
// Make newform 1600.2.s.c in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1600_s();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1600_s_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_s_c();".
// 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_s_c();".
// 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, 0, -128, 0, 96, 0, -48, 0, 36, 0, -12, 0, 6, 0, -2, 0, 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], [-48, 0, 48, 0, -20, 0, 12, 0, -6, 0, 2, 0, -1, 0, 0, 0], [0, 320, 0, -432, 0, 224, 0, -60, 0, 68, 0, -42, 0, 18, 0, -1], [0, 1088, 0, 480, 0, 0, 0, 132, 0, 108, 0, -42, 0, -6, 0, 5], [0, -96, 0, -72, 0, 0, 0, 32, 0, -12, 0, 0, 0, -4, 0, 1], [0, -128, 0, 144, 0, -24, 0, 36, 0, -24, 0, 6, 0, 0, 0, 1], [0, 32, 0, 276, 0, 84, 0, 78, 0, 18, 0, 21, 0, 6, 0, 2], [0, -384, 0, 48, 0, -224, 0, 44, 0, -20, 0, -14, 0, -2, 0, -3], [0, 64, 0, -176, 0, 280, 0, -124, 0, 64, 0, -42, 0, 12, 0, -3], [-832, 0, 384, 0, -336, 0, 156, 0, -132, 0, 42, 0, -30, 0, 11, 0], [1600, 0, -256, 0, 448, 0, -244, 0, 116, 0, -14, 0, 6, 0, -5, 0], [64, 0, 48, 0, -84, 0, 156, 0, -6, 0, 42, 0, -9, 0, 4, 0], [0, 0, -64, 0, 0, 0, -4, 0, 4, 0, -6, 0, -2, 0, -1, 0], [0, 1504, 0, -216, 0, 420, 0, -156, 0, 90, 0, -42, 0, 9, 0, -11], [-640, 0, 1536, 0, 672, 0, 540, 0, -108, 0, 42, 0, 6, 0, 23, 0], [832, 0, 2304, 0, 0, 0, 348, 0, 132, 0, 42, 0, 30, 0, 31, 0]];
    Rf_basisdens := [1, 16, 448, 1344, 224, 192, 336, 448, 448, 1344, 448, 336, 64, 672, 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_s();"
function MakeCharacter_1600_s()
    N := 1600;
    order := 4;
    char_gens := [1151, 901, 577];
    v := [2, 1, 1];
    // 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_s_Hecke();"
function MakeCharacter_1600_s_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, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 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, -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, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0], [0, 0, 0, -1, 0, -1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -1, 1, -1, -1, 0, 0, 0, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0], [0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, -1, 1], [-1, 1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 1], [0, 0, 0, -2, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0], [-1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, -1, 1], [0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, -2, 0, 0], [0, 0, -1, 0, 3, 0, 1, 0, 0, 0, 0, 0, 0, -1, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 1, 1], [-2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 1, 0, 1, 0], [0, 0, 0, 2, 0, 1, 1, 0, -2, 0, 0, 0, 0, 1, 0, 0], [1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 2, 0, 0, 1], [0, 0, 1, 1, 0, 0, -1, 1, -1, 0, 0, 0, 0, 0, 0, 0], [6, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 1, -2, 0, 0, 0], [0, 0, 0, 0, 1, 0, 1, 2, 2, 0, 0, 0, 0, -1, 0, 0], [-1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 1], [0, 0, 1, 1, 2, -2, -3, -1, -1, 0, 0, 0, 0, 0, 0, 0], [3, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 1, -1, 0, 1, 1], [0, 0, -1, 1, 2, 2, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0], [0, 0, 2, 0, -1, 0, -2, 0, 0, 0, 0, 0, 0, 2, 0, 0], [-2, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -2, -2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0], [0, 0, -3, 3, 2, 2, 0, 0, 4, 0, 0, 0, 0, -4, 0, 0], [-5, 0, 0, 0, 0, 0, 0, 0, 0, -4, 0, 0, 1, 0, -1, 1], [0, 0, -3, 3, -1, -1, 0, 0, 1, 0, 0, 0, 0, -2, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 2, 0, -2, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 2, 0, -1, 0, -1, 0], [-6, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, -1, -2, 0, 0, 0], [0, 0, 2, 0, 6, 0, 2, 0, 0, 0, 0, 0, 0, -2, 0, 0], [0, 0, 4, 0, -1, 0, 0, -2, -2, 0, 0, 0, 0, 0, 0, 0], [0, 0, -3, -3, 2, -2, 2, 3, 3, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 2, 0, -4, -1, -1, -1, 0, 0, 0, 0, -1, 0, 0], [3, 2, 0, 0, 0, 0, 0, 0, 0, -6, 1, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, -1, 1, 0, -1, -1], [2, -2, 0, 0, 0, 0, 0, 0, 0, -6, 0, 0, 0, 0, 4, -2], [0, 0, -3, 3, 4, 4, 0, 0, -2, 0, 0, 0, 0, 1, 0, 0], [0, 0, 0, -1, 0, 3, 2, -1, 2, 0, 0, 0, 0, 2, 0, 0], [1, -1, 0, 0, 0, 0, 0, 0, 0, -6, -3, -3, 0, 0, 2, -1], [7, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 2, 1, 0, -1, -1], [0, 0, -1, -1, 4, -4, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-4, 1, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, -3, 0, -3, 0], [0, 0, 1, 1, 2, -2, 1, 3, -1, 0, 0, 0, 0, 0, 0, 0], [-1, -1, 0, 0, 0, 0, 0, 0, 0, -3, 4, -4, 0, 0, 0, 1], [-2, 0, 0, 0, 0, 0, 0, 0, 0, 2, -2, 2, 3, 0, 0, 0], [9, -2, 0, 0, 0, 0, 0, 0, 0, -6, -1, 0, 0, 0, 0, 0], [0, 0, 1, 1, 3, -3, -2, -4, -1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -2, -2, 0, 0, -3, 0, 0, 0, 0, 2, 0, 0], [6, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, -2, -1, 0, 1, 0], [2, -2, 0, 0, 0, 0, 0, 0, 0, 11, -3, -3, 0, 0, 2, -2], [0, 0, 0, -4, 0, -6, -1, 1, 3, 0, 0, 0, 0, -1, 0, 0], [2, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -2], [0, 0, 0, 2, 0, -3, 2, 2, -4, 0, 0, 0, 0, 2, 0, 0], [0, 0, 0, 0, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 2, 0, 1, 1, -6, 4, 0, 0, 0, 0, 1, 0, 0], [-13, -1, 0, 0, 0, 0, 0, 0, 0, 3, -2, 2, 0, 0, 0, 1], [0, 0, -3, -3, -1, 1, -2, 4, 3, 0, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, 3, 0, -2, -3, -3, 0, 0, 0, 0, 2, 0, 0], [-8, 1, 0, 0, 0, 0, 0, 0, 0, 11, -4, 0, -1, 0, -1, 0], [0, 0, -1, -1, -5, 5, 0, 2, 1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 5, 0, 1, -2, -2, 0, 0, 0, 0, -1, 0, 0], [-7, 0, 0, 0, 0, 0, 0, 0, 0, -10, 0, -3, 1, 0, -1, -3], [0, 0, 4, -4, 0, 0, 0, 0, -4, 0, 0, 0, 0, 0, 0, 0], [-3, 3, 0, 0, 0, 0, 0, 0, 0, -1, 2, 2, 0, 0, 0, 3], [0, 0, -3, 3, -4, -4, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0], [0, 0, 2, 0, 8, 0, 1, 1, 1, 0, 0, 0, 0, -1, 0, 0], [-9, 0, 0, 0, 0, 0, 0, 0, 0, -9, 0, -1, 2, 0, -2, 0], [0, 0, 1, 1, -4, 4, 2, 3, -1, 0, 0, 0, 0, 0, 0, 0], [4, 4, 0, 0, 0, 0, 0, 0, 0, -10, 2, 0, 1, 0, 1, 0], [0, 0, -5, 0, 1, 0, 2, 3, 3, 0, 0, 0, 0, -2, 0, 0], [1, -1, 0, 0, 0, 0, 0, 0, 0, 4, -3, 3, 0, 0, 0, 1], [-1, 1, 0, 0, 0, 0, 0, 0, 0, -2, 1, -1, -1, 0, 0, -1], [-13, 2, 0, 0, 0, 0, 0, 0, 0, 10, 1, 0, 2, 0, 2, 0], [10, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, -2, -5, 0, 5, -4], [1, -1, 0, 0, 0, 0, 0, 0, 0, 19, 2, 2, 0, 0, 0, -1], [0, 0, -4, 4, 5, 5, 0, 0, 1, 0, 0, 0, 0, -5, 0, 0], [1, -1, 0, 0, 0, 0, 0, 0, 0, 11, 2, 2, 0, 0, -4, -1], [0, 0, 0, 6, 0, 5, -2, -8, 2, 0, 0, 0, 0, -2, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 2, 2, 0, 0, 3, 0], [0, 0, 5, -5, -7, -7, 0, 0, -1, 0, 0, 0, 0, 4, 0, 0], [2, -5, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 3, 0, 3, 0], [0, 0, 0, 0, 8, -8, 4, -3, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 2, 0, -3, -2, 2, -4, 0, 0, 0, 0, -2, 0, 0], [-1, 1, 0, 0, 0, 0, 0, 0, 0, -4, 3, -3, -6, 0, 0, -1], [0, 0, 4, 4, -2, 2, -2, -1, -4, 0, 0, 0, 0, 0, 0, 0], [-6, -1, 0, 0, 0, 0, 0, 0, 0, 5, 2, 0, -5, 0, -5, 0], [7, -4, 0, 0, 0, 0, 0, 0, 0, 0, -3, 0, 2, 0, 2, 0], [0, 0, 1, -1, 8, 8, 0, 0, 4, 0, 0, 0, 0, 2, 0, 0], [3, 0, 0, 0, 0, 0, 0, 0, 0, 10, 0, -1, 1, 0, -1, 7], [-2, 2, 0, 0, 0, 0, 0, 0, 0, -10, 0, 0, 0, 0, -6, 2], [0, 0, 0, 6, 0, 5, -2, 0, -6, 0, 0, 0, 0, -2, 0, 0], [0, -1, 0, 0, 0, 0, 0, 0, 0, 2, -1, 0, -3, 0, -3, 0], [0, 0, 0, -6, 0, -11, 2, 2, 4, 0, 0, 0, 0, 2, 0, 0], [0, 0, -2, 0, -2, 0, -2, 0, 0, 0, 0, 0, 0, 2, 0, 0], [0, 0, -2, 0, 3, 0, 2, -4, -4, 0, 0, 0, 0, -2, 0, 0], [9, -1, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, -5, 0, 0, 1], [9, 4, 0, 0, 0, 0, 0, 0, 0, -16, 3, 0, 0, 0, 0, 0], [0, 0, 4, 4, 4, -4, 4, 0, -4, 0, 0, 0, 0, 0, 0, 0], [0, 0, -6, 0, 9, 0, 3, -2, -2, 0, 0, 0, 0, -3, 0, 0], [0, 0, -1, 1, -1, -1, 0, 0, 1, 0, 0, 0, 0, -4, 0, 0], [-2, 2, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, -8, 2], [-2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 4, 4, 0, 0, -3, 2], [0, 0, -4, 4, 6, 6, 0, 0, -1, 0, 0, 0, 0, 2, 0, 0], [0, 0, 1, 0, -11, 0, -1, 0, 0, 0, 0, 0, 0, 1, 0, 0], [0, 0, 4, -4, -7, -7, 0, 0, -3, 0, 0, 0, 0, -1, 0, 0], [9, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 2, -3, 0, 3, -3], [-16, 2, 0, 0, 0, 0, 0, 0, 0, -1, -1, 1, 2, 0, 0, -2], [2, -4, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 7, 0, 0, 4], [0, 0, 10, 0, -5, 0, -7, -10, -10, 0, 0, 0, 0, 7, 0, 0], [0, 0, 0, 0, -2, 2, -2, 5, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 7, 0, 1, -1, -10, 3, 0, 0, 0, 0, -1, 0, 0], [13, -2, 0, 0, 0, 0, 0, 0, 0, -18, 7, 0, 0, 0, 0, 0], [-2, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, -4, -2, 0, 2, 4], [0, 0, -5, 5, 0, 0, 0, 0, 6, 0, 0, 0, 0, -5, 0, 0], [0, 0, 0, 5, 0, 5, -4, -5, 0, 0, 0, 0, 0, -4, 0, 0], [0, 0, 0, -4, 0, -7, -1, 2, 2, 0, 0, 0, 0, -1, 0, 0], [11, 0, 0, 0, 0, 0, 0, 0, 0, 14, 0, -2, 5, 0, -5, 3], [4, -4, 0, 0, 0, 0, 0, 0, 0, -4, 4, 0, 4, 0, 4, 0], [12, -5, 0, 0, 0, 0, 0, 0, 0, -10, 3, 0, 5, 0, 5, 0], [12, -2, 0, 0, 0, 0, 0, 0, 0, 3, -1, 1, 2, 0, 0, 2], [0, 0, 4, 4, 0, 0, -10, -9, -4, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -2, 0, 0, 7, 7, -5, 0, 0, 0, 0, 7, 0, 0], [-3, 0, 0, 0, 0, 0, 0, 0, 0, 8, -5, 0, 2, 0, 2, 0], [0, 0, 8, -8, -2, -2, 0, 0, -7, 0, 0, 0, 0, 2, 0, 0], [-1, 1, 0, 0, 0, 0, 0, 0, 0, 22, 3, 3, 0, 0, 2, 1], [0, 0, 0, 0, 0, 2, -5, -7, 7, 0, 0, 0, 0, -5, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -4, 2, 2, 0, 0, 6, 0], [1, -1, 0, 0, 0, 0, 0, 0, 0, -12, -3, -3, 0, 0, -4, -1], [0, 0, -1, 0, -5, 0, -3, -8, -8, 0, 0, 0, 0, 3, 0, 0], [0, 0, 0, -8, 0, 5, -1, 8, 0, 0, 0, 0, 0, -1, 0, 0], [0, 0, 3, 0, -9, 0, -3, -8, -8, 0, 0, 0, 0, 3, 0, 0], [2, -2, 0, 0, 0, 0, 0, 0, 0, -2, 4, -4, 2, 0, 0, 2], [-15, 2, 0, 0, 0, 0, 0, 0, 0, 10, 3, 0, -2, 0, -2, 0], [6, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0], [0, 0, 1, -1, 12, 12, 0, 0, -4, 0, 0, 0, 0, -2, 0, 0], [0, 0, 2, 0, -5, 0, 1, 0, 0, 0, 0, 0, 0, -1, 0, 0], [16, 0, 0, 0, 0, 0, 0, 0, 0, 12, 0, 2, -3, 0, 3, -4], [-2, 2, 0, 0, 0, 0, 0, 0, 0, 1, -1, -1, 0, 0, 6, 2], [0, 0, 9, 0, -9, 0, 0, -1, -1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 5, -5, -3, -3, 0, 0, 7, 0, 0, 0, 0, 0, 0, 0], [-3, 0, 0, 0, 0, 0, 0, 0, 0, -4, 0, 4, -3, 0, 3, -1], [0, 0, 8, 8, 0, 0, -4, -13, -8, 0, 0, 0, 0, 0, 0, 0], [0, 0, -6, 0, 10, 0, 4, -2, -2, 0, 0, 0, 0, -4, 0, 0], [-11, -1, 0, 0, 0, 0, 0, 0, 0, -6, 7, -7, -1, 0, 0, 1], [0, 0, 6, 0, -3, 0, -6, -8, -8, 0, 0, 0, 0, 6, 0, 0], [0, 0, -4, -4, -4, 4, 6, 9, 4, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0, 4, 0, 0, 0, 0, 0, 0, -4, 0, 0], [-1, 1, 0, 0, 0, 0, 0, 0, 0, 25, 2, 2, 0, 0, -4, 1], [4, -4, 0, 0, 0, 0, 0, 0, 0, 20, 0, 0, 0, 0, 4, -4], [-2, 2, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 0, 0, 5, 2], [0, 0, -2, 2, -1, -1, 0, 0, -1, 0, 0, 0, 0, 3, 0, 0], [-6, -3, 0, 0, 0, 0, 0, 0, 0, 4, 5, 0, -1, 0, -1, 0], [0, 0, 0, -6, 0, -5, -1, 0, 6, 0, 0, 0, 0, -1, 0, 0], [0, 0, 1, 1, 1, -1, 0, -6, -1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 3, 3, 0, 0, 0, -7, -3, 0, 0, 0, 0, 0, 0, 0], [-2, 5, 0, 0, 0, 0, 0, 0, 0, 3, -6, 0, -3, 0, -3, 0], [0, 0, 0, 0, 1, -1, 1, 9, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -4, 4, -14, -14, 0, 0, 3, 0, 0, 0, 0, 4, 0, 0], [4, -4, 0, 0, 0, 0, 0, 0, 0, -2, 2, 2, 0, 0, 12, -4], [0, 0, 0, 3, 0, -11, -1, -2, -1, 0, 0, 0, 0, -1, 0, 0], [-5, 5, 0, 0, 0, 0, 0, 0, 0, -8, 3, 3, 0, 0, -12, 5], [0, 0, -3, 0, 1, 0, -5, 0, 0, 0, 0, 0, 0, 5, 0, 0], [-13, 0, 0, 0, 0, 0, 0, 0, 0, -13, 0, -1, 0, 0, 0, 0], [-2, 5, 0, 0, 0, 0, 0, 0, 0, -8, 5, 0, 1, 0, 1, 0], [30, 4, 0, 0, 0, 0, 0, 0, 0, -5, 1, -1, -6, 0, 0, -4], [0, 0, 5, 5, -8, 8, -9, -15, -5, 0, 0, 0, 0, 0, 0, 0], [11, -3, 0, 0, 0, 0, 0, 0, 0, 0, 3, -3, 2, 0, 0, 3], [1, 3, 0, 0, 0, 0, 0, 0, 0, 0, -3, 3, 4, 0, 0, -3], [23, -4, 0, 0, 0, 0, 0, 0, 0, -16, -3, 0, 2, 0, 2, 0], [21, 0, 0, 0, 0, 0, 0, 0, 0, 22, 0, 9, 3, 0, -3, 1], [0, 0, -3, 3, 6, 6, 0, 0, 12, 0, 0, 0, 0, -6, 0, 0], [-12, 0, 0, 0, 0, 0, 0, 0, 0, -16, 0, 4, -2, 0, 2, -4], [0, 0, 1, -1, 0, 0, 0, 0, -4, 0, 0, 0, 0, -2, 0, 0], [-9, 0, 0, 0, 0, 0, 0, 0, 0, -8, 0, 4, -1, 0, 1, 1], [0, 0, 9, 0, -3, 0, 3, 0, 0, 0, 0, 0, 0, -3, 0, 0], [0, 0, -6, 6, 5, 5, 0, 0, -7, 0, 0, 0, 0, -3, 0, 0], [0, 0, 0, 0, 8, -8, -6, -7, 0, 0, 0, 0, 0, 0, 0, 0], [-6, 1, 0, 0, 0, 0, 0, 0, 0, -4, 9, 0, 1, 0, 1, 0], [0, 0, -8, 0, -6, 0, 1, 11, 11, 0, 0, 0, 0, -1, 0, 0], [0, 0, 0, 0, -13, 0, -1, 2, 2, 0, 0, 0, 0, 1, 0, 0], [-1, -3, 0, 0, 0, 0, 0, 0, 0, 6, -3, 3, 1, 0, 0, 3], [1, -1, 0, 0, 0, 0, 0, 0, 0, -30, 5, 5, 0, 0, -2, -1], [0, 0, -1, 1, 1, 1, 0, 0, -5, 0, 0, 0, 0, -6, 0, 0], [0, 0, 0, 4, 0, -1, -4, -10, 6, 0, 0, 0, 0, -4, 0, 0], [13, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, -2, 1, 0, -1, -7], [-8, 4, 0, 0, 0, 0, 0, 0, 0, 8, -4, 0, -2, 0, -2, 0], [0, 0, 0, 12, 0, -1, -3, -6, -6, 0, 0, 0, 0, -3, 0, 0], [0, 0, 0, 0, -9, 9, -5, 3, 0, 0, 0, 0, 0, 0, 0, 0], [5, -5, 0, 0, 0, 0, 0, 0, 0, 4, 1, -1, -1, 0, 0, 5], [0, 0, 0, -9, 0, 1, -1, -2, 11, 0, 0, 0, 0, -1, 0, 0], [0, 0, -11, -11, 7, -7, 2, 12, 11, 0, 0, 0, 0, 0, 0, 0], [0, 0, 1, -1, -10, -10, 0, 0, -4, 0, 0, 0, 0, 10, 0, 0], [15, 0, 0, 0, 0, 0, 0, 0, 0, 18, 0, -5, 5, 0, -5, 3], [-3, 3, 0, 0, 0, 0, 0, 0, 0, 8, -1, -1, 0, 0, -10, 3], [0, 0, 0, 1, 0, 3, 7, 0, -1, 0, 0, 0, 0, 7, 0, 0], [2, -2, 0, 0, 0, 0, 0, 0, 0, -8, -6, -6, 0, 0, -4, -2], [21, 0, 0, 0, 0, 0, 0, 0, 0, 20, 0, -8, -1, 0, 1, -1], [0, 0, 2, 0, -6, 0, 6, 0, 0, 0, 0, 0, 0, -6, 0, 0], [-24, 2, 0, 0, 0, 0, 0, 0, 0, -5, 3, -3, 2, 0, 0, -2], [0, 0, 0, 0, 9, 0, 5, 8, 8, 0, 0, 0, 0, -5, 0, 0], [1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, -4, 0, 0, 1], [-22, -1, 0, 0, 0, 0, 0, 0, 0, 17, 6, 0, -3, 0, -3, 0], [0, 0, 8, 8, -7, 7, -7, -15, -8, 0, 0, 0, 0, 0, 0, 0], [-16, 0, 0, 0, 0, 0, 0, 0, 0, -12, 0, -2, 1, 0, -1, 4], [0, 0, -4, 4, -8, -8, 0, 0, 7, 0, 0, 0, 0, -12, 0, 0], [0, 0, 6, 0, -13, 0, -3, -4, -4, 0, 0, 0, 0, 3, 0, 0], [-1, 1, 0, 0, 0, 0, 0, 0, 0, -10, -5, -5, 0, 0, -6, 1], [3, -3, 0, 0, 0, 0, 0, 0, 0, 14, -1, -1, 0, 0, 5, -3], [0, 0, -3, 3, -6, -6, 0, 0, -8, 0, 0, 0, 0, -6, 0, 0], [17, 3, 0, 0, 0, 0, 0, 0, 0, 0, -3, 3, -4, 0, 0, -3], [0, 0, -5, -5, 8, -8, 8, 1, 5, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 15, 0, 13, -7, -12, -3, 0, 0, 0, 0, -7, 0, 0], [8, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 6, 0, -6, 0], [-1, 1, 0, 0, 0, 0, 0, 0, 0, 17, -6, -6, 0, 0, -4, 1], [-1, 1, 0, 0, 0, 0, 0, 0, 0, -10, 3, 3, 0, 0, -4, 1], [0, 0, -1, -1, 6, -6, -8, 1, 1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 2, 0, 13, 5, 2, -4, 0, 0, 0, 0, 5, 0, 0], [6, 0, 0, 0, 0, 0, 0, 0, 0, -6, 0, 0, 2, 0, 2, 0], [0, 0, -9, -9, 14, -14, 3, 9, 9, 0, 0, 0, 0, 0, 0, 0], [11, 7, 0, 0, 0, 0, 0, 0, 0, -9, 2, -2, -8, 0, 0, -7], [0, 0, -9, -9, 8, -8, -2, 1, 9, 0, 0, 0, 0, 0, 0, 0], [-6, -11, 0, 0, 0, 0, 0, 0, 0, 19, -2, 0, 5, 0, 5, 0], [0, 0, 0, -9, 0, -11, 5, 4, 5, 0, 0, 0, 0, 5, 0, 0], [-9, 4, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 4, 0, 4, 0], [-2, 2, 0, 0, 0, 0, 0, 0, 0, -16, -6, -6, 0, 0, 0, 2], [4, -4, 0, 0, 0, 0, 0, 0, 0, 26, 0, 0, 0, 0, 9, -4], [0, 0, 0, -4, 0, 5, 3, 8, -4, 0, 0, 0, 0, 3, 0, 0], [0, 0, 3, -3, 6, 6, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0], [2, -2, 0, 0, 0, 0, 0, 0, 0, 4, 4, 4, 0, 0, 5, -2], [0, 0, -7, 0, 7, 0, 8, 7, 7, 0, 0, 0, 0, -8, 0, 0], [-3, 0, 0, 0, 0, 0, 0, 0, 0, -10, 0, 2, -11, 0, 11, -7], [17, -3, 0, 0, 0, 0, 0, 0, 0, 5, -2, 2, 16, 0, 0, 3], [0, 0, -4, 0, 3, 0, 7, 2, 2, 0, 0, 0, 0, -7, 0, 0], [17, -4, 0, 0, 0, 0, 0, 0, 0, 0, -13, 0, 4, 0, 4, 0], [0, 0, -8, 8, 2, 2, 0, 0, -1, 0, 0, 0, 0, -12, 0, 0], [17, 0, 0, 0, 0, 0, 0, 0, 0, 18, 0, 1, 5, 0, -5, 1], [0, 0, -6, 6, 6, 6, 0, 0, 14, 0, 0, 0, 0, 4, 0, 0], [-7, 7, 0, 0, 0, 0, 0, 0, 0, -11, 4, 4, 0, 0, -8, 7], [0, 0, 5, -5, 4, 4, 0, 0, 0, 0, 0, 0, 0, -6, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, -9, 0, -9, -4, 0, 4, -8], [-7, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, -6, 3, 0, -3, 5], [0, 0, 0, 0, -10, 10, -4, -5, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -7, 0, -19, 0, 7, 8, 8, 0, 0, 0, 0, -7, 0, 0], [-12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -12, 0, 0, 0], [0, 0, -4, -4, 0, 0, 0, -1, 4, 0, 0, 0, 0, 0, 0, 0], [8, 6, 0, 0, 0, 0, 0, 0, 0, -10, 4, -4, 1, 0, 0, -6], [0, 0, 0, -2, 0, 2, -4, 6, -4, 0, 0, 0, 0, -4, 0, 0], [14, -5, 0, 0, 0, 0, 0, 0, 0, -3, -6, 0, 3, 0, 3, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, -8, 0, 1, 3, 0, -3, -7], [0, 0, 0, 0, 11, 0, -1, 8, 8, 0, 0, 0, 0, 1, 0, 0], [0, 0, 0, -6, 0, -14, 8, -2, 8, 0, 0, 0, 0, 8, 0, 0], [0, 0, -4, 4, -8, -8, 0, 0, -16, 0, 0, 0, 0, 0, 0, 0], [0, 0, -4, -4, -2, 2, -6, 11, 4, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 12, 0, 1, -4, -10, -2, 0, 0, 0, 0, -4, 0, 0], [28, -1, 0, 0, 0, 0, 0, 0, 0, -26, -1, 0, 3, 0, 3, 0], [0, 0, 0, -10, 0, -26, 10, 0, 10, 0, 0, 0, 0, 10, 0, 0], [0, 0, 12, 12, -5, 5, -5, -29, -12, 0, 0, 0, 0, 0, 0, 0], [-1, -6, 0, 0, 0, 0, 0, 0, 0, 6, 1, 0, -2, 0, -2, 0], [-33, 0, 0, 0, 0, 0, 0, 0, 0, -28, 0, -3, -3, 0, 3, 5], [-5, 5, 0, 0, 0, 0, 0, 0, 0, 10, -1, -1, 0, 0, -12, 5], [0, 0, 0, -8, 0, 1, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -14, 0, -6, 0, 4, 10, 10, 0, 0, 0, 0, -4, 0, 0], [18, -3, 0, 0, 0, 0, 0, 0, 0, -20, 5, 0, 1, 0, 1, 0], [0, 0, 0, -10, 0, -19, 6, 8, 2, 0, 0, 0, 0, 6, 0, 0], [0, 0, 8, 8, 7, -7, -9, -17, -8, 0, 0, 0, 0, 0, 0, 0], [-2, -4, 0, 0, 0, 0, 0, 0, 0, 7, -3, 3, 6, 0, 0, 4], [0, 0, 9, 9, -5, 5, -10, -16, -9, 0, 0, 0, 0, 0, 0, 0], [0, 0, 4, -4, 12, 12, 0, 0, 7, 0, 0, 0, 0, -2, 0, 0], [0, 0, -10, 0, 11, 0, -6, 2, 2, 0, 0, 0, 0, 6, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, -9, 9, 0, -9, 7], [6, -6, 0, 0, 0, 0, 0, 0, 0, -24, -4, -4, 0, 0, -3, -6], [-19, 0, 0, 0, 0, 0, 0, 0, 0, -17, 0, -9, 2, 0, -2, 2], [0, 0, 4, 4, 6, -6, -8, 1, -4, 0, 0, 0, 0, 0, 0, 0], [-4, 4, 0, 0, 0, 0, 0, 0, 0, -16, 12, -12, -8, 0, 0, -4], [0, 0, 5, 5, -10, 10, 0, -5, -5, 0, 0, 0, 0, 0, 0, 0], [-19, 0, 0, 0, 0, 0, 0, 0, 0, -10, 0, 1, -3, 0, 3, 9], [0, 0, 4, 0, 3, 0, -5, 2, 2, 0, 0, 0, 0, 5, 0, 0], [2, -2, 0, 0, 0, 0, 0, 0, 0, 38, 4, 4, 0, 0, 0, -2], [0, 0, -1, 1, 8, 8, 0, 0, 6, 0, 0, 0, 0, 7, 0, 0], [0, 0, 0, -7, 0, 7, 5, 10, -3, 0, 0, 0, 0, 5, 0, 0], [-1, 1, 0, 0, 0, 0, 0, 0, 0, 20, 1, 1, 0, 0, 6, 1], [-5, 5, 0, 0, 0, 0, 0, 0, 0, 0, -3, -3, 0, 0, -13, 5], [-8, 8, 0, 0, 0, 0, 0, 0, 0, -2, 2, 0, -7, 0, -7, 0], [0, 0, 0, 6, 0, 7, 3, 4, -10, 0, 0, 0, 0, 3, 0, 0], [0, 0, 8, 8, 1, -1, 5, -11, -8, 0, 0, 0, 0, 0, 0, 0], [-18, -1, 0, 0, 0, 0, 0, 0, 0, 9, 10, 0, 9, 0, 9, 0], [0, 0, 0, 4, 0, -2, 3, 1, -5, 0, 0, 0, 0, 3, 0, 0], [-8, 0, 0, 0, 0, 0, 0, 0, 0, -16, 0, 2, -7, 0, 7, -8], [1, -1, 0, 0, 0, 0, 0, 0, 0, -4, 7, 7, 0, 0, -6, -1], [0, 0, 4, 0, 6, 0, -5, -7, -7, 0, 0, 0, 0, 5, 0, 0], [-23, 0, 0, 0, 0, 0, 0, 0, 0, -26, 0, 6, 1, 0, -1, -3], [0, 0, 0, 2, 0, 7, -2, 4, -6, 0, 0, 0, 0, -2, 0, 0], [0, 0, -2, -2, -2, 2, -8, -4, 2, 0, 0, 0, 0, 0, 0, 0], [0, 0, 4, 0, 16, 0, 2, 6, 6, 0, 0, 0, 0, -2, 0, 0], [-7, 3, 0, 0, 0, 0, 0, 0, 0, -6, 3, -3, 10, 0, 0, -3]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1600_s_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1600_2_s_c();".
// 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_s_c(:prec:=16)
    chi := MakeCharacter_1600_s();
    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_s_c();".
// 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_s_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1600_s();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![4, 0, -96, 0, 72, 0, -16, 0, 1]>],Snew);
    return Vf;
end function;