
// Make newform 1600.2.l.f in Magma, downloaded from the LMFDB on 17 June 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_1600_l();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1600_l_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_l_f();".
// 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_l_f();".
// 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 := [64, -128, 112, -32, -32, 48, -38, 24, -8, -4, 7, -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], [-8, -8, 0, 6, 0, 0, 0, 1, 0, -1, 0, 0], [-12, 20, -10, 4, 0, -2, 3, -2, 1, 0, 0, 0], [64, -144, 64, 40, -60, 38, -24, 12, 2, -11, 6, -1], [8, 16, -36, 18, 4, -10, 6, -5, 4, 0, -2, 1], [-192, 48, 144, -120, 20, 2, 0, 12, -22, 7, 6, -3], [-96, 272, -160, -24, 92, -66, 56, -36, 6, 17, -14, 3], [64, -88, 56, 12, -32, 26, -20, 10, 0, -5, 4, -1], [-144, 208, -96, -20, 72, -58, 40, -22, 0, 11, -8, 3], [-64, 304, -336, 80, 124, -118, 80, -56, 22, 15, -22, 9], [-352, 496, -224, -56, 164, -142, 104, -52, -6, 31, -18, 5], [352, -544, 288, 72, -188, 150, -104, 60, 2, -31, 22, -5]];
    Rf_basisdens := [1, 8, 4, 32, 8, 32, 32, 8, 16, 32, 32, 16];
    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_l();"
function MakeCharacter_1600_l()
    N := 1600;
    order := 4;
    char_gens := [1151, 901, 577];
    v := [4, 3, 4];
    // 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_l_Hecke();"
function MakeCharacter_1600_l_Hecke( : Kf := false)
    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, 1, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := ISA(Type(Kf), Fld), 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]];
    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, 1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 1, 0, 0, -1, 2, -1, 0], [0, 1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0], [-1, -1, 0, 1, -1, -1, -1, -1, 1, -1, 1, 0], [1, 0, 1, -1, -1, -1, 0, -1, 0, 0, 0, -1], [-1, -1, 0, -1, 0, 0, 0, 0, -1, 0, 0, -2], [1, 1, -1, -1, 3, 1, 0, -1, 0, 0, 0, -1], [0, 0, 0, 0, -1, 0, 1, 0, 0, -1, -1, 0], [1, 0, -1, 1, 0, -1, 0, 0, 1, -1, 2, 0], [-1, -1, -2, -2, 1, 0, 0, 0, -1, -1, 0, 0], [0, 0, 1, 0, 0, 1, 0, 0, -1, 3, 1, 0], [-1, 0, 0, 0, -1, -1, -1, -2, 0, -1, 1, 0], [-1, 0, 2, -1, 0, 2, 0, 1, 0, 2, 0, -1], [-3, 0, 2, -3, 0, 2, 0, 0, -3, 1, 1, 0], [1, 1, -2, -1, 1, 2, 0, 1, 0, 0, 0, 1], [-5, -1, 1, 5, -2, -1, -1, -1, 0, 0, 0, -1], [0, 0, -1, -1, -1, 0, -1, 0, 0, 1, -1, 0], [-1, -1, 1, 3, 1, 0, 0, 0, -1, -1, 0, 1], [-2, -5, 0, 5, -3, -3, -1, -2, 5, -3, 1, 0], [2, 3, 0, -2, 2, 0, -1, 1, 0, 0, 0, 1], [-1, -1, 1, 3, -3, 0, 1, 0, -1, 3, 1, 1], [1, 1, 0, -1, 3, 4, 0, -2, -1, 3, 0, 0], [1, 0, 3, 1, 0, 3, 0, 0, -1, 1, -2, 0], [4, 4, -5, 3, 3, 0, 1, 0, 4, -3, 1, 2], [4, 0, -1, 4, 0, -1, 0, 1, 1, -2, 1, -1], [-2, 3, -1, 2, -1, 1, -2, 0, 0, 0, 0, 0], [3, -2, 0, 2, 0, -2, -1, 0, 2, 0, 1, 0], [6, 0, 0, 0, 1, -2, -1, -2, 0, 1, 1, 0], [3, -3, 1, -3, -1, -1, -1, -1, 0, 0, 0, -1], [2, 2, 0, 1, -2, 0, -1, 0, 2, 2, -1, 0], [2, 0, 3, 2, 0, 3, 0, -1, -1, 4, 1, 1], [-2, 0, 1, -2, 0, 1, 0, -1, 3, 3, 0, 1], [-1, -1, -3, 4, 1, 0, 1, 0, -1, -1, 1, 0], [1, -2, 0, -1, -4, 0, 2, -1, 0, 0, 0, -1], [-2, -3, 4, 2, -4, -4, 1, -2, 0, 0, 0, -2], [-1, -1, 1, 4, 2, 0, 0, 0, -1, -2, 0, 0], [3, 0, -3, -3, 0, 3, -2, -1, 0, 0, 0, -1], [5, 2, 1, -5, 1, -1, 0, 1, 0, 0, 0, 1], [1, 0, -4, 1, 0, -4, 0, -3, 6, -4, 0, 3], [-7, -3, 0, 3, 0, -2, -2, 0, 3, 0, 2, 0], [-6, 2, 0, -2, -2, -3, 0, 5, -2, -2, 0, 0], [3, 0, 0, 3, 0, 0, 0, -1, 0, 2, -2, 1], [-2, -2, -2, 4, -2, 0, 0, 0, -2, 2, 0, -4], [-2, -1, 0, 2, 2, 0, -1, -3, 0, 0, 0, -3], [5, 1, 0, -1, 7, 6, 3, 0, -1, 7, -3, 0], [-4, -3, -2, 4, -3, 2, 1, 0, 0, 0, 0, 0], [-4, 0, 4, -4, 0, 4, 0, 4, -6, -2, -4, -4], [-3, -3, 6, -1, -5, 0, -1, 0, -3, 5, -1, -4], [-6, 1, 0, -1, -2, -1, 2, 0, -1, -2, -2, 0], [-3, 3, 0, -3, 1, -3, 0, -1, -3, 1, 0, 0], [5, 0, 1, 5, 0, 1, 0, 1, -2, -1, 0, -1], [-2, 0, 0, 0, 4, 4, 1, 4, 0, 4, -1, 0], [-2, -2, 2, 8, -1, 0, -3, 0, -2, 1, -3, 0], [5, 2, -1, -5, -6, 1, 4, -1, 0, 0, 0, -1], [11, -1, 0, 1, 2, -2, 0, -2, 1, 2, 0, 0], [3, 0, -4, 3, 0, -4, 0, -2, 3, -9, -2, 2], [-1, -1, 2, 3, -3, 0, 0, 0, -1, 3, 0, -4], [-8, 0, -2, -8, 0, -2, 0, 1, -2, 1, 0, -1], [0, 0, 2, 0, 0, 2, 0, 3, -3, -1, -4, -3], [7, 4, -1, -7, -1, 1, 0, 4, 0, 0, 0, 4], [-1, -1, 7, 10, -7, 0, -3, 0, -1, 7, -3, -4], [0, 0, -2, 4, -4, 0, -1, 0, 0, 4, -1, 2], [3, -2, -1, -3, -6, 1, -2, 1, 0, 0, 0, 1], [-2, 0, -1, -2, 0, -1, 0, -4, 7, -8, 3, 4], [3, 4, 0, -4, 6, 6, 3, 4, -4, 6, -3, 0], [-10, 0, 5, -10, 0, 5, 0, 2, -7, 6, -5, -2], [-1, -5, 2, 1, -1, -2, -4, 1, 0, 0, 0, 1], [4, 0, 0, 0, 0, 6, 1, 2, 0, 0, -1, 0], [0, 0, -1, 3, 4, 0, 0, 0, 0, -4, 0, 4], [-7, -1, 0, 1, -3, -4, 3, 0, 1, -3, -3, 0], [-10, 0, 1, -10, 0, 1, 0, 0, -6, 8, -2, 0], [1, 0, 5, 1, 0, 5, 0, 2, 2, 5, -4, -2], [8, -1, 0, 1, -7, -5, -3, 4, 1, -7, 3, 0], [0, 0, -5, 0, 0, -5, 0, -4, 2, -8, 2, 4], [-4, -8, -2, 4, 0, 2, 2, -4, 0, 0, 0, -4], [0, -8, 0, 8, -2, -5, -4, -3, 8, -2, 4, 0], [2, 2, 0, -3, -4, 0, -2, 0, 2, 4, -2, 2], [-16, 0, 4, 16, -7, -4, 2, 2, 0, 0, 0, 2], [4, 0, -8, 4, 0, -8, 0, 0, 8, -12, 4, 0], [5, 2, 0, -2, -6, 3, 0, 0, -2, -6, 0, 0], [1, 1, 0, -1, 7, 5, 3, -1, -1, 7, -3, 0], [1, 1, 1, -10, -2, 0, 2, 0, 1, 2, 2, 4], [2, 0, 4, 2, 0, 4, 0, 4, 2, 3, 0, -4], [-4, 0, 0, 0, -4, 5, 0, 1, 0, -4, 0, 0], [1, 1, -4, 0, -1, 0, 3, 0, 1, 1, 3, 2], [-1, 0, -1, 1, 0, 1, -4, 3, 0, 0, 0, 3], [-6, 0, 0, 0, -4, 2, 0, -6, 0, -4, 0, 0], [11, 1, 1, -11, -5, -1, -1, 0, 0, 0, 0, 0], [7, -1, 0, 1, 0, -2, -2, -2, 1, 0, 2, 0], [0, 0, -4, 16, 5, 0, 1, 0, 0, -5, 1, 0], [1, 0, -4, 1, 0, -4, 0, -4, 7, -3, -1, 4], [5, -3, -5, -5, -1, 5, -1, -1, 0, 0, 0, -1], [-5, -5, 8, -13, -3, 0, -5, 0, -5, 3, -5, -4], [-6, -6, 5, 6, 4, -5, 2, 0, 0, 0, 0, 0], [0, 0, 2, 9, 0, 0, -4, 0, 0, 0, -4, -2], [15, 0, -4, 15, 0, -4, 0, -5, 9, -2, 3, 5], [-11, 4, -3, 11, 4, 3, 4, 5, 0, 0, 0, 5], [-8, 8, -4, 8, 7, 4, -2, 1, 0, 0, 0, 1], [14, 13, -6, -14, 15, 6, 2, 0, 0, 0, 0, 0], [-6, -9, 0, 6, -3, 0, -5, 0, 0, 0, 0, 0], [8, 8, -4, 7, 0, 0, 2, 0, 8, 0, 2, 8], [6, 0, 5, 6, 0, 5, 0, 2, 1, 5, -1, -2], [3, -2, 0, 2, 0, -10, -3, -4, 2, 0, 3, 0], [9, 0, 1, 9, 0, 1, 0, 2, -2, 11, 2, -2], [-3, 2, 0, -2, 2, 2, -3, 2, -2, 2, 3, 0], [0, 0, 9, -7, -4, 0, 0, 0, 0, 4, 0, 2], [5, 5, -3, -11, 1, 0, 1, 0, 5, -1, 1, 7], [11, 1, 0, -1, -5, -4, -1, 2, -1, -5, 1, 0], [-2, 0, 3, -2, 0, 3, 0, 0, -10, 4, 2, 0], [-2, -2, 6, 10, 0, 0, 0, 0, -2, 0, 0, 0], [1, 0, -6, 1, 0, -6, 0, 4, 5, -5, 5, -4], [-5, -5, 2, 5, -6, 0, 2, 0, -5, 6, 2, 2], [3, -1, 0, 1, -3, 0, 4, -2, 1, -3, -4, 0], [-1, 5, -7, 1, 7, 7, 7, 4, 0, 0, 0, 4], [-5, -5, 4, 27, -7, 0, -3, 0, -5, 7, -3, -6], [-7, -9, 6, 7, -13, -6, -2, -5, 0, 0, 0, -5], [8, 3, -2, -8, 0, 2, 3, 1, 0, 0, 0, 1], [15, 0, 4, 15, 0, 4, 0, 2, -1, 7, -6, -2], [-4, -6, 0, 6, -8, -8, -4, -6, 6, -8, 4, 0], [0, 0, 4, 0, 0, 4, 0, 0, -4, 0, 0, 0], [2, 0, -1, 2, 0, -1, 0, 2, 7, -8, -3, -2], [-3, 8, -11, 3, 19, 11, 6, 1, 0, 0, 0, 1], [3, -5, 9, -3, -11, -9, -2, -1, 0, 0, 0, -1], [8, 0, -2, 8, 0, -2, 0, 0, 2, -2, -2, 0], [-5, 6, 0, -6, 10, 1, 0, -6, -6, 10, 0, 0], [0, 0, -5, -13, 5, 0, 3, 0, 0, -5, 3, 0], [-8, -1, -2, 8, 9, 2, 4, -2, 0, 0, 0, -2], [-17, -5, 5, 17, 1, -5, 1, -6, 0, 0, 0, -6], [-3, -3, 10, 15, -12, 0, 0, 0, -3, 12, 0, -6], [16, -5, 0, 5, -8, -13, -2, -2, 5, -8, 2, 0], [3, 0, 5, 3, 0, 5, 0, -1, 6, -2, 4, 1], [-2, -2, 0, -5, -4, 0, -5, 0, -2, 4, -5, -10], [-6, 6, 0, -6, 10, 7, 0, -1, -6, 10, 0, 0], [4, 0, -3, 4, 0, -3, 0, 1, 1, -7, -2, -1], [19, 9, -3, -19, 11, 3, 3, 1, 0, 0, 0, 1], [-10, -4, 1, 10, -6, -1, -2, -4, 0, 0, 0, -4], [6, 6, -6, -6, 6, 0, 3, 0, 6, -6, 3, -2], [-8, 0, -3, -8, 0, -3, 0, 0, 7, -13, -3, 0], [-4, 0, 7, -4, 0, 7, 0, 0, 0, -2, -4, 0], [4, 4, 2, -2, 0, 0, 0, 0, 4, 0, 0, 6], [-10, 0, -4, -10, 0, -4, 0, -1, 6, -1, 8, 1], [-15, -9, 9, 15, -23, -9, -6, -5, 0, 0, 0, -5], [3, 3, 0, -1, -3, 0, 5, 0, 3, 3, 5, -2], [8, 0, 5, 8, 0, 5, 0, -2, -12, 14, -2, 2], [5, 5, -12, 10, 5, 0, 3, 0, 5, -5, 3, 0], [-10, 0, -3, -10, 0, -3, 0, 3, 1, -4, -1, -3], [20, 3, 0, -3, 10, 7, -2, -2, -3, 10, 2, 0], [-3, 0, -1, 3, 0, 1, -6, 1, 0, 0, 0, 1], [-8, 2, 0, -2, 2, 8, 3, 8, -2, 2, -3, 0], [7, 4, 3, -7, -9, -3, 0, -2, 0, 0, 0, -2], [12, 12, -4, -20, 4, 0, 2, 0, 12, -4, 2, 2], [4, 0, 1, 4, 0, 1, 0, 4, -9, -5, 1, -4], [5, 7, 0, -7, 13, 16, -1, 12, -7, 13, 1, 0], [0, 0, -6, -18, 11, 0, 3, 0, 0, -11, 3, 4], [10, 0, 0, 0, 2, 6, 0, 0, 0, 2, 0, 0], [3, 3, 3, -1, 5, 0, 1, 0, 3, -5, 1, 3], [18, 7, -8, -18, 21, 8, 6, 2, 0, 0, 0, 2], [4, -3, 6, -4, -9, -6, -7, -2, 0, 0, 0, -2], [-3, -3, 8, 8, -3, 0, -1, 0, -3, 3, -1, -8], [-5, -5, 0, 15, -11, 0, 3, 0, -5, 11, 3, 0], [-11, 0, 2, -11, 0, 2, 0, 8, -3, 4, -3, -8], [4, 2, 0, -2, -4, 1, -2, 7, -2, -4, 2, 0], [-7, -7, 1, -10, -5, 0, -3, 0, -7, 5, -3, 2], [-21, 2, 0, -2, -10, -11, 4, 4, -2, -10, -4, 0], [3, 0, -11, 3, 0, -11, 0, 3, 10, -10, 4, -3], [13, -3, 0, 3, -1, 3, -3, -3, 3, -1, 3, 0], [-10, 0, 1, -10, 0, 1, 0, -5, 5, -11, 0, 5], [-14, 0, 6, -14, 0, 6, 0, -2, -4, 11, 2, 2], [7, -7, -4, -7, -3, 4, 0, -1, 0, 0, 0, -1], [11, 11, -3, 8, -3, 0, 1, 0, 11, 3, 1, 10], [7, 7, -7, -9, -5, 0, 5, 0, 7, 5, 5, 3], [-15, -4, 0, 4, -5, -11, -1, 0, 4, -5, 1, 0], [2, 2, -7, -6, 14, 0, 2, 0, 2, -14, 2, -1], [1, 0, 3, 1, 0, 3, 0, 1, 4, -3, -2, -1], [-2, 0, 6, -2, 0, 6, 0, 5, -5, 17, -2, -5], [6, 6, -8, 2, -2, 0, -2, 0, 6, 2, -2, 4], [-5, -2, -2, 5, 8, 2, 6, -1, 0, 0, 0, -1], [20, 0, 0, 0, 12, 4, 2, 2, 0, 12, -2, 0], [16, -2, 0, -16, 1, 0, 6, 1, 0, 0, 0, 1], [-3, 0, 3, -3, 0, 3, 0, -2, 1, -5, 2, 2], [-6, -6, 11, 0, -4, 0, -1, 0, -6, 4, -1, -9], [-10, 1, 0, -1, 6, 11, 0, 6, -1, 6, 0, 0], [-15, 0, 11, -15, 0, 11, 0, 1, -4, 0, -4, -1], [-9, -1, -4, 9, 7, 4, 2, -1, 0, 0, 0, -1], [-10, -10, 2, 10, 5, -2, 2, -1, 0, 0, 0, -1], [-5, -5, 10, 10, -11, 0, -5, 0, -5, 11, -5, 4], [-6, -5, 0, 5, 10, 1, 0, -6, 5, 10, 0, 0], [-2, -17, 0, 17, -13, -14, -1, -6, 17, -13, 1, 0], [-2, 0, 6, -2, 0, 6, 0, 1, -8, 11, 0, -1], [-8, 13, -2, 8, 2, 2, -5, 4, 0, 0, 0, 4], [-9, 0, 8, 9, -10, -8, -2, -1, 0, 0, 0, -1], [-4, -12, 4, 4, -9, -4, 4, -4, 0, 0, 0, -4], [0, 0, 10, 2, -10, 0, -6, 0, 0, 10, -6, 8], [16, 3, 0, -3, 15, 2, -1, -2, -3, 15, 1, 0], [30, 12, -1, -30, 2, 1, 4, -2, 0, 0, 0, -2], [0, 7, 0, -7, 3, 12, 4, 4, -7, 3, -4, 0], [-2, -2, -3, 27, 6, 0, 0, 0, -2, -6, 0, 0], [9, -6, -2, -9, 8, 2, -10, -3, 0, 0, 0, -3], [13, 1, 0, -1, 11, -6, 3, 4, -1, 11, -3, 0], [7, 0, 1, 7, 0, 1, 0, 3, -12, 12, -8, -3], [6, -1, 0, 1, 5, -2, 4, -4, 1, 5, -4, 0], [18, 0, -3, 18, 0, -3, 0, -5, 9, -3, 3, 5], [22, 10, -9, -22, 12, 9, 4, 6, 0, 0, 0, 6], [16, 2, 0, -2, 3, 2, -1, 12, -2, 3, 1, 0], [5, 1, -5, -5, 11, 5, -5, -5, 0, 0, 0, -5], [-8, -8, -8, -14, -6, 0, 2, 0, -8, 6, 2, 2], [-11, 0, 2, -11, 0, 2, 0, -4, 1, -3, 5, 4], [-13, 5, 0, -5, 7, -4, -3, -2, -5, 7, 3, 0], [6, 0, -2, 6, 0, -2, 0, 3, -3, -17, 6, -3], [3, 3, -1, 2, 6, 0, -2, 0, 3, -6, -2, -4], [-4, 0, -13, -4, 0, -13, 0, 1, 5, -18, -3, -1], [4, 4, -16, 0, 15, 0, 5, 0, 4, -15, 5, 2], [8, 8, 0, -1, 8, 0, 3, 0, 8, -8, 3, 0], [-1, 7, 0, -7, 14, 18, 8, 10, -7, 14, -8, 0], [0, 0, 0, 0, -6, -7, 1, 3, 0, -6, -1, 0], [-5, -5, 9, -8, -7, 0, -5, 0, -5, 7, -5, 4], [-28, 5, -1, 28, -3, 1, -2, 2, 0, 0, 0, 2], [11, 0, -4, 11, 0, -4, 0, -3, 20, -10, 10, 3], [-3, -3, 3, -8, -12, 0, -6, 0, -3, 12, -6, -2], [4, 4, 0, -4, 2, 1, 3, 11, -4, 2, -3, 0], [-8, 1, 0, -1, -13, -7, -3, -2, -1, -13, 3, 0], [12, 6, -12, -12, 11, 12, 4, 10, 0, 0, 0, 10], [-18, 0, 2, -18, 0, 2, 0, 7, -1, -7, -6, -7], [-14, -14, 17, -6, 0, 0, 2, 0, -14, 0, 2, -3], [-4, 9, 0, -9, 14, 17, 6, 10, -9, 14, -6, 0], [-13, -13, 17, 12, -12, 0, -2, 0, -13, 12, -2, 2], [-13, 0, 3, -13, 0, 3, 0, 3, -2, 11, 12, -3], [29, 7, -4, -29, 15, 4, 4, 3, 0, 0, 0, 3], [8, -5, 6, -8, 10, -6, -7, -5, 0, 0, 0, -5], [2, -11, 0, 11, -1, -17, -5, -12, 11, -1, 5, 0], [-5, -5, -13, -13, 11, 0, 4, 0, -5, -11, 4, 1], [-1, 0, 12, -1, 0, 12, 0, 4, -19, 17, -11, -4], [-27, 12, 0, -12, 8, 9, 4, 6, -12, 8, -4, 0], [5, 5, 0, -5, -7, -3, 7, 1, -5, -7, -7, 0], [20, 0, 3, 20, 0, 3, 0, 7, -7, 1, -2, -7], [-11, 0, 9, -11, 0, 9, 0, 1, -14, 9, -10, -1], [2, 2, -4, -14, 14, 0, -2, 0, 2, -14, -2, -4], [11, 22, -13, -11, 15, 13, 8, 6, 0, 0, 0, 6], [5, 0, -5, 5, 0, -5, 0, 0, 8, -3, 4, 0], [2, 2, -12, -14, 13, 0, -1, 0, 2, -13, -1, -4], [14, 16, -12, -14, 28, 12, 2, 2, 0, 0, 0, 2], [10, -9, 0, 9, -7, -10, 6, -4, 9, -7, -6, 0], [4, 4, -18, 2, 4, 0, 0, 0, 4, -4, 0, -2], [20, -3, 0, 3, -17, -11, -3, 0, 3, -17, 3, 0], [17, -7, 7, -17, -13, -7, -1, -4, 0, 0, 0, -4], [-19, 0, 15, -19, 0, 15, 0, 10, -28, 9, -10, -10], [-18, -1, 0, 1, -8, -19, 2, 0, 1, -8, -2, 0], [-5, -6, -3, 5, -6, 3, 4, 9, 0, 0, 0, 9], [6, 3, 0, -3, 3, 14, 1, -6, -3, 3, -1, 0], [11, 11, -15, 8, 5, 0, 1, 0, 11, -5, 1, 10], [-5, -5, 5, 1, -9, 0, 3, 0, -5, 9, 3, -5], [-8, 9, -8, 8, 11, 8, 0, 4, 0, 0, 0, 4], [19, -5, 1, -19, -2, -1, 5, -2, 0, 0, 0, -2], [7, 0, 4, 7, 0, 4, 0, 10, -3, -3, -10, -10], [1, 0, 3, 1, 0, 3, 0, -6, 8, 13, -6, 6], [1, 0, 15, 1, 0, 15, 0, 2, -11, 27, -10, -2], [6, 6, 2, 9, 0, 0, -2, 0, 6, 0, -2, -4], [-8, 9, 0, -9, 3, 21, 9, 4, -9, 3, -9, 0], [-7, 1, -13, 7, 18, 13, 3, 3, 0, 0, 0, 3], [4, 0, 11, 4, 0, 11, 0, -1, -3, 21, -10, 1], [-15, 7, -9, 15, 9, 9, 4, 3, 0, 0, 0, 3], [32, 2, 0, -2, 2, 9, 3, -7, -2, 2, -3, 0], [10, 4, -2, -10, -3, 2, -6, -1, 0, 0, 0, -1], [-5, -4, -2, 5, 14, 2, -2, 1, 0, 0, 0, 1], [3, 3, 8, 21, -7, 0, -2, 0, 3, 7, -2, 10], [-6, 0, 5, -6, 0, 5, 0, -4, 1, 0, -1, 4], [-3, 0, -11, -3, 0, -11, 0, 0, 7, 1, -4, 0], [-19, -3, 9, 19, -5, -9, 8, -3, 0, 0, 0, -3], [12, 3, -6, -12, 8, 6, -7, -8, 0, 0, 0, -8], [7, 7, -19, 15, 9, 0, -2, 0, 7, -9, -2, 5], [0, -3, 0, 3, 0, -7, 2, 10, 3, 0, -2, 0], [17, 9, 0, -9, 9, 10, 7, 2, -9, 9, -7, 0], [5, 5, -8, 13, 3, 0, 1, 0, 5, -3, 1, 0], [6, 0, 3, 6, 0, 3, 0, -4, -9, -8, -5, 4], [-15, 0, -11, 15, 8, 11, 4, -3, 0, 0, 0, -3], [-5, -5, 8, 19, -3, 0, -3, 0, -5, 3, -3, 14], [-9, -18, -1, 9, -1, 1, -6, -6, 0, 0, 0, -6], [14, -10, 0, 10, -2, -20, -4, -2, 10, -2, 4, 0], [-4, -4, -4, 0, 15, 0, -1, 0, -4, -15, -1, 0], [-9, -9, 2, -35, 1, 0, 5, 0, -9, -1, 5, -4], [18, 0, -2, 18, 0, -2, 0, -2, 16, 0, -4, 2], [10, 0, 2, 10, 0, 2, 0, 5, 8, 3, -4, -5], [-19, -13, 0, 13, -3, -12, -7, -6, 13, -3, 7, 0], [4, 10, 0, -10, 8, 22, 0, 16, -10, 8, 0, 0], [-1, 0, -4, -1, 0, -4, 0, 2, -5, 19, 6, -2], [1, 1, -10, -17, 11, 0, -1, 0, 1, -11, -1, -8], [-11, -15, 0, 15, -11, -14, -7, -14, 15, -11, 7, 0], [24, 25, -7, -24, 17, 7, 8, 4, 0, 0, 0, 4], [8, 3, 2, -8, 15, -2, 7, 2, 0, 0, 0, 2], [14, 14, -9, 4, 10, 0, 6, 0, 14, -10, 6, 5], [-11, 0, -7, -11, 0, -7, 0, -8, 6, -1, -6, 8], [-13, -2, -7, 13, -6, 7, -4, -7, 0, 0, 0, -7], [2, 2, 7, -2, -20, -7, -10, 4, 0, 0, 0, 4], [13, 0, 0, 0, -4, 5, 0, -6, 0, -4, 0, 0], [-27, 0, -6, -27, 0, -6, 0, -2, 13, -15, 0, 2], [2, 0, -9, 2, 0, -9, 0, 2, 15, -29, 5, -2], [-13, -21, 5, 13, -4, -5, -5, -4, 0, 0, 0, -4], [-6, -6, 16, 27, -22, 0, -1, 0, -6, 22, -1, -4], [-3, 19, -14, 3, 23, 14, 6, 3, 0, 0, 0, 3], [-33, -7, 0, 7, -9, -12, -7, 0, 7, -9, 7, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1600_l_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1600_2_l_f();".
// 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_l_f(:prec:=12)
    chi := MakeCharacter_1600_l();
    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_l_f();".
// 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_l_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1600_l();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![1, 18, 162, 282, 243, -20, 36, 60, 51, -6, 2, 2, 1]>,<7,R![256, 0, 2304, 0, 3776, 0, 2144, 0, 484, 0, 40, 0, 1]>],Snew);
    return Vf;
end function;