# q-expansion of newform 7350.2.a.bh, 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 = 7350
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], [0], [2], [-2], [4], [0], [-8], [0], [-2], [-8], [-2], [2], [-10], [2], [4], [-10], [-2], [-12], [-10], [16], [-16], [14], [-6], [-14], [20], [-12], [-2], [14], [12], [12], [2], [-4], [-16], [0], [10], [10], [-18], [6], [-2], [-22], [0], [18], [-18], [10], [-4], [-16], [12], [10], [14], [-8], [-20], [-20], [-12], [0], [-10], [-14], [28], [14], [-4], [-30], [20], [-20], [26], [-6], [-4], [2], [4], [10], [-24], [-20], [28], [36], [8], [-14], [-24], [-14], [-14], [-32], [36], [38], [12], [-38], [-26], [-28], [6], [42], [-18], [-16], [-8], [4], [-28], [6], [40], [6], [18], [30], [-20], [2], [14], [-30], [-24], [-38], [12], [2], [16], [20], [32], [4], [-8], [8], [-6], [-24], [-8], [42], [36], [2], [-2], [-34], [-26], [-10], [-26], [-24], [-28], [16], [42], [-48], [-32], [-30], [48], [48], [-8], [20], [-30], [16], [-30], [12], [-2], [-10], [-28], [24], [16], [-8], [-10], [12], [-38], [12], [-4], [-24], [-12], [-46], [-34], [2], [-10], [16], [56], [-30], [-42], [50], [0], [58], [-32], [-20], [30], [46], [-24], [-6], [-2], [14], [18], [50], [0], [-30], [58], [-30], [-16], [-40], [40], [-26], [-4], [-12], [-52], [60], [40], [-18], [40], [44], [42], [0], [-54], [-12], [-24], [-42], [8], [12], [-50], [32], [-4], [-6], [-36], [2], [-26], [-56], [-12], [-30], [34], [-48], [-38], [52], [38], [54], [-16], [-32], [-4], [-8], [4], [-10], [72], [-2], [18], [50], [-18], [20], [-64], [10], [-12], [-48], [64], [-66], [48], [56], [-40], [50], [52], [2], [64], [-42], [30], [20], [12], [64], [48], [50], [-40], [56], [24], [44], [4], [24], [-12], [6], [32], [-20], [26], [22], [30], [12], [26], [-30], [24], [44], [-10], [30], [-16], [-60], [52], [30], [-30], [30], [-22], [62], [82], [-16], [2], [-12], [52], [-14], [22], [-44], [30], [24], [26], [62], [4], [28], [-70], [50], [70], [26], [28], [12], [-34], [76], [32], [18], [-78], [-42], [4], [76], [14], [-74], [8], [24], [68], [-2], [-12], [-66], [60], [-68], [-34], [16], [74], [30], [-72], [8], [36], [20], [-14], [-50], [-68], [-2], [18], [-56], [26], [52], [-4], [-28], [8], [2], [14], [-50], [-34], [60], [-40], [-60], [18], [48], [-32], [20], [-28], [90], [-38], [48], [-50], [66], [26], [-62], [36], [-46], [36], [2], [96], [8], [58], [24], [-40], [-60], [6], [-16], [-60], [90], [-48], [-10], [4], [-62], [-18], [-52], [-14], [-6], [-84], [32], [36], [10], [12], [52], [-100], [-70], [70], [-26], [-34], [-34], [28], [-50], [-28], [18], [-40], [-22], [94], [66], [-50], [50], [86], [20], [-60], [-66], [70], [-62], [80], [68], [-42], [66], [-68], [16], [-38], [78], [12], [-10], [-20], [-12], [22], [58], [-12], [-80], [58], [-14], [8], [12], [-8], [-38], [48], [-82], [-8], [36], [-30], [-62], [-56], [86], [-56], [-60], [16], [6], [-96], [-54], [82], [70], [-14], [-18], [-78], [74], [-24], [-58], [-46], [-80], [12], [74], [-28], [94], [-72], [-10], [-26], [-48], [108], [70], [-34], [-12], [-18], [-36], [-10], [-38], [-92], [32], [6], [-26], [60], [-86], [56], [-4], [-30], [-88], [100], [-88], [-104], [-58]]