
// Make newform 1600.2.q.g in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1600_q();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1600_q_Hecke();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_1600_2_q_g();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_1600_2_q_g();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
function ConvertToHeckeField(input: pass_field := false, Kf := [])
    if not pass_field then
        poly := [256, -512, 256, 0, 112, -64, -112, 56, 17, 28, -28, -8, 7, 0, 4, -4, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-60352, 52128, 18176, 7384, -26360, -17886, 13624, 9201, 2932, -6494, -2160, 1271, 720, 534, -604, 65], [-256, -5920, 3968, 2872, 1536, -2378, -2482, 673, 1108, 654, -454, -337, 16, 66, 74, -39], [-12672, 3712, 10912, 4256, -5336, -7812, 390, 4001, 2156, -968, -1398, -129, 304, 232, -38, -55], [-576, -1744, 2048, 940, 8, -1239, -672, 580, 472, 95, -276, -108, 44, 41, 20, -20], [83072, -108736, 1504, 5840, 42864, 9900, -32015, -6320, 2228, 11700, -489, -3508, -460, -256, 1201, -396], [-109568, 141024, -7232, -7264, -53512, -10386, 42059, 6168, -3820, -15166, 1317, 4780, 492, 234, -1613, 556], [-65920, -24384, 77600, 43096, -14288, -54624, -16367, 23382, 18652, -8, -9945, -3082, 1572, 1644, 281, -526], [176384, -296512, 44320, 43856, 107784, -4752, -95888, -5477, 17228, 31596, -6240, -11155, -664, 236, 3292, -1269], [-32384, 52032, -6976, -6376, -18888, -132, 16455, 1306, -2596, -5588, 977, 1914, 148, -16, -593, 222], [-196096, 324288, -51040, -44192, -115064, 3072, 102370, 5967, -16892, -34004, 6366, 11729, 720, -12, -3586, 1367], [64864, -94144, 5520, 8700, 36556, 5004, -28915, -4547, 3156, 10264, -837, -3209, -400, -124, 1041, -355], [33600, -36032, -4448, -1120, 15784, 7116, -10011, -3694, -508, 4060, 483, -1014, -260, -200, 397, -106], [36160, -153824, 73664, 46600, 37560, -39234, -53377, 14480, 19972, 13746, -9999, -7004, 988, 1258, 1703, -996], [342400, -386240, -35584, -3704, 164544, 67332, -108253, -36019, -2696, 42912, 3909, -11105, -2612, -1916, 4247, -1167], [168960, -279808, 37088, 36088, 100472, 192, -87051, -6659, 13576, 29156, -5133, -9945, -676, 8, 3065, -1175]];
    Rf_basisdens := [1, 1344, 448, 384, 96, 1344, 1344, 1344, 2688, 448, 2688, 672, 192, 1344, 1344, 1344];
    Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf);
    Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]];
    inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf);
    return Eltseq(inp_vec);
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_1600_q();"
function MakeCharacter_1600_q()
    N := 1600;
    order := 4;
    char_gens := [1151, 901, 577];
    v := [4, 1, 2];
    // chi(gens[i]) = zeta^v[i]
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    F := CyclotomicField(order);
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),[F|F.1^e:e in v]);
    return MinimalBaseRingCharacter(chi);
end function;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1600_q_Hecke();"
function MakeCharacter_1600_q_Hecke(Kf)
    N := 1600;
    order := 4;
    char_gens := [1151, 901, 577];
    char_values := [[1, 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], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := true, Kf := Kf); // the value of chi on the gens as elements in the Hecke field
    F := Universe(values);// the Hecke field
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values);
    return chi;
end function;


function ExtendMultiplicatively(weight, aps, character)
    prec := NextPrime(NthPrime(#aps)) - 1; // we will able to figure out a_0 ... a_prec
    primes := PrimesUpTo(prec);
    prime_powers := primes;
    assert #primes eq #aps;
    log_prec := Floor(Log(prec)/Log(2)); // prec < 2^(log_prec+1)
    F := Universe(aps);
    FXY<X, Y> := PolynomialRing(F, 2);
    // 1/(1 - a_p T + p^(weight - 1) * char(p) T^2) = 1 + a_p T + a_{p^2} T^2 + ...
    R<T> := PowerSeriesRing(FXY : Precision := log_prec + 1);
    recursion := Coefficients(1/(1 - X*T + Y*T^2));
    coeffs := [F!0: i in [1..(prec+1)]];
    coeffs[1] := 1; //a_1
    for i := 1 to #primes do
        p := primes[i];
        coeffs[p] := aps[i];
        b := p^(weight - 1) * F!character(p);
        r := 2;
        p_power := p * p;
        //deals with powers of p
        while p_power le prec do
            Append(~prime_powers, p_power);
            coeffs[p_power] := Evaluate(recursion[r + 1], [aps[i], b]);
            p_power *:= p;
            r +:= 1;
        end while;    
    end for;
    Sort(~prime_powers);
    for pp in prime_powers do
        for k := 1 to Floor(prec/pp) do
            if GCD(k, pp) eq 1 then
                coeffs[pp*k] := coeffs[pp]*coeffs[k];
            end if;
        end for;
    end for;
    return coeffs;
end function;


function qexpCoeffs()
    // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
    weight := 2;
    raw_aps := [[0, 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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -1, 0, 0, 1, 1, 0, 0, 0, 0, 1], [0, 0, 1, 1, 1, 0, 1, -1, 1, 0, 0, 0, 0, 0, 0, 0], [-1, 1, 0, 1, 0, 0, 1, 0, 1, -1, 0, 0, 0, 0, 0, 0], [0, 0, 1, -1, 0, 0, 1, 1, 0, 0, -1, -1, -1, 0, 1, -1], [-1, 0, -1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, -1, -1, 1], [-2, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0], [2, 0, 0, 1, 0, -2, 0, 0, 1, 0, 2, 0, 1, 1, 1, 1], [0, 0, 0, -1, 2, 0, 1, -1, 0, 0, -1, 1, 0, 0, 0, 0], [-2, 0, 0, 2, 0, 2, 0, 0, 0, 0, 2, 2, 0, -2, -1, 1], [1, -1, 0, 0, 0, -2, 0, 0, 2, 0, 0, 0, 0, 1, 1, -1], [0, 0, 1, -1, -1, 0, 0, 0, -1, 0, -2, 1, 0, 0, 0, 0], [-1, 1, 2, 1, 0, -2, 1, 0, 1, -1, 1, 0, 1, -1, -1, 2], [-2, 0, -1, 0, 1, 2, 0, 0, -1, 0, -1, 2, -2, -1, 1, 0], [0, -1, 1, 0, 1, -1, -1, -1, 0, 0, 0, 0, 0, 0, -1, 1], [0, 0, -1, 0, 1, 0, 0, 0, 0, 0, 0, 2, -2, -2, 1, 1], [2, 0, -3, 1, -3, 2, 1, 2, 1, -2, 0, 0, 0, 0, 2, -2], [0, 0, -4, 0, 0, 0, 2, 2, 2, -2, 0, 0, 0, 0, 2, 0], [-1, 1, 0, 2, 1, 0, 0, -1, 0, 1, 0, 1, 1, 1, 0, 1], [1, 1, 0, -1, 2, 0, -1, -1, 0, 0, -1, -1, 1, 1, 0, 0], [-4, 2, -1, 1, -1, -2, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0], [1, -1, -2, -2, 0, 0, -2, 0, -2, 2, -2, 0, 1, 1, -2, 0], [1, -1, -3, -2, 0, 0, -2, 3, -2, -1, -2, 3, 1, 1, 1, 0], [-3, 1, 0, -1, 0, -2, -6, -4, -1, 2, 0, 0, 0, 0, -3, 3], [2, 0, 0, 1, 2, 0, -1, 0, 0, 0, 0, -1, -1, 0, 0, 0], [2, 0, -2, -2, 2, -2, 0, 0, 0, 0, -2, 1, 2, 0, -2, 0], [-2, 0, -1, 0, 1, 2, 0, 0, -2, 0, -2, 0, 0, -2, -2, 0], [0, 0, 4, 2, 0, -2, 2, 0, 4, -2, 2, 0, 0, 0, 2, 0], [-1, 1, -4, 0, 0, 0, -1, 0, -3, 0, 0, 1, -2, -1, 1, -1], [-3, 0, -1, 1, 1, 3, 0, 0, -2, 0, -1, 1, -1, -3, -3, -1], [-3, 1, 0, -1, 4, 0, 5, -2, 0, -1, -3, 4, 0, 1, 0, -1], [0, 0, -3, 1, -3, 0, 1, 1, 1, 0, 0, 0, 0, 0, 2, -2], [-2, 0, 1, 2, 1, -2, 6, -2, 2, -4, 0, 0, 0, 0, 1, -1], [2, -2, 2, -1, 0, -8, 3, 3, 0, -2, -1, -1, 0, 2, 4, -2], [0, 0, 1, 0, 1, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0], [4, 2, 4, 0, 4, 6, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-2, 0, 0, -1, -2, 0, 1, -4, 0, 0, -4, 1, -3, 0, 0, 0], [2, 0, 2, 2, 2, 2, 4, -2, 2, -2, 0, 0, 0, 0, 3, -3], [6, 0, -1, 0, 1, -6, 0, 0, 1, 0, 1, -3, 2, 2, 0, 0], [-3, 3, 2, 1, 2, 0, 4, -4, 1, 0, 0, 0, 0, 0, -3, 3], [6, 2, 0, -2, 4, 0, -2, 0, 0, 0, 0, -2, 0, 2, 0, 0], [1, -1, 1, 0, 0, -4, 0, -1, -2, 1, 0, -1, 1, 1, -1, 0], [4, 0, 4, -2, -4, -4, 0, 0, 1, 0, -1, 0, -2, -1, 0, -1], [0, 0, 0, 2, 0, -4, -2, 0, 0, -2, 2, 4, 2, 0, 0, 2], [5, 0, 5, 3, -5, -5, 0, 0, 0, 0, 3, 1, 3, 1, -1, 1], [-1, 1, 4, 4, 0, -4, 3, -4, 3, 0, 4, -3, 0, -1, -1, 1], [4, -4, -2, -2, -2, 0, -1, 2, -2, 4, 0, 0, 0, 0, 0, 0], [1, 1, 0, -1, 0, 2, -8, -4, -1, 4, 0, 0, 0, 0, -3, 3], [3, -3, 0, -2, 3, 0, -4, 1, 0, 3, 4, -1, -1, -3, 0, 3], [0, 0, 0, 1, 6, 0, 5, 1, 0, -2, -1, 3, 0, 0, 0, -2], [4, 2, 0, 0, 4, 0, 4, 2, 0, -2, 0, 2, 1, 2, 0, -2], [-4, 4, -1, 3, -1, 0, 5, -1, 3, -4, 0, 0, 0, 0, 2, -2], [-1, 1, 0, 5, 0, -2, 3, -2, 6, -3, 5, 0, -2, -1, 4, -1], [8, 0, 0, -2, 6, 0, 5, 0, 0, -1, -1, 4, 0, 0, 0, -1], [4, 0, 1, -4, -1, -4, 0, 0, 0, 0, -4, -6, -2, 2, 1, -3], [-7, 1, 0, 1, 2, 0, -1, 1, 0, 2, 3, 1, 5, 1, 0, 2], [-6, 0, 2, 0, -2, 6, 0, 0, 0, 0, 0, -6, 2, -2, -3, 1], [2, -2, 6, 0, 0, -2, 4, 0, 0, 0, 0, -4, 1, 2, 1, -1], [0, 0, -4, -2, 4, 0, 0, 0, 2, 0, 0, -3, 4, 6, 2, 0], [0, 0, 1, -2, -1, 0, 0, 0, 0, 0, -2, -4, -6, 0, 2, -4], [-2, 2, 5, 1, 5, 0, 5, 2, 1, -2, 0, 0, 0, 0, 2, -2], [-1, 1, 2, -1, 0, -2, -3, 1, -2, 0, -1, 3, 1, -1, -2, 2], [5, 1, 0, -1, 10, 0, 1, 0, 0, 1, 1, 2, 2, 1, 0, 1], [5, -1, -2, -1, -2, 4, 1, -4, -1, 3, 0, 0, 0, 0, -2, 2], [8, -3, 3, 0, 3, 5, 1, 3, 0, -4, 0, 0, 0, 0, 3, -3], [1, -1, 8, 3, 0, -2, 5, -2, 2, -1, 3, -4, 4, 1, -2, 3], [-6, 0, 1, -3, -1, 6, 0, 0, -1, 0, -4, 1, -2, -6, -4, 0], [4, 0, 0, 3, 0, -4, 0, 0, 1, 0, 4, 4, -1, -3, -1, 1], [2, -2, -2, 2, 0, -2, 2, -2, 4, 0, 2, -2, 4, 2, -2, 2], [1, -1, 6, 3, 0, -2, 7, -3, 4, 0, 3, -7, 1, 1, 0, 0], [3, -3, -2, -2, 0, 4, -1, 2, -3, 0, -2, 1, 0, 3, 3, -3], [2, 0, 2, 0, -2, -2, 0, 0, 4, 0, 4, -6, -6, 2, 3, -5], [4, -6, 3, 1, 3, -2, 5, 5, 1, -4, 0, 0, 0, 0, 4, -4], [3, -3, 2, 2, 0, -16, -3, -2, -1, 0, 2, 3, 2, 3, 1, -1], [0, -2, -3, 0, -3, -2, 0, 2, 0, -2, 0, 0, 0, 0, 1, -1], [-5, 1, -2, -3, -2, -4, -7, 0, -3, -1, 0, 0, 0, 0, -2, 2], [3, 3, 0, 0, -8, 0, -2, -2, 0, 4, 2, 2, 3, 3, 0, 4], [-1, 1, -12, -2, 0, 2, 2, 4, -2, -2, -2, 0, -4, -1, 5, -3], [0, 0, -3, 4, 3, 0, 0, 0, 5, 0, 9, 1, 4, -4, 0, 8], [0, 0, -3, 2, -3, 0, -6, 2, 2, -4, 0, 0, 0, 0, -1, 1], [-6, -2, 0, 1, -6, 0, -3, 3, 0, 2, 5, -1, 0, -2, 0, 2], [-4, 4, 7, 3, 0, 4, 1, -1, 0, -2, 3, 1, 1, -4, -3, 5], [-3, 3, 2, -3, 0, -2, -3, 3, -4, 0, -3, 3, 1, -3, -4, 4], [-10, 0, 4, -2, -4, 10, 0, 0, 2, 0, 0, -5, -2, -4, -2, 0], [10, 0, 0, -4, -6, 0, -4, 0, 0, 0, 0, -4, 1, 0, 0, 0], [9, 1, 0, 1, 6, 0, -1, 0, 0, 3, 3, 2, 4, 1, 0, 3], [0, 0, 0, -5, 0, 0, 0, 0, -5, 0, -10, -4, 3, 3, -3, -3], [1, -1, 4, 5, 0, 10, -1, -4, 3, -1, 5, 2, 3, 1, -1, 2], [-6, 2, 0, 2, 0, -4, -1, -10, 2, 2, 0, 0, 0, 0, -2, 2], [5, -3, 0, -4, -8, 0, 4, 4, 0, -2, 2, 2, 1, -3, 0, -2], [10, 2, 0, 6, 6, 0, 5, 4, 0, -1, 3, 4, 6, 2, 0, -1], [8, 1, -3, -6, -3, 9, -5, 1, -6, 4, 0, 0, 0, 0, -3, 3], [-6, 0, -3, -2, 3, 6, 0, 0, -1, 0, -3, 1, -2, 2, 0, -4], [-16, 2, 0, -3, -2, 0, 1, -2, 0, 0, -2, 1, -1, 2, 0, 0], [0, 0, -2, -1, 2, 0, 0, 0, 1, 0, 0, 0, 3, 5, 1, -1], [-2, 2, 8, -6, 0, -4, -6, 2, -8, 4, -6, 2, -4, -2, -2, -2], [6, 0, 0, -6, 0, -6, 0, 0, -2, 0, -8, 1, -2, 4, 2, -4], [2, 0, 2, -1, -2, -2, 0, 0, 3, 0, 2, 0, -1, -1, -3, -3], [2, 2, -4, -6, -4, 4, -19, -6, -6, 10, 0, 0, 0, 0, -10, 10], [0, 0, -5, 8, -5, 0, 12, 0, 8, -12, 0, 0, 0, 0, 6, -6], [-10, 0, 6, 4, 6, -10, 3, -4, 4, 2, 0, 0, 0, 0, 2, -2], [-3, 3, 8, 10, 0, 6, 2, -6, 6, -4, 10, 2, 2, -3, -1, 5], [-8, 2, -7, 3, -7, -6, 5, 3, 3, -12, 0, 0, 0, 0, 6, -6], [-2, 2, 0, -6, 0, 6, -10, 0, -8, 6, -6, 4, 2, -2, -10, 4], [4, 0, 1, -3, -1, -4, 0, 0, 1, 0, -2, 5, -8, -2, 6, 0], [3, -3, -12, -9, 0, -10, -3, 6, -6, 3, -9, 0, -2, 3, 2, -5], [5, -5, 4, -2, 0, -2, -2, -4, -2, 6, -2, -4, 5, 5, -6, 0], [-2, 2, 14, 0, 0, 2, 4, 0, 0, 0, 0, -4, -1, -2, -1, 1], [3, -3, 10, 1, 0, -10, -1, -2, 3, 1, 1, 0, 5, 3, -3, 2], [-6, 0, -3, 2, 3, 6, 0, 0, -3, 0, -1, 10, 0, 1, 1, 0], [8, -4, 0, 1, -7, 0, 5, 1, 0, 0, 1, 5, 3, -4, 0, 0], [-8, -3, -3, 2, -3, -11, 7, 11, 2, -4, 0, 0, 0, 0, 3, -3], [-2, 2, 2, 7, 0, -20, 5, -7, 8, 0, 7, -5, -2, -2, 0, 0], [0, -2, 0, -2, 8, 0, 2, -4, 0, -2, -6, 0, -1, -2, 0, -2], [6, 2, -3, -3, -3, 8, -3, 8, -3, -2, 0, 0, 0, 0, -2, 2], [-4, -4, 0, 0, -2, 0, 5, 2, 0, -3, -1, 2, 0, -4, 0, -3], [-11, 5, 1, -1, 1, -6, -7, -6, -1, 5, 0, 0, 0, 0, -7, 7], [4, 0, 3, 4, -3, -4, 0, 0, 3, 0, 7, -5, -4, 4, 6, -2], [6, 0, -5, 0, -5, 6, 0, 4, 0, 4, 0, 0, 0, 0, 2, -2], [-3, 3, 3, 6, 0, -8, 10, 1, 6, -7, 6, -3, -3, -3, 7, 0], [-8, 0, -2, -6, 2, 8, 0, 0, 1, 0, -5, 4, -6, -5, 0, -1], [10, 0, -5, -7, 5, -10, 0, 0, -7, 0, -14, 3, -10, 2, 4, -8], [11, 0, -1, -3, 1, -11, 0, 0, 4, 0, 1, 3, 1, 3, 1, -1], [-4, 0, -3, 0, 3, 4, 0, 0, 6, 0, 6, 4, -4, -6, 0, 2], [7, -1, 8, 5, 8, 6, 2, 0, 5, -2, 0, 0, 0, 0, 3, -3], [7, 3, 0, -1, -10, 0, -3, 1, 0, -6, -5, -9, -3, 3, 0, -6], [-12, 4, 0, 4, -14, 0, 3, 2, 0, -3, -1, 0, 0, 4, 0, -3], [0, -10, -4, 4, -4, -10, 6, 10, 4, -8, 0, 0, 0, 0, 7, -7], [2, 0, 3, -8, -3, -2, 0, 0, -3, 0, -11, 1, -6, 6, 4, -8], [6, -2, 0, -2, -2, 0, -9, -4, 0, 1, -3, -8, -6, -2, 0, 1], [-3, -3, 0, 4, 16, 0, 2, 0, 0, 4, 4, 6, 5, -3, 0, 4], [14, 0, 6, -4, -6, -14, 0, 0, 0, 0, -4, -6, -2, 10, 5, -7], [-6, 6, -8, 8, 0, 2, 4, -2, 8, -6, 8, 2, -2, -6, 2, 4], [-17, 1, 0, 2, 10, 0, 12, -4, 0, -4, -8, 8, -1, 1, 0, -4], [6, 0, 5, -2, -5, -6, 0, 0, -1, 0, -3, 6, 4, 3, 1, 2], [-4, 2, 5, -3, 5, -2, -1, 2, -3, 4, 0, 0, 0, 0, 2, -2], [-17, 7, -3, -3, -3, -10, -5, -2, -3, 7, 0, 0, 0, 0, -3, 3], [-2, 2, 12, 2, 0, -4, 2, -6, 0, 4, 2, -6, 4, -2, -10, 6], [4, 2, 3, 5, 3, 6, 9, 3, 5, 0, 0, 0, 0, 0, 2, -2], [-8, 2, 5, 10, 5, -6, 18, 0, 10, -10, 0, 0, 0, 0, 4, -4], [-18, -4, 0, 9, 14, 0, 5, 4, 0, 0, 4, 5, 3, -4, 0, 0], [-2, 0, -3, 1, 3, 2, 0, 0, 5, 0, 6, 1, -2, 0, 6, 4], [-6, 0, 11, 0, -11, 6, 0, 0, 0, 0, 0, -2, 2, -2, -7, -3], [2, -2, -2, 5, 0, -12, -1, 1, 10, -6, 5, 7, -2, 2, 10, -4], [-8, 0, 6, -6, -6, 8, 0, 0, -5, 0, -11, -4, -2, -3, -4, -3], [-3, 1, 0, 1, 12, 0, -1, -6, 0, 5, -1, 4, -6, 1, 0, 5], [16, -7, -5, -8, -5, 9, -15, 9, -8, 8, 0, 0, 0, 0, 1, -1], [-1, 1, -4, -1, 0, -10, 13, 6, 3, -5, -1, -8, -7, -1, 11, -6], [14, -2, 1, -10, 1, 12, -16, 4, -10, 8, 0, 0, 0, 0, -4, 4], [9, -3, 0, -4, 14, 0, -10, -2, 0, 8, 6, -2, -2, -3, 0, 8], [-18, 6, -6, -2, -6, -12, 5, 12, -2, -6, 0, 0, 0, 0, 4, -4], [6, -2, 0, -2, -10, 0, -1, 0, 0, -1, -1, -2, -2, -2, 0, -1], [-2, 0, -4, 2, 4, 2, 0, 0, 2, 0, 4, -1, 2, -2, 2, 6], [-20, 4, 0, 3, -2, 0, 5, -5, 0, -4, -9, 1, 0, 4, 0, -4], [-9, 9, 16, 10, 0, 10, 6, -4, 6, -6, 10, 0, -5, -9, 2, 4], [0, -6, 0, 2, 2, 0, 6, 10, 0, -4, 6, 2, -3, -6, 0, -4], [-16, 0, 0, 7, 1, 0, 3, -3, 0, 2, -1, 5, -3, 0, 0, 2], [8, 0, 0, -7, 0, -8, 0, 0, -1, 0, -8, -8, -7, 7, 3, -11], [-10, -8, -1, 3, -1, -18, 11, -2, 3, 2, 0, 0, 0, 0, 2, -2], [-7, -3, 0, 3, 4, 0, 9, 6, 0, -5, 1, 4, -2, -3, 0, -5], [10, 4, 0, -5, 8, 0, -1, -2, 0, -4, -6, -5, -3, 4, 0, -4], [0, 3, -9, 6, -9, 3, -1, -3, 6, -8, 0, 0, 0, 0, -1, 1], [-2, 2, 11, 3, 0, -8, -3, -1, 4, -2, 3, 5, 7, -2, -7, 9], [-16, 0, 0, 2, -4, 0, 3, -2, 0, -7, -9, -4, -4, 0, 0, -7], [24, 0, 0, -8, -16, 0, -2, -4, 0, 2, -2, 0, -4, 0, 0, 2], [2, 0, -1, -4, 1, -2, 0, 0, -5, 0, -9, -6, -6, 3, 5, -4], [-3, 3, 0, -4, -10, 0, -6, 4, 0, 2, 6, -4, 3, 3, 0, 2], [6, 0, -7, 2, 7, -6, 0, 0, -1, 0, 1, -6, -8, -9, 1, 2], [-8, 4, -1, -1, -1, -4, 5, 1, -1, 4, 0, 0, 0, 0, 0, 0], [-8, 0, 0, 3, 0, 8, 0, 0, -11, 0, -8, 8, -1, -3, -5, -3], [0, 0, 6, -1, 0, 20, -5, -1, 0, 2, -1, 3, 2, 0, -4, 2], [-5, 1, 0, -2, 3, 0, -8, -11, 0, 7, -4, -1, -3, 1, 0, 7], [-19, -3, 0, 4, -4, 0, -8, -4, 0, 6, 2, -2, -3, -3, 0, 6], [3, -3, 6, 4, 0, -6, 8, -4, 10, 0, 4, -8, -7, 3, 10, -10], [-4, 0, -7, -7, -7, -4, -9, 1, -7, 8, 0, 0, 0, 0, -2, 2], [-7, 3, -2, -3, -2, -4, 0, 12, -3, 4, 0, 0, 0, 0, -1, 1], [12, 2, 0, 3, -14, 0, -3, -2, 0, -4, -6, -7, 1, 2, 0, -4], [-8, 0, 13, 0, -13, 8, 0, 0, -4, 0, -4, -4, -8, -4, 2, -2], [1, -1, 14, 6, 0, -8, -1, -6, 1, 0, 6, 1, 2, 1, -1, 1], [1, 0, 13, 3, -13, -1, 0, 0, 4, 0, 7, -15, 7, 1, -5, 1], [6, 0, 1, -4, -1, -6, 0, 0, 1, 0, -3, -6, -2, 5, -1, -8], [6, -2, 0, -1, -1, 0, 3, 3, 0, -8, -5, -5, -3, -2, 0, -8], [9, -9, 2, -3, 0, -6, 5, 10, 3, -7, -3, 2, 3, 9, 13, -6], [-11, -1, -8, -3, -8, -12, 2, 0, -3, -2, 0, 0, 0, 0, 1, -1], [-18, 4, 4, 2, 4, -14, -10, -6, 2, 0, 0, 0, 0, 0, -1, 1], [4, -2, 3, 1, 3, 2, 13, 4, 1, -8, 0, 0, 0, 0, 4, -4], [5, -5, 4, 4, 0, -2, 12, 0, 2, -4, 4, -8, -2, 5, 11, -7], [1, -7, 0, 2, 0, 0, -4, 6, 0, 0, 6, -4, -3, -7, 0, 0], [-1, 1, -8, -5, 0, 6, 1, 6, -6, -1, -5, 0, 0, -1, 0, 1], [12, 0, -2, 10, 2, -12, 0, 0, 0, 0, 10, 3, 4, -4, -6, 2], [-5, 0, -1, -1, 1, 5, 0, 0, -4, 0, -5, 5, -11, -9, -1, -3], [0, 0, -1, 4, 1, 0, 0, 0, 4, 0, 8, -12, -4, 0, 8, 4], [8, 6, 3, 5, 3, 14, 1, -2, 5, -4, 0, 0, 0, 0, 0, 0], [-8, 0, 0, 3, 5, 0, 7, 1, 0, -6, -5, 1, -7, 0, 0, -6], [-5, 3, 0, -4, -2, 0, -6, 4, 0, -6, -2, -12, 2, 3, 0, -6], [11, -1, 3, -3, 3, 10, 7, 2, -3, -1, 0, 0, 0, 0, -1, 1], [3, -3, -10, -3, 0, -6, 3, 8, 2, -5, -3, 2, 4, 3, 4, 1], [10, -4, 0, -3, 2, 0, 3, -2, 0, 8, 6, 11, -3, -4, 0, 8], [16, 0, 4, -7, -4, -16, 0, 0, 1, 0, -6, -8, -3, 9, 3, -9], [-17, 3, 0, -6, 12, 0, -6, 0, 0, -2, -2, -8, -7, 3, 0, -2], [-14, 0, -3, 4, 3, 14, 0, 0, 1, 0, 5, 6, 6, 1, -5, 0], [14, 6, 0, -12, -12, 0, -8, -4, 0, 4, 0, -4, 2, 6, 0, 4], [0, -3, 13, 2, 13, -3, 5, -11, 2, 4, 0, 0, 0, 0, -1, 1], [12, 0, 5, 4, 5, 12, 2, 0, 4, -2, 0, 0, 0, 0, -2, 2], [-3, 5, 0, -11, 14, 0, -15, -7, 0, 0, -7, -15, -3, 5, 0, 0], [16, -2, -11, -7, -11, 14, -1, 0, -7, 0, 0, 0, 0, 0, 2, -2], [-1, 1, -2, -6, 0, 2, 2, 4, 2, 2, -6, -4, -1, -1, -2, 0], [20, -5, 3, -6, 3, 15, 1, 7, -6, 4, 0, 0, 0, 0, -3, 3], [8, -8, 3, -5, 0, 4, -3, 3, 0, 2, -5, 1, -5, 8, 11, -13], [-6, 8, -3, 6, -3, 2, 2, -6, 6, -12, 0, 0, 0, 0, -1, 1], [2, 2, 0, -12, -4, 0, -21, 0, 0, 9, 9, -12, 2, 2, 0, 9], [-8, 0, -5, -11, 5, 8, 0, 0, 1, 0, -10, -9, -4, 10, 4, -10], [-4, 4, -8, -10, 0, 4, -2, 4, -8, 6, -10, -4, 2, -4, -12, 6], [7, -7, -4, -10, 0, -2, 6, 10, -6, 0, -10, -6, -6, 7, 13, -13], [-3, 3, -10, 4, 0, 4, 5, 0, 3, -4, 4, -1, 4, -3, -3, 7], [-9, 3, 0, 12, -12, 0, 2, 10, 0, -4, 6, -2, 3, 3, 0, -4], [14, 4, 0, -5, 4, 0, -1, 0, 0, 4, 4, 3, 11, 4, 0, 4], [7, -1, -9, -11, -9, 6, -11, 2, -11, 9, 0, 0, 0, 0, -5, 5], [8, 0, 7, -8, 7, 8, 0, 0, -8, 8, 0, 0, 0, 0, 2, -2], [-3, 3, 22, 7, 0, -6, 13, -1, 12, -6, 7, -7, -5, -3, 8, -2], [5, 5, 0, 2, -6, 0, 4, 0, 0, 0, 0, 4, -1, 5, 0, 0], [-1, 1, 24, -2, 0, 16, -7, -6, 1, 8, -2, -1, 2, -1, -11, 3], [-14, 6, -13, 3, -13, -8, 3, 6, 3, -6, 0, 0, 0, 0, 2, -2], [-14, -4, 17, 0, 17, -18, 8, -4, 0, 4, 0, 0, 0, 0, 4, -4], [-5, 1, 0, -8, -3, 0, 2, -5, 0, -5, -10, -3, 3, 1, 0, -5], [0, 0, 0, 3, -2, 0, 3, -1, 0, 10, 9, 13, 4, 0, 0, 10], [-4, -2, 0, -5, 22, 0, -1, 0, 0, 4, 4, 3, -5, -2, 0, 4], [16, -1, 1, 0, 1, 15, -13, -11, 0, 8, 0, 0, 0, 0, -5, 5], [1, -7, 7, 3, 7, -6, 7, -6, 3, -9, 0, 0, 0, 0, -1, 1], [-10, 0, -1, 0, 1, 10, 0, 0, 5, 0, 5, -3, -10, -6, 2, -2], [4, 12, 0, 2, 0, 0, -8, -2, 0, 2, 0, -6, 0, 12, 0, 2], [-2, 2, 2, 0, 0, 26, 0, -6, -4, 6, 0, -6, -1, -2, -7, 1], [10, 0, -3, -7, 3, -10, 0, 0, 1, 0, -6, -3, -2, 14, 10, -6], [-1, 1, 14, 0, 0, -12, -1, -4, -7, 4, 0, -3, 0, -1, -5, 1], [-10, 4, 0, 2, -2, 0, 10, 4, 0, -14, -10, -4, 7, 4, 0, -14], [-4, 0, 4, 8, -4, 4, 0, 0, 9, 0, 17, -4, 4, 7, 8, 5], [-12, -1, 1, 8, 1, -13, 11, -9, 8, -8, 0, 0, 0, 0, 1, -1], [-3, 3, -6, -3, 0, 30, 1, 5, -6, -2, -3, 1, -11, -3, 10, -8], [-12, 12, -3, 1, -3, 0, 3, -8, 1, -8, 0, 0, 0, 0, -2, 2], [0, -6, 1, 3, 1, -6, 11, -3, 3, 4, 0, 0, 0, 0, -2, 2], [28, -4, 0, 0, -12, 0, 11, 4, 0, -5, -1, 6, -4, -4, 0, -5], [12, 0, -4, 17, 4, -12, 0, 0, 3, 0, 20, 4, 13, -5, -9, 9], [2, -2, -12, 4, 0, 18, 0, 0, 8, -4, 4, 4, -2, 2, 8, -4], [5, -5, -10, -5, 0, 14, 11, 13, 0, -8, -5, -3, -11, 5, 24, -16], [-5, 5, 6, -4, 0, -8, 7, 12, -1, -8, -4, 1, -6, -5, 9, -1], [-8, 0, 7, 2, -7, 8, 0, 0, -1, 0, 1, -5, 8, -8, -8, 8], [8, -7, -13, 0, -13, 1, 5, 1, 0, -4, 0, 0, 0, 0, 5, -5], [-3, 3, 0, 3, 0, 2, 7, -8, 5, 5, 3, -12, 3, -3, -11, 6], [17, -3, -3, -1, -3, 14, -9, -14, -1, 7, 0, 0, 0, 0, -5, 5], [12, 0, 0, 0, 0, 0, 8, 4, 0, -4, 0, 4, 0, 0, 0, -4], [-2, -2, 0, 10, 8, 0, -5, 2, 0, 7, 9, 2, 2, -2, 0, 7], [0, 0, -4, 12, 0, 2, 16, -2, 20, -10, 12, -6, 1, 0, 9, 1], [-6, -2, 7, -2, 7, -8, -12, -8, -2, 0, 0, 0, 0, 0, 2, -2], [9, 0, -3, 7, 3, -9, 0, 0, -4, 0, 3, 11, 11, 9, 3, 5], [-5, -5, 16, 1, 16, -10, -10, -4, 1, 6, 0, 0, 0, 0, -3, 3], [0, 0, 11, -1, -11, 0, 0, 0, 5, 0, 4, 3, 0, 0, 0, 0], [-16, 0, 1, 0, -1, 16, 0, 0, -8, 0, -8, -4, 4, 0, -8, -4], [2, 8, 0, -6, -14, 0, -2, -6, 0, 4, -2, 2, -2, 8, 0, 4], [7, -7, -10, -3, 0, -26, -5, 4, 1, -1, -3, 6, 5, 7, 3, -2], [24, -8, -6, -4, -6, 16, 1, 6, -4, 4, 0, 0, 0, 0, 2, -2], [3, -5, -2, -3, -2, -2, -16, 0, -3, 16, 0, 0, 0, 0, -5, 5], [-12, -4, 3, 0, 3, -16, 4, 8, 0, 8, 0, 0, 0, 0, 4, -4], [0, 0, 1, 1, 0, 16, 11, 5, 8, -6, 1, -5, -7, 0, 13, -7], [2, 0, 11, 6, -11, -2, 0, 0, 1, 0, 7, -5, -2, -2, -6, -6], [-14, 0, 0, -1, 0, 14, 0, 0, -7, 0, -8, 4, 11, 1, -7, 3], [1, -1, 0, -2, 0, 6, -10, 2, 6, 0, -2, 10, 5, 1, -4, 4], [-2, 0, -10, -2, 10, 2, 0, 0, 12, 0, 10, -11, 2, 6, 6, 2], [-4, 0, -15, 2, 15, 4, 0, 0, 4, 0, 6, 0, -6, -4, 2, 0], [0, 0, -4, 7, 4, 0, 0, 0, -7, 0, 0, 8, 3, 5, 1, -1], [2, 6, 8, 8, 8, 8, 3, -2, 8, -2, 0, 0, 0, 0, 0, 0], [10, -2, 0, 3, -11, 0, 15, 1, 0, -10, -9, 5, -1, -2, 0, -10], [9, 5, 0, 5, 18, 0, 15, -5, 0, -6, -11, 9, -5, 5, 0, -6], [1, -1, 4, 5, 0, -18, 7, 6, 14, -11, 5, 4, -4, 1, 16, -5], [-2, -2, 0, 14, 18, 0, 3, -4, 0, 7, 3, 10, 6, -2, 0, 7], [-18, 0, -6, 0, 6, 18, 0, 0, -2, 0, -2, -9, -2, 4, 0, -6], [-28, 0, -2, 5, 2, 28, 0, 0, 1, 0, 6, 4, -3, -11, -3, 5], [0, 0, 2, 2, 0, -14, 18, 4, 12, -6, 2, -12, -5, 0, 11, -5], [8, 0, 3, 14, -3, -8, 0, 0, -1, 0, 13, -1, 0, -12, -4, 8], [5, -5, 4, -19, 0, 2, -11, 6, -23, 13, -19, -2, -1, 5, -7, -6], [5, -5, 0, -12, 0, 22, -8, 4, -16, 8, -12, 0, -3, 5, 0, -8], [4, 10, 0, -3, 16, 0, -5, -14, 0, 4, -10, -1, 1, 10, 0, 4], [-5, 5, -2, -1, -2, 0, -4, -4, -1, 8, 0, 0, 0, 0, -3, 3], [-12, 0, -7, 7, 7, 12, 0, 0, -7, 0, 0, 17, 4, -4, -10, -2], [-6, -6, 0, 0, -4, 0, -2, 0, 0, 2, 2, 0, 6, -6, 0, 2], [-9, 9, 9, -6, 0, 0, -10, 3, -6, 3, -6, 7, -7, -9, -5, 2], [-26, 0, -5, 2, 5, 26, 0, 0, -3, 0, -1, -2, 4, 1, -3, 0], [2, -2, -4, -2, 0, -12, -8, -6, -12, 8, -2, 0, 10, 2, -16, 8], [-5, -5, 0, 6, 10, 0, 12, -6, 0, 2, -4, 14, 4, -5, 0, 2], [2, 0, -1, 0, 1, -2, 0, 0, -6, 0, -6, 14, 10, -8, -9, 9], [-22, -2, -3, -1, -3, -24, -3, 10, -1, -10, 0, 0, 0, 0, 4, -4], [4, 0, 0, 3, -9, 0, 15, 3, 0, -8, -5, 7, -1, 0, 0, -8], [0, -9, 1, -4, 1, -9, 11, 1, -4, 8, 0, 0, 0, 0, 1, -1], [14, -12, -6, -6, -6, 2, 0, 22, -6, 2, 0, 0, 0, 0, 13, -13], [18, 0, 1, 0, -1, -18, 0, 0, 2, 0, 2, 6, -10, 4, 11, -3], [-9, 3, 0, -4, -4, 0, 0, -2, 0, 6, 4, 6, -3, 3, 0, 6], [22, 0, 18, -2, -18, -22, 0, 0, 4, 0, 2, -14, 4, 6, 5, 3], [-4, 1, -1, -12, -1, -3, -29, -5, -12, 20, 0, 0, 0, 0, -13, 13], [-18, 2, -8, 4, -8, -16, -7, 2, 4, 2, 0, 0, 0, 0, -4, 4], [22, 4, 0, -10, 10, 0, -2, -6, 0, 4, -2, 2, 8, 4, 0, 4], [-12, 12, 1, 0, 1, 0, 6, -8, 0, -6, 0, 0, 0, 0, 0, 0], [5, -3, 0, 0, 4, 0, -4, 0, 0, -2, -2, -6, -3, -3, 0, -2]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1600_q_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1600_2_q_g();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
function MakeNewformModFrm_1600_2_q_g(:prec:=16)
    chi := MakeCharacter_1600_q();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(1999) - 1;
    while true do
        trunc_vec := Vector(Kf, [0] cat [f_vec[i]: i in [1..prec]]);
        B := Basis(S, prec + 1);
        S_basismat := Matrix([AbsEltseq(g): g in B]);
        if Rank(S_basismat) eq Min(NumberOfRows(S_basismat), NumberOfColumns(S_basismat)) then
            S_basismat := ChangeRing(S_basismat,Kf);
            f_lincom := Solution(S_basismat,trunc_vec);
            f := &+[f_lincom[i]*Basis(S)[i] : i in [1..#Basis(S)]];
            return f;
        end if;
        error if prec eq maxprec, "Unable to distinguish newform within newspace";
        prec := Min(Ceiling(1.25 * prec), maxprec);
    end while;
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_1600_2_q_g();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
function MakeNewformModSym_1600_2_q_g( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1600_q();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![16, -64, 128, 1088, 5824, -2656, 1024, 2560, 2632, 176, 32, 80, 112, 8, 0, 0, 1]>],Snew);
    return Vf;
end function;