], [-7], [12], [-30], [-48], [39], [-31], [3], [-18], [-40], [-41], [-10], [-6], [1], [30], [16], [12], [26], [-5], [-28], [-52], [-14], [28], [0], [-22], [2], [-48], [-7], [6], [14], [7], [20], [-3], [0], [8], [-6], [15], [12], [1], [19], [-15], [-56], [9], [2], [-14], [-54], [30], [39], [60], [-64], [-57], [55], [21], [16], [52], [-40], [-36], [-10], [67], [69], [-46], [-27], [39], [-1], [5], [-22], [-6], [20], [-14], [36], [20], [5], [-48], [-24], [-27], [-6], [-49], [-64], [-14], [48], [-34], [-55], [0], [0], [-11], [27], [-1], [54], [60], [-26], [54], [34], [-36], [54], [33], [52], [-33], [29], [46], [-23], [-44], [0], [62], [6], [-46], [-52], [47], [66], [53], [8], [-12], [19], [-10], [-69], [23], [-24], [-70], [70], [3], [6], [68], [-61], [10], [-25], [-25], [46], [-53], [-31], [64], [46], [72], [20], [52], [43], [5], [-78], [56], [33], [62], [-10], [-57], [-12], [-10], [-62], [42], [-38], [26], [-5], [38], [-8], [66], [3], [67], [-36], [-38], [76], [48], [-52], [-30], [-39], [77], [10], [-51], [32], [51], [22], [36], [2], [-38], [-14], [22], [-80], [24], [10], [-13], [66], [40], [12], [-68], [-76], [-41], [28], [-27], [-25], [81], [11], [-16], [43], [-42], [-28], [-13], [46], [-46], [-26], [-10], [25], [-67], [-38], [41], [73], [-44], [28], [54], [36], [-31], [-90], [10], [-8], [-79], [-42], [44], [48], [13], [-15], [-70], [-50], [30], [-6], [91], [-72], [54], [53], [49], [81], [-54], [2], [-45], [26], [22], [-65], [24], [-63], [26], [-2], [34], [86], [-43], [17], [64], [-20], [-41], [36], [-94], [-38], [48], [-62], [-42], [80], [32], [-97], [21], [-32], [-1], [56], [74], [-77], [-42], [-23], [82], [-86], [-24], [-39], [-59], [-77], [-2], [96], [6], [22], [-42], [49], [-3], [-98], [-52], [-16], [-24], [-76], [15], [58], [-37], [79], [18], [-26], [48], [67], [-28], [-36], [-39], [19], [-103], [-64], [10]]
