
# q-expansion of newform 1900.2.i.f, downloaded from the LMFDB on 17 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, ZZ
    R = PolynomialRing(QQ, "x")
    f = R(poly_data)
    K = NumberField(f, "a")
    betas = [K([c/ZZ(den) for c in num]) for num, den in basis_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 = 1900
weight = 2
poly_data = [9, 33, 145, 14, 233, 13, 298, -65, 79, -16, 15, -3, 1]

# The entries in the following list give a basis for the
# coefficient ring in terms of a root of the defining polynomial above.
# Each line consists of the coefficients of the numerator, and a denominator.
basis_data  = [[[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], 1], [[0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], 1], [[-2681192682, -207837960, -892972135, -3200881200, -1233567830, -445497855, -313073128, -86473310, -61632615, 2185864, -5059670, -638495], 2135068398], [[-1951111485, -105738663, -452450816, 772282665, -606181345, -259923180, 19282261, -89367331, 8533470, -24678133, 5607653, -2112220], 711689466], [[1619981121, 521414583, 2238395086, 4917790269, 3073317005, 1150918890, 606863995, 262313951, 114731760, 20306405, 4511687, 3389210], 711689466], [[-5196018312, -2369519571, -10151561138, -422891238, -13728352537, -5600322462, -1136318216, -1696717843, -77727516, -385595812, 70497029, -35281678], 2135068398], [[-1794617748, -7905696885, 2351354536, -12904983078, -343348333, -17769708720, 4030704250, -4732124989, 968701752, -905147470, 182679923, -60680476], 2135068398], [[-397480590, 17820058356, -2952954247, 14173622286, -3449941442, 19505876175, -5139259390, 5086161106, -1201195899, 980933620, -203695424, 62867497], 2135068398], [[17785843362, 7628969433, 32695561541, 9517460142, 44330893465, 17827730625, 4686279104, 5185680031, 493877715, 1095480130, -177003953, 102677425], 2135068398], [[924096825, -16532556735, 7259257192, -31258077591, 14406379883, -42996086940, 12770142025, -11360311729, 2925836544, -2136716515, 444681245, -131028322], 2135068398], [[-1015605495, 21056968308, -7169319793, 37830916029, -2022066866, 52083858765, -12347339095, 13841799664, -2942137461, 2643593875, -536276480, 175066273], 2135068398], [[-965971179, 19342466968, -7561467697, 32789448329, -14256401492, 45104060701, -13293030539, 11871320512, -3049379221, 2243299983, -469201302, 137992489], 711689466]]

hecke_ring_character_values = [[951, [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]], [77, [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]], [401, [0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0]]]
aps_data = [[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, -1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0], [-1, 1, -1, 0, 0, 0, -2, 0, -1, 0, 0, -1], [0, 0, 0, 0, 0, 0, 0, -1, 0, -1, 1, 0], [1, -1, 1, 0, 0, -1, 1, 0, 0, -1, 1, -1], [0, 0, 0, -1, 1, 0, 0, 1, -1, 0, 1, -1], [-1, 1, -1, 0, 0, 0, -1, 0, -1, 0, 0, -1], [0, 0, 0, 2, 1, -1, 0, 0, 0, 0, 0, 0], [2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0], [-2, 1, 0, 0, 0, 0, -2, 0, 0, 2, -3, 0], [5, -3, 0, 0, 0, 0, 5, -1, 0, -2, 2, 0], [1, -1, 1, 2, 0, -1, 3, 0, 0, -1, -2, 0], [-1, 1, -1, -2, -1, -1, -2, -1, -2, -1, 2, -2], [-1, 1, 0, 0, 0, 0, -1, 2, 0, -1, 0, -2], [0, 0, 0, 0, 3, 1, 1, 3, 1, 1, 0, 1], [-1, 1, -1, -1, 1, -1, -3, 1, -1, -1, 1, -1], [1, 1, 0, 0, 0, 0, 1, 2, 0, 1, -1, 1], [-2, 0, 0, 0, 0, 0, -2, -1, 0, -1, -1, -2], [2, -4, 0, 0, 0, 0, 2, 0, 0, -1, 1, -1], [-2, 0, -1, -1, 1, 2, 0, 0, -1, 0, 0, 0], [0, 0, 0, 1, -2, 0, 1, -2, -1, 0, -1, -1], [-3, 1, 0, 0, 0, 0, -3, -2, 0, 0, 1, -2], [-3, 3, -3, 1, -1, 1, -3, -1, 1, 1, -1, 1], [0, 0, 2, -1, -2, 2, 0, 0, -1, 0, 0, 0], [0, 0, 1, -1, -3, -4, 0, 0, -1, 0, 0, 0], [0, -1, 0, 0, 0, 0, 0, -1, 0, -1, -1, 1], [1, 0, -1, -3, 0, 2, 0, 0, 0, 0, 0, 0], [0, 0, 0, -1, 2, -1, -6, 2, -1, -1, 1, -1], [-7, 5, 0, 0, 0, 0, -7, 4, 0, 0, -3, 1], [2, -2, 2, -2, -2, 0, 1, -2, 0, 0, 2, 0], [-4, 4, -4, -3, -2, -3, -4, -2, -1, -3, 3, -1], [5, 1, 0, 0, 0, 0, 5, 1, 0, -3, 1, 1], [4, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0], [6, 2, 0, 0, 0, 0, 6, 3, 0, -1, 0, -3], [0, 0, 3, -2, 0, 1, 0, 0, 0, 0, 0, 0], [7, -7, 7, 0, 2, -4, 5, 2, 2, -4, 0, 2], [0, 1, 0, 0, 0, 0, 0, 3, 0, -1, -4, 0], [2, 0, -7, -2, -1, 1, 0, 0, 2, 0, 0, 0], [-1, 1, -1, 1, -4, 2, -5, -4, 0, 2, -1, 0], [-6, 0, 2, -3, 3, -3, 0, 0, -1, 0, 0, 0], [0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 5, -1], [-5, 0, -1, 0, 2, 0, 0, 0, -1, 0, 0, 0], [3, -3, 3, 6, 1, 1, -3, 1, 0, 1, -6, 0], [-1, -1, 0, 0, 0, 0, -1, 4, 0, 0, -2, 0], [-4, 2, 0, 0, 0, 0, -4, 1, 0, 3, -2, 0], [8, 0, 8, -3, 1, -2, 0, 0, -1, 0, 0, 0], [1, 0, -5, 0, 1, -3, 0, 0, -2, 0, 0, 0], [3, -1, 0, 0, 0, 0, 3, -3, 0, 3, -1, 1], [-2, 0, -7, 1, 2, -1, 0, 0, 1, 0, 0, 0], [-2, 2, -2, -4, -1, -1, -8, -1, -3, -1, 4, -3], [-1, 1, -1, 3, 2, -2, -5, 2, -1, -2, -3, -1], [-6, 6, -6, 3, 3, -1, -12, 3, 0, -1, -3, 0], [0, -2, 0, 0, 0, 0, 0, 1, 0, 4, 1, 3], [7, -9, 0, 0, 0, 0, 7, 0, 0, 0, 1, 0], [10, -2, 0, 0, 0, 0, 10, -1, 0, 1, -1, -1], [4, 0, -5, -2, -3, 3, 0, 0, 2, 0, 0, 0], [1, -1, 1, 1, 1, -1, -4, 1, -3, -1, -1, -3], [5, -3, 0, 0, 0, 0, 5, 2, 0, 2, -2, 2], [-5, 0, 7, 0, 0, -4, 0, 0, -1, 0, 0, 0], [7, -5, 0, 0, 0, 0, 7, 0, 0, 3, -3, 1], [2, 0, -2, -1, -3, -4, 0, 0, -5, 0, 0, 0], [-9, 9, -9, 4, 1, 5, -6, 1, 2, 5, -4, 2], [5, -5, 5, 4, -3, -1, -11, -3, 2, -1, -4, 2], [-10, 0, -5, -2, -3, -3, 0, 0, 0, 0, 0, 0], [8, 3, 0, 0, 0, 0, 8, -3, 0, -1, 1, -3], [15, 3, 0, 0, 0, 0, 15, 0, 0, 0, 0, 6], [5, 0, 9, -4, -1, -7, 0, 0, 0, 0, 0, 0], [5, 0, 10, 2, 4, 0, 0, 0, 2, 0, 0, 0], [-5, 11, 0, 0, 0, 0, -5, -1, 0, 0, 4, -2], [-5, 5, -5, -4, -2, -4, -5, -2, 2, -4, 4, 2], [-1, 0, -2, -3, -1, -5, 0, 0, 0, 0, 0, 0], [-6, 0, 6, 3, 0, 0, 0, 0, 1, 0, 0, 0], [-2, 8, 0, 0, 0, 0, -2, -1, 0, 4, -2, 4], [10, -10, 10, -1, -5, -3, -1, -5, 2, -3, 1, 2], [-10, 5, 0, 0, 0, 0, -10, 1, 0, 1, 8, -2], [-4, -2, 0, 0, 0, 0, -4, -2, 0, -2, 5, -1], [-1, 1, -1, -2, 1, -3, 5, 1, 4, -3, 2, 4], [6, 0, -3, 1, 3, 5, 0, 0, 5, 0, 0, 0], [7, -5, 0, 0, 0, 0, 7, -3, 0, -3, 3, -5], [14, -14, 14, -2, 0, -1, 4, 0, 2, -1, 2, 2], [-3, 3, -3, 5, 1, 7, 2, 1, 5, 7, -5, 5], [2, 0, 0, 0, 0, 0, 2, 5, 0, 6, -2, 2], [7, -7, 7, 8, -1, -1, 1, -1, 4, -1, -8, 4], [-5, 0, 0, 5, 2, -6, 0, 0, 1, 0, 0, 0], [-14, 0, -1, 0, -1, -1, 0, 0, 6, 0, 0, 0], [11, 5, 0, 0, 0, 0, 11, 2, 0, 0, -4, 1], [-6, 0, -1, -3, 2, 0, 0, 0, -1, 0, 0, 0], [8, 0, 2, -4, 2, -8, 0, 0, -2, 0, 0, 0], [-2, 2, -2, -4, -5, 4, -2, -5, 0, 4, 4, 0], [4, 0, -5, 4, 2, 5, 0, 0, -2, 0, 0, 0], [-6, 0, 0, 0, 0, 0, -6, -3, 0, -6, -2, 4], [-11, 1, 0, 0, 0, 0, -11, 0, 0, -1, -5, 5], [0, 0, 0, -1, 1, -2, 12, 1, 5, -2, 1, 5], [0, 0, 0, 3, 6, 0, 2, 6, -5, 0, -3, -5], [-6, 0, -12, 0, -4, 2, 0, 0, -4, 0, 0, 0], [2, -2, 2, -2, -3, -2, -6, -3, 2, -2, 2, 2], [-3, 3, -3, 6, 0, 2, 9, 0, -1, 2, -6, -1], [6, -6, 6, -5, -2, 4, -10, -2, -3, 4, 5, -3], [-4, 4, -4, 2, 5, -7, -4, 5, -1, -7, -2, -1], [0, 0, -1, 8, 0, 2, 0, 0, 0, 0, 0, 0], [-12, 0, 2, 5, 0, 6, 0, 0, 3, 0, 0, 0], [-10, 0, -3, 8, -3, 7, 0, 0, 4, 0, 0, 0], [-6, 0, -9, 5, -1, 1, 0, 0, 1, 0, 0, 0], [-7, 7, 0, 0, 0, 0, -7, -1, 0, 5, -4, 6], [1, -1, 1, -3, -3, -1, 6, -3, 5, -1, 3, 5], [-1, 1, -1, 1, -2, 2, 13, -2, 3, 2, -1, 3], [2, 0, -4, -2, -4, 6, 0, 0, 0, 0, 0, 0], [26, 0, 4, 1, -2, -3, 0, 0, 1, 0, 0, 0], [12, 3, 0, 0, 0, 0, 12, 1, 0, -1, -3, -1], [2, -2, 2, 3, -2, -4, 17, -2, -3, -4, -3, -3], [-10, 0, -6, -4, 7, 5, 0, 0, 2, 0, 0, 0], [-4, 4, -4, -3, -2, 1, -8, -2, -3, 1, 3, -3], [11, -4, 0, 0, 0, 0, 11, 1, 0, 1, -2, 5], [-3, -3, 0, 0, 0, 0, -3, 3, 0, -7, 1, -1], [-8, 0, -1, -5, 4, 0, 0, 0, -3, 0, 0, 0], [-7, 0, -3, -6, 1, -5, 0, 0, -6, 0, 0, 0], [4, -4, 4, 5, -1, 2, 22, -1, 1, 2, -5, 1], [5, -5, 5, -2, -2, 2, -6, -2, 3, 2, 2, 3], [-2, 0, -1, 7, -2, 4, 0, 0, 4, 0, 0, 0], [-7, 0, 10, 0, 4, -4, 0, 0, -2, 0, 0, 0], [16, 0, -6, -3, -5, -2, 0, 0, -1, 0, 0, 0], [-10, 0, 1, -5, 1, -4, 0, 0, 1, 0, 0, 0], [-9, 2, 0, 0, 0, 0, -9, -9, 0, 5, -2, -3], [-9, 9, -9, -6, -3, -3, -1, -3, 0, -3, 6, 0], [7, 5, 0, 0, 0, 0, 7, -2, 0, 1, 8, -8], [24, 8, 0, 0, 0, 0, 24, -3, 0, -2, -2, -2], [6, 0, 4, -3, 2, 8, 0, 0, -5, 0, 0, 0], [3, 7, 0, 0, 0, 0, 3, 0, 0, -3, 2, 4], [-3, -3, 0, 0, 0, 0, -3, -8, 0, -5, -1, -3], [-6, 6, -6, 7, 6, 8, -12, 6, 1, 8, -7, 1], [-14, -3, 0, 0, 0, 0, -14, -1, 0, -9, 1, -3], [-2, 0, 9, 3, -1, -9, 0, 0, -3, 0, 0, 0], [9, -9, 9, -5, 1, 5, -13, 1, -3, 5, 5, -3], [2, -2, 2, -8, 6, -8, -15, 6, -6, -8, 8, -6], [-20, 0, -7, 1, 6, 11, 0, 0, -3, 0, 0, 0], [17, 0, 13, -11, 2, -8, 0, 0, -2, 0, 0, 0], [-5, 0, -6, 0, -7, 5, 0, 0, 3, 0, 0, 0], [12, -12, 12, 3, 3, -1, -12, 3, 5, -1, -3, 5], [0, 0, 0, -4, 0, 2, 4, 0, 6, 2, 4, 6], [3, -3, 3, 0, 7, -5, -1, 7, -8, -5, 0, -8], [10, -10, 10, -7, -5, 0, 0, -5, 1, 0, 7, 1], [2, 0, -4, 5, 1, 1, 0, 0, -2, 0, 0, 0], [-2, -4, 0, 0, 0, 0, -2, 5, 0, 1, -2, -2], [0, -15, 0, 0, 0, 0, 0, -8, 0, -2, 0, -1], [8, -9, 0, 0, 0, 0, 8, 1, 0, -7, 3, 5], [-2, 2, -2, 2, 1, -3, 2, 1, -8, -3, -2, -8], [-12, 0, -8, 8, 6, 2, 0, 0, -4, 0, 0, 0], [11, 2, 0, 0, 0, 0, 11, -6, 0, -8, 2, 0], [-4, 0, 7, 7, 6, -4, 0, 0, -4, 0, 0, 0], [-3, 3, -3, -12, -7, -3, 13, -7, -6, -3, 12, -6], [4, -4, 4, -5, -2, -1, -20, -2, 1, -1, 5, 1], [14, 2, 0, 0, 0, 0, 14, -3, 0, -3, 4, -4], [16, 0, 4, -3, 2, -8, 0, 0, -5, 0, 0, 0], [8, 0, -4, -5, 1, 3, 0, 0, -1, 0, 0, 0], [-2, 2, 0, 0, 0, 0, -2, -7, 0, -1, 7, -8], [2, -2, 2, 1, -9, 3, 1, -9, 4, 3, -1, 4], [4, -4, 4, 9, 12, 8, -15, 12, 5, 8, -9, 5], [-9, -5, 0, 0, 0, 0, -9, -6, 0, -5, 0, 4], [23, 0, 0, 0, 0, 0, 23, 5, 0, -5, -5, -2], [-7, 1, 0, 0, 0, 0, -7, 9, 0, -5, -2, -8], [-22, 10, 0, 0, 0, 0, -22, 1, 0, 8, -10, -4], [-28, 0, -2, -1, -3, -1, 0, 0, -2, 0, 0, 0], [-5, 3, 0, 0, 0, 0, -5, -5, 0, -1, 3, -3], [18, 2, 0, 0, 0, 0, 18, 8, 0, 1, -2, 0], [0, 0, 0, 0, -4, 4, 27, -4, -2, 4, 0, -2], [8, 0, 0, 0, 0, 0, 8, -8, 0, 8, -1, -1], [4, -4, 4, -2, -9, -9, 3, -9, -5, -9, 2, -5], [-4, 0, 0, 3, 3, 10, 0, 0, 5, 0, 0, 0], [3, 2, 0, 0, 0, 0, 3, -6, 0, 8, 3, -3], [34, -8, 0, 0, 0, 0, 34, 3, 0, 1, -4, 2], [9, 0, 3, 11, 9, 5, 0, 0, 5, 0, 0, 0], [7, -7, 7, 3, 6, -5, 35, 6, -1, -5, -3, -1], [3, -3, 3, -3, -3, -1, 11, -3, 1, -1, 3, 1], [7, -7, 7, -7, -4, 1, 5, -4, -1, 1, 7, -1], [-20, 11, 0, 0, 0, 0, -20, 5, 0, 5, -5, 7], [20, 0, -5, -3, 1, 7, 0, 0, -1, 0, 0, 0], [-14, -5, 0, 0, 0, 0, -14, 3, 0, 1, 8, -6], [2, -2, 2, -4, 6, 3, 12, 6, 4, 3, 4, 4], [-16, 0, -5, -3, 6, -5, 0, 0, -3, 0, 0, 0], [4, -4, 4, -3, -2, -8, -12, -2, -7, -8, 3, -7], [26, 7, 0, 0, 0, 0, 26, 5, 0, 1, 1, 5], [12, 0, 4, -7, -8, 2, 0, 0, 3, 0, 0, 0], [11, 0, 2, -1, -2, 8, 0, 0, -3, 0, 0, 0], [1, -1, 1, 0, 4, -2, -4, 4, -3, -2, 0, -3], [4, -16, 0, 0, 0, 0, 4, -4, 0, 3, 6, -2], [11, 0, 5, -4, 3, 11, 0, 0, 6, 0, 0, 0], [-2, 0, 2, -11, -6, 2, 0, 0, 5, 0, 0, 0], [-10, 10, -10, 1, 2, 8, -35, 2, 3, 8, -1, 3], [-3, 3, -3, 6, 0, 5, 3, 0, 0, 5, -6, 0], [-12, 0, 8, 10, -2, 8, 0, 0, 6, 0, 0, 0], [9, -5, 0, 0, 0, 0, 9, 6, 0, 8, -2, 5], [1, -1, 1, -1, -4, -2, 39, -4, -1, -2, 1, -1], [-5, 5, -5, 0, 0, -2, -14, 0, -5, -2, 0, -5], [-6, 6, -6, 4, -9, 5, -7, -9, -1, 5, -4, -1], [-10, 5, 0, 0, 0, 0, -10, 4, 0, -6, 11, -2], [-12, 0, 0, 2, 10, -8, 0, 0, -4, 0, 0, 0], [-28, 0, 4, 8, 0, 3, 0, 0, 4, 0, 0, 0], [-5, 5, -5, -8, -10, 2, 12, -10, -3, 2, 8, -3], [-13, 13, -13, 5, 0, -1, -11, 0, 3, -1, -5, 3], [-5, -3, 0, 0, 0, 0, -5, -1, 0, 7, -6, 2], [-13, 7, 0, 0, 0, 0, -13, -6, 0, 4, 6, 3], [-20, -8, 0, 0, 0, 0, -20, -1, 0, -4, 2, -2], [13, -13, 13, -7, -5, -11, 12, -5, 1, -11, 7, 1], [0, 0, 0, 2, -4, -6, -4, -4, -6, -6, -2, -6], [-6, 6, -6, 2, 11, 5, 4, 11, 2, 5, -2, 2], [9, 6, 0, 0, 0, 0, 9, -1, 0, 3, -7, 4], [6, 0, -3, 7, 1, 2, 0, 0, 1, 0, 0, 0], [30, 6, 0, 0, 0, 0, 30, -4, 0, 6, -4, 2], [-3, 3, -3, 6, -4, 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