
// Make newform 1260.2.c.e in Magma, downloaded from the LMFDB on 29 December 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_1260_c();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1260_c_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_1260_2_c_e();".
// 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_1260_2_c_e();".
// 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, -256, 192, -128, 48, 16, -28, 24, -28, 12, -7, 2, 3, -4, 3, -2, 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], [0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1152, 320, 256, 256, 296, -196, 56, -104, 26, 19, 0, -3, -26, 1, 0, 3], [-384, 32, -112, 72, 68, 8, 20, -18, 17, 0, 7, 2, -5, 0, -3, 0], [-256, 192, -128, 48, 16, -28, 24, -28, 12, -7, 2, 3, -4, 3, -2, 1], [-384, -960, -64, -480, -120, -12, -8, 112, 34, 25, 12, 31, 14, 3, 4, -7], [-2176, 576, -448, 800, 280, -28, 120, -128, 102, -3, 32, 3, -38, 15, -16, 5], [1408, -64, 704, -224, -40, 60, 40, 112, -58, -5, -28, -19, 10, -7, 12, -5], [64, -64, 80, -64, 12, 4, 4, 16, -5, 5, -5, 1, 1, -1, 1, -1], [704, -352, 240, -280, -64, 44, -36, 50, -36, 7, -10, 1, 12, -3, 6, -3], [128, 2112, 320, 1056, -312, -300, -168, -144, 82, -79, -20, -57, -34, 11, 4, 17], [-3456, 2368, -960, 1120, 264, -236, 184, -256, 178, -31, 44, -9, -66, 27, -28, 17], [-4224, 1472, -1088, 1376, 504, 12, 264, -224, 206, -33, 84, 9, -62, 5, -36, 15], [0, 384, -32, 112, -72, -68, -8, -20, 18, -17, 0, -7, -2, 5, 0, 3], [-1408, 1152, -288, 784, 0, -164, 64, -164, 104, -37, 18, -15, -32, 9, -10, 11], [5504, -1984, 2176, -1792, -312, 156, -264, 440, -302, 11, -104, -11, 78, -23, 40, -21]];
    Rf_basisdens := [1, 1, 512, 128, 128, 512, 512, 512, 64, 128, 512, 512, 512, 64, 256, 512];
    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_1260_c();"
function MakeCharacter_1260_c()
    N := 1260;
    order := 2;
    char_gens := [631, 281, 757, 1081];
    v := [1, 2, 2, 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_1260_c_Hecke();"
function MakeCharacter_1260_c_Hecke( : Kf := false)
    N := 1260;
    order := 2;
    char_gens := [631, 281, 757, 1081];
    char_values := [[-1, 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, 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 := 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, 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, -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, -1], [0, 0, 1, -1, 1, 1, 0, 0, -1, -1, 1, 0, 1, -1, -1, 0], [0, 1, 0, 1, -1, -1, 1, -1, 1, 0, -1, -1, -1, 1, 1, 0], [0, 0, -1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0], [1, 0, 0, 0, 1, 1, 0, 1, -1, -1, 0, 1, -1, 0, 1, 0], [0, 0, 1, 0, 1, 0, 0, 0, 2, 0, 1, -1, 0, 1, -2, -1], [-2, 0, 1, 0, -1, 0, 0, 2, -2, 0, 1, 1, 0, 0, -2, -1], [-1, 0, -1, 0, 1, 1, -1, 2, -3, -1, -1, 0, -1, -2, -1, 1], [2, -1, 1, 0, -1, 0, -2, 0, -2, 0, -1, 1, 0, -1, -2, 1], [0, 1, -2, 0, -2, -2, 2, -2, 2, -1, -1, 0, -2, 0, 2, 1], [0, 1, -1, 0, 2, 0, 1, -1, 0, 2, 1, 1, 0, 0, 0, 1], [0, 1, 1, 0, -2, 2, -1, 1, -2, -2, -1, 3, -2, 0, 2, 1], [3, -1, 0, 0, 2, -1, 1, 0, 1, 3, 0, -1, 1, -1, -1, 0], [0, 1, -2, 0, 1, 2, -1, 1, -2, -4, -2, 0, -2, -1, 2, 2], [0, 0, 3, -2, 0, 2, 1, -1, -2, 0, 1, -1, 2, 0, -2, -1], [0, 1, 1, -2, -1, 0, 0, 0, -2, 1, -2, -1, 0, 1, 2, 0], [0, 0, 2, 1, 0, 1, -2, 2, -3, -3, -2, -1, 1, -1, 1, -1], [0, 1, -2, 1, 1, -1, -1, 1, -1, -1, -2, 1, -1, -1, 3, 1], [0, 1, 0, 0, 1, 0, -1, 1, -2, -3, -1, 0, 0, -2, 2, 1], [0, 1, 0, 0, -1, 2, -1, -1, 0, 3, -1, 2, -2, -2, 4, 1], [0, 1, 0, 4, -1, 0, 1, -1, -2, 2, -2, 0, 0, 1, 2, 0], [0, 1, -2, -3, 3, -1, 1, -1, 3, 3, 2, 1, -1, 1, -1, 1], [0, 2, -2, 2, -1, -2, 1, -1, 0, 3, -1, 0, -2, 1, 4, 3], [0, -2, 4, 0, -2, 0, -2, -2, 0, 1, 1, 4, 0, 2, 0, -1], [0, 0, 1, 0, -1, 0, 2, -2, 2, 4, 1, -1, 0, 3, -2, -1], [0, -1, 2, 0, -1, 0, -1, 1, -2, -3, 1, 2, 0, 2, -2, -1], [3, -1, 0, 0, 0, -1, -1, -2, 1, 3, 0, -1, 1, -1, -1, 0], [0, 0, 0, -2, 3, 0, -1, 1, 2, 0, 0, 0, 0, 1, -2, -2], [0, -3, 2, 0, 1, -2, -1, -3, 2, 2, 4, 0, 2, 3, -2, -4], [-3, 1, -2, 0, 0, 1, -1, 0, -1, 3, 0, -1, -1, -3, 1, 0], [-3, 0, -2, 0, 3, 1, 0, -3, 3, 1, 0, -1, -1, 0, 5, 0], [0, 0, -1, 0, 5, 2, 2, 2, 0, -4, 1, 1, -2, 2, 4, -1], [0, -3, 2, 2, -1, 2, -1, 1, 0, 5, 3, 0, 2, 2, -4, -1], [0, 1, -2, -3, -5, -1, 3, -3, -3, 2, -1, 1, -1, -1, 5, 4], [0, -1, 1, 0, -2, 0, -1, 1, 0, -4, 1, -1, 0, -2, 0, 1], [0, -2, -1, 0, 5, 0, 0, 2, -2, 1, 0, -1, 0, -2, -2, 0], [0, 2, 4, -6, -2, 0, 2, -2, 0, -2, -2, -4, 0, 2, 0, -2], [0, 2, -1, 11, -3, -3, 2, -2, 3, -3, -3, -2, -3, 1, 3, 0], [0, -2, -1, 8, 0, 0, 1, -1, 4, 2, 1, 1, 0, 2, -4, -3], [0, 2, 0, 1, 4, 1, 0, 0, -3, 3, -2, 1, 1, 1, 1, -1], [4, -2, -2, 0, 6, -2, 2, 0, 2, -4, 0, -4, 2, 2, -2, 0], [-1, 3, -2, 0, 0, 3, 1, 4, -3, -1, -2, 1, -3, -3, 3, 2], [5, 2, -3, 0, -1, -1, 1, -2, 3, -3, -3, -4, 1, 0, 1, 3], [0, -1, 4, 6, 1, 2, -1, 1, 0, 1, 1, -2, 2, 2, -4, -3], [6, -2, 0, 0, 6, -2, 4, -2, 6, 3, 1, -2, 2, 2, 2, -1], [0, 1, 0, 0, -3, 2, -3, 5, -8, -5, -1, 2, -2, -2, -4, 1], [0, 0, -3, -2, -4, -2, 3, -3, 2, 4, -1, 1, -2, 2, 2, 1], [1, 1, -2, 0, 0, 1, -1, 0, -1, -3, 2, -1, -1, 1, 1, -2], [0, 2, -2, 1, 0, -3, 2, -2, 5, -1, 0, -1, -3, 1, 1, 1], [0, -4, 2, 2, 2, 2, -4, 4, 2, 0, 4, 0, 2, 0, -6, -2], [8, 1, -2, 0, 1, -2, 3, -3, 6, 4, 2, -4, 2, 1, 2, -2], [0, -2, -1, 4, -3, 0, 0, 0, 0, 7, -2, 1, 0, 4, 0, -2], [0, 0, 3, -6, 3, 2, 0, 0, 0, -6, 3, -1, 2, -3, -4, -1], [0, -4, -2, 2, -1, 2, -3, 3, -4, 1, 1, 4, 2, -3, 0, 1], [-5, 4, -3, 0, 1, 1, 5, 4, 1, -3, -1, -2, -1, 0, 3, 1], [0, 3, -1, 0, -5, 2, -2, -2, 0, -6, -3, 1, -2, 1, 4, 3], [4, -4, 0, 0, 2, -2, -4, -2, -2, 0, -2, -2, 2, -2, -6, 2], [8, 0, 2, 0, -6, 0, -4, -4, 0, 0, -2, 2, 0, 0, 0, 2], [0, 0, -4, 0, 4, -2, -4, 4, 2, -2, -4, 2, -2, 0, 2, -2], [2, 2, 2, 0, -6, -2, 2, -4, 6, 8, 4, 0, 2, 2, 2, -4], [6, 2, -2, 0, 0, -2, 4, -6, 10, 4, -4, -4, 2, 0, 6, 4], [0, -1, 0, 3, 7, 1, -5, 5, 1, 1, 0, 1, 1, 1, -3, -3], [1, -1, 2, 0, -2, -3, 1, -2, 3, -1, 0, -1, 3, 3, -3, 0], [0, -1, -2, 6, 1, 2, 1, -1, -4, 1, -1, 4, 2, -2, 0, -1], [0, 0, 2, 0, -2, 2, -2, 4, -6, -2, 2, 4, -2, 0, -2, -2], [0, -2, 1, 2, -1, 2, -2, 2, -4, -8, -1, 1, 2, -5, 0, -1], [0, -4, 1, 2, -2, 2, 1, -1, 2, -2, 5, 1, 2, -2, -6, -1], [0, 0, -1, -2, -3, -2, 0, 0, 2, -1, 0, -1, -2, 0, 2, 2], [0, -2, -4, 5, -8, -3, 0, 0, 1, 3, -2, 1, -3, 1, 5, 3], [-2, -2, 2, 0, 4, -2, 4, 2, 2, 3, -1, 0, 2, 0, -2, 1], [4, -5, 1, 0, 5, -2, -2, -6, 4, 8, 5, -1, 2, 1, 0, -5], [0, 3, 2, -6, -9, -2, 7, -7, 0, -1, -3, -4, -2, 2, 4, 1], [0, 2, 1, 0, -1, 0, 4, 2, 2, -7, 2, 1, 0, 6, 2, -2], [0, -4, 5, 0, -1, -6, 2, -2, 4, 8, 3, -1, 6, 2, -8, -3], [0, -1, 0, -1, 1, -3, -1, 1, 11, 0, 1, -3, -3, 5, -5, -4], [-2, 4, 0, 0, -6, 0, 2, -2, 4, 0, -4, 0, 0, 0, 4, 4], [0, 4, 0, 8, 2, 0, -2, 2, -8, 0, -4, 0, 0, -2, 8, 4], [-6, 5, -2, 0, -7, 0, 1, 1, 0, 0, 0, -2, 0, -1, 0, 0], [-10, 2, -2, 0, -4, 4, -6, 2, -8, -8, -6, 2, -4, -4, 0, 6], [0, -4, 0, -11, -2, 1, -2, 2, 1, -3, 0, 1, 1, -1, -3, -3], [0, -1, 4, -17, -3, 1, 1, -1, 1, -9, -2, -3, 1, -1, -3, -5], [-9, -2, -1, 0, 5, -3, 3, 2, 1, -1, 3, -4, 3, 2, -5, -3], [0, -2, -3, -10, 9, 2, -4, 4, 0, 2, 5, 5, 2, -3, -4, 1], [6, 4, -2, 0, -2, 0, 4, 0, 4, -4, 2, -2, 0, 4, 4, -2], [-4, 0, 0, 0, 4, 2, 2, 0, 2, 0, -2, 2, -2, 0, 6, 2], [0, 0, 6, -10, 2, 4, 4, -4, 0, -2, 8, -2, 4, -2, -8, 0], [0, 0, -6, 2, 1, -4, -5, 5, 2, 2, -4, 2, -4, 1, 6, 2], [-4, -1, -2, 0, 1, -6, 3, -9, 12, 1, -1, -8, 6, 4, 0, 1], [-6, -4, -2, 0, 8, -2, 0, 2, -2, -2, 2, -4, 2, 0, -6, -2], [0, 0, 0, 6, 9, 4, -3, 3, -6, 2, 2, 4, 4, -3, -2, 0], [0, -2, 1, 11, 1, 1, 0, 0, 3, 1, 3, 0, 1, 1, -5, -2], [0, 3, 0, 2, -1, -2, 5, -5, 4, -1, -1, -2, -2, 2, 0, -1], [4, -2, 7, 0, -5, -4, 2, -4, 6, 9, 4, 3, 4, 4, -2, -4], [0, 0, 0, -10, 5, 2, 1, -1, 0, 11, 5, 2, 2, 3, -4, 1], [0, -3, 4, -8, -2, 6, 0, 0, -10, 3, 1, 2, 6, -2, -2, -1], [2, -4, 4, 0, -2, -2, -4, -6, 2, 4, 2, 2, 2, 2, -2, -2], [-2, 2, 4, 0, -8, 0, 0, -4, 4, 2, -2, 4, 0, 2, 4, 2], [0, -5, -3, -2, 3, 4, -4, 4, -6, 3, 4, 7, 4, -5, -2, 2], [-3, 7, -4, 0, -6, 5, -1, 0, -1, -3, -4, 1, -5, -3, 9, 4], [-4, 0, 4, 0, -4, 0, 0, 0, 0, 0, 4, 4, 0, 4, 0, -4], [-10, 2, 0, 0, -6, 4, -6, 2, -8, -6, -6, 4, -4, -4, 0, 6], [0, -3, 4, -16, -7, 4, 3, -3, -6, -3, -1, 0, 4, -2, -2, -3], [0, -1, 4, 3, -9, 1, 3, -3, -3, 7, 2, -3, 1, 3, 1, 3], [0, -6, 2, 0, 12, 0, 2, -2, 4, 6, 4, 2, 0, 2, 4, -4], [0, 2, -1, -2, 1, -2, 0, 0, 2, 9, 2, -1, -2, 4, 2, 4], [0, 0, 0, -5, 8, 3, -4, 4, -5, 7, 4, 3, 3, -1, -1, 3], [0, -4, 0, -10, 6, 2, -2, 2, 6, 2, 12, 2, 2, -2, -10, 2], [-6, 4, -4, 0, 4, 6, 2, 8, -6, -11, -1, 2, -6, 0, 6, 1], [0, -1, 1, 0, -3, -6, 2, -6, 8, 8, 9, -5, 6, 5, -4, -9], [-5, -1, 6, 0, -4, 3, -3, 0, -3, 1, 4, 9, -3, 3, 3, -4], [-13, 8, 0, 0, -15, 3, -2, 1, -3, -1, -2, 3, -3, -2, 3, 2], [0, -3, -2, 8, 5, 4, -7, 7, -10, -1, -1, 6, 4, -6, 2, 1], [-4, 2, -2, 0, 0, 2, 0, 2, -2, -6, -6, 0, -2, -2, 2, 6], [-2, 0, 2, 0, 0, 6, -4, 2, -6, 0, 0, 8, -6, -2, 6, 0], [-4, -2, 1, 0, -1, -4, 0, 2, -2, 1, 6, -3, 4, 2, -10, -6], [1, -5, 2, 0, -2, -3, -7, -2, -5, -1, 0, -1, 3, -1, -11, 0], [0, 0, 3, -7, 3, 3, 0, 0, -3, -3, -1, 0, 3, -1, -3, -4], [0, 0, -3, -2, -4, -6, 3, -3, 14, -8, -1, -3, -6, 2, -2, -3], [8, -4, 4, 0, -4, -6, -2, -12, 10, 4, -2, -2, 6, 4, -2, 2], [0, 4, 4, 0, 0, 2, 0, 0, -10, 6, 0, -2, 2, 0, 6, 6], [0, 4, -3, 10, -5, -6, 6, -6, 8, 4, -1, -3, -6, 5, 4, 3], [1, 2, -4, 0, 1, -3, 4, -7, 11, 1, -2, -7, 3, 2, 5, 2], [-14, -2, -1, 0, 3, 0, -2, 0, -2, 2, 5, -1, 0, 0, -2, -5], [-10, 0, -4, 0, 2, -4, 2, -6, 8, 2, 2, -8, 4, 2, 0, -2], [2, 4, 0, 0, -6, -2, 4, -2, 6, -2, 4, -2, 2, 6, 2, -4], [-10, 10, -6, 0, -8, 2, 4, 2, 2, -1, -5, -4, -2, -4, 6, 5], [0, 3, -6, -5, -3, -3, 1, -1, -5, 8, -5, 3, -3, 1, 11, 6], [0, -3, -2, -8, 11, 4, -5, 5, -2, 3, 3, 6, 4, -2, -6, -3], [0, 4, -7, -2, -5, -6, 2, -2, 0, 6, -11, 1, -6, 5, 12, 1], [0, 3, 12, -6, 13, 6, -5, 5, -4, -1, 3, -6, 6, 2, -8, -5], [-10, 5, -3, 0, -3, 4, 0, 2, -2, -2, -7, 1, -4, -5, 6, 7], [0, -1, 6, 8, 7, 4, 3, -3, 6, 0, 6, -2, 4, 3, -14, -8], [0, -4, -4, -6, -10, -2, 2, -2, 2, 6, -4, 2, -2, 4, 2, -2], [0, -4, 4, -6, -2, 4, -6, 6, -8, -6, -6, 0, 4, -2, 0, -6], [-10, 2, -4, 0, 4, 6, -2, 0, -2, -2, -8, 2, -6, -6, 10, 8], [0, -6, -2, 2, -14, -4, 0, 0, 8, -6, 0, -2, -4, 0, 0, 0], [12, -4, 2, 0, 0, -2, -4, -6, 2, -4, -4, 0, 2, 2, -2, 4], [-21, 0, -2, 0, 1, -1, 0, -5, 5, 3, 4, -3, 1, 2, 3, -4], [2, 6, -1, 0, -11, 0, 0, -2, 2, 6, -3, -1, 0, -4, 2, 3], [0, 6, 8, 10, 3, 0, 3, -3, 2, -10, 2, -8, 0, -1, -2, 0], [0, 2, -3, 8, -3, -4, 4, -4, 6, -2, -5, -1, -4, 3, 2, -3], [0, 2, -5, -12, 10, 0, -1, 1, 0, 8, -1, 5, 0, 2, 0, -1], [4, 8, -2, 0, -8, 4, 2, 2, 0, 10, 4, 2, -4, -4, 8, -4], [0, -3, -4, 9, 3, 3, -3, 3, -7, 2, 5, 7, 3, -7, 1, 6], [0, 4, -5, -6, 1, -6, 0, 0, 6, -1, -8, -1, -6, 4, 6, -2], [11, -2, 2, 0, -3, 3, -8, -1, -7, -15, -2, 5, -3, 4, -1, 2], [0, -4, -3, 8, -1, -4, -6, 6, 10, -10, -1, -1, -4, -1, -2, -3], [8, -3, 1, 0, 9, -2, 6, 6, 0, -2, 3, -1, 2, 3, -4, -3], [0, -1, 0, 16, 3, 0, -7, 7, -2, -4, -8, 0, 0, 1, 2, -6], [0, 1, -5, 2, 3, 0, 2, -2, -2, 5, 4, 5, 0, -3, 2, 6], [12, -1, -3, 0, 14, 2, 7, 1, 6, -8, 1, -1, -2, 6, 10, -1], [0, 5, -1, -10, 3, -4, 6, -6, 10, -7, 0, -3, -4, 1, -2, -2], [0, -6, 0, 1, -2, 1, -2, 2, 5, -1, 2, 1, 1, 1, -7, -5], [0, -5, 0, -26, -3, 2, -1, 1, 0, 7, 3, 2, 2, 2, -4, -1], [0, 1, -2, 4, 5, 0, -3, 3, -2, 8, 2, 2, 0, 1, 2, 4], [0, 5, 4, -7, 7, -1, -1, 1, 3, -1, -4, -5, -1, 5, -1, -5], [0, 6, -2, 18, -20, -8, 10, -10, 0, 6, -12, -6, -8, 8, 16, 4], [0, 4, 1, -6, -1, -2, 2, -2, 0, -6, -3, -3, -2, -1, 4, 1], [17, 1, 4, 0, -8, -3, 1, -10, 11, 7, 2, 1, 3, 5, 5, -2], [0, 6, 4, 20, -2, -4, 6, -6, 8, -1, -1, -8, -4, 6, 0, -1], [6, 3, -6, 0, 1, 4, -3, 5, -8, -16, -4, -2, -4, -1, 0, 4], [-5, 3, 4, 0, -4, 3, 3, 6, -3, -7, -2, 7, -3, 3, 3, 2], [-12, 1, -3, 0, 6, 6, -1, 5, -6, -6, -5, 3, -6, -4, 6, 5], [0, 1, -4, 26, -17, -10, 1, -1, 8, -13, -13, -6, -10, 2, 12, -1], [0, -7, 4, 1, -13, 3, -3, 3, -7, -6, -7, -1, 3, -1, 1, -6], [-4, 3, -12, 0, 7, 0, 1, 3, -2, -9, -3, -12, 0, -4, -2, 3], [0, -2, -8, -4, 4, 2, 0, 0, -6, 10, -4, 10, 2, 0, 2, -2], [0, -4, 3, 15, 7, 5, -4, 4, -1, 3, -1, 2, 5, 3, -9, -10], [0, 2, -5, 8, -10, -8, 3, -3, 8, -4, -9, -3, -8, 4, 8, -1], [0, 4, -2, 25, 8, 1, 2, -2, -3, 5, 2, 3, 1, -1, 1, 3], [-2, 8, -6, 0, -2, 4, 4, 0, 4, -6, -4, -2, -4, 0, 12, 4], [3, 2, 3, 0, -3, 1, 3, 6, -3, 9, 5, 4, -1, -2, -1, -5], [0, 7, -6, 0, 4, -6, 0, 0, 2, -9, -9, 0, -6, -2, 10, 1], [0, -3, -8, -4, -9, -8, 1, -1, 14, -1, -7, 0, -8, 6, 2, -5], [-8, -2, -5, 0, 11, -2, 4, 4, 0, -6, 3, -7, 2, 2, -4, -3], [-10, -4, 6, 0, -2, -2, -2, -4, 2, 3, 3, 4, 2, 4, -2, -3], [0, -6, 5, -8, 4, 4, -9, 9, 0, 2, 3, -1, 4, 2, -8, -5], [0, -4, -6, -4, -8, 0, -2, 2, -8, 11, -3, 6, 0, 0, 8, 5], [-2, 5, 6, 0, -13, 4, -1, -1, 0, 6, 2, 10, -4, 1, 8, -2], [-8, 3, 1, 0, -3, 6, -2, 2, -4, 8, 5, 7, -6, -3, 8, -5], [0, -2, -7, 0, -9, -4, 4, -4, 4, 13, 2, 3, -4, 4, 4, 6], [0, 0, 1, -22, -1, -2, -2, 2, 4, 2, -3, -3, -2, 5, 0, -3], [0, -6, 0, -10, -13, -2, 3, -3, 8, 3, 1, -2, -2, 5, -4, -3], [10, 7, -9, 0, -1, 0, 4, -2, 6, 6, -5, -9, 0, -7, 6, 5], [-10, 2, 4, 0, -4, -2, 6, 0, 6, 8, 10, 2, 2, 6, 2, -10], [8, -3, 0, 0, 5, 0, -1, 5, -6, -3, 3, 0, 0, 0, -6, -3], [10, -2, 8, 0, -2, 2, 0, 6, -6, -6, 0, 10, -2, 4, -2, 0], [0, -1, 2, 3, -11, 1, 3, -3, -7, -9, -8, -1, 1, -3, 5, -3], [0, 4, 1, 20, -5, 0, 2, -2, -10, 2, -1, -1, 0, -3, 10, 9], [0, 11, -6, 14, -5, -6, 3, -3, -12, 7, -15, 0, -6, 2, 24, 9], [0, -4, -6, -14, -8, 0, -2, 2, -8, 12, -2, 6, 0, 0, 8, 6], [0, 0, 5, 18, 9, 6, -4, 4, -8, 10, 1, 1, 6, 3, -4, -3], [0, 2, -3, -2, -1, -2, 4, -4, 2, 11, 2, 1, -2, 4, 2, 4], [8, -7, 0, 0, 9, 0, -5, -7, 2, 15, 1, 0, 0, -6, 2, -1], [-12, -3, 7, 0, -3, 2, -4, 8, -12, -8, 7, 9, -2, 5, -8, -7], [0, -2, -5, 0, -9, -4, -2, 2, 0, -7, -8, 1, -4, -2, 8, 0], [0, -2, 3, 0, -7, 0, -8, -6, -2, -5, -10, 3, 0, -2, -2, 10], [2, 4, -1, 0, -9, 4, -6, 0, -6, -4, 3, 3, -4, 0, 2, -3], [17, 2, -9, 0, 11, -1, 7, 8, -1, -5, -3, -10, 1, -4, -3, 3], [0, 5, -2, 1, 3, -1, -1, 1, -7, 12, -11, 1, -1, 7, 9, -2], [0, 4, 12, -2, 0, 2, 8, -8, 6, -10, 4, -10, 2, 2, -10, -6], [6, -1, 6, 0, -15, -4, -7, -11, 4, 10, -6, 2, 4, -3, -4, 6], [0, 0, -2, 2, 10, 4, 0, 0, -4, 2, 4, 6, 4, -4, -4, 0], [-3, -2, 1, 0, 3, -1, 1, -2, 3, 11, 3, 0, 1, -2, 1, -3], [0, -2, -3, 0, -5, -8, 2, -2, 22, -8, 1, -5, -8, 5, -6, -5], [12, 2, -6, 0, 6, 2, 2, 8, -6, -6, -6, -4, -2, -6, -2, 6], [25, -4, 4, 0, -3, 1, -8, -7, -1, 11, 0, 5, -1, -4, 1, 0], [-14, -4, -6, 0, 6, -4, -4, -8, 4, 6, 0, -10, 4, -4, -4, 0], [0, 6, -2, 14, -1, -2, 1, -1, -8, -1, -5, 0, -2, -3, 12, 7], [0, 2, -6, 32, -2, -4, 0, 0, 0, 11, -7, 2, -4, 6, 8, 1], [-4, -1, 6, 0, -9, 2, -7, 5, -12, -7, -1, 8, -2, 0, -8, 1], [-4, -8, 6, 0, 6, -4, 0, -8, 8, 14, 8, 2, 4, 4, 0, -8], [0, -2, -12, 22, -6, -6, -2, 2, 2, 14, -4, 6, -6, 4, 10, 6], [0, 6, -4, 18, 9, -4, -3, 3, 2, -4, -4, 0, -4, -1, 6, 2], [0, -1, 2, 4, 17, 8, -7, 7, -10, -2, 4, 6, 8, -7, -6, -2], [0, -4, -3, 22, -5, 2, -2, 2, -8, -2, -7, 5, 2, -3, 4, -3], [-1, 5, -2, 0, -10, -1, -1, -6, 5, -3, -8, -3, 1, -1, 3, 8], [10, 6, -4, 0, -8, 4, -4, 8, -12, -16, -12, 0, -4, -6, -4, 12], [-9, -4, -3, 0, 13, 5, -3, 0, -3, 5, -5, 2, -5, -8, 7, 5], [8, -4, 0, 0, 8, -6, 6, 0, 6, 12, 2, -6, 6, -2, -6, -2], [0, 2, 4, 8, 6, 0, -10, 10, -4, -9, -9, -4, 0, 0, 4, -5], [-20, -6, 0, 0, 10, 4, -6, 6, -12, -4, -4, 4, -4, -6, -4, 4], [8, 7, -2, 0, -3, 4, 5, 3, 2, 1, 1, 2, -4, 0, 10, -1], [0, -2, -1, 32, 5, 4, -8, 8, -12, 7, 2, 5, 4, -4, 4, 6], [0, 0, -2, -27, 2, -3, 0, 0, 9, 5, 6, -1, -3, 3, -3, 3], [4, -6, 6, 0, -4, -4, -6, 2, -8, -5, 1, 2, 4, 2, -16, -1], [0, 8, -5, -15, -3, -5, 6, -6, -3, 5, -9, 0, -5, 3, 13, 4], [14, -1, -3, 0, 11, 0, 6, 16, -10, 0, 3, -3, 0, -5, -10, -3], [5, -2, 6, 0, -5, -3, 0, -7, 7, -5, -2, 3, 3, 8, 1, 2], [0, -7, 8, -6, 13, 10, -5, 5, 0, -3, 13, 2, 10, -4, -20, -7], [8, -6, 0, 0, 14, -2, 4, -2, 6, -6, 0, -2, 2, 6, 2, 0], [10, -8, -2, 0, 4, -6, -8, -6, -2, 0, -4, -8, 6, -4, -14, 4], [8, 5, 3, 0, -12, 2, -1, -3, 2, -6, -3, 5, -2, 4, 6, 3], [0, 0, -10, 20, -2, -4, 4, -4, 4, 4, -2, 6, -4, 0, 4, 2], [5, -11, 4, 0, 12, 1, -7, 2, -9, -9, 2, 5, -1, 3, -7, -2], [0, -4, 1, 5, -9, 3, 2, -2, -7, -7, -3, 2, 3, -5, 1, -2], [-10, 8, -6, 0, -4, 10, -4, 2, -6, -10, -6, 4, -10, -4, 14, 6], [0, -4, 5, 28, 5, 4, -8, 8, -2, 0, 5, -1, 4, -1, -6, -1], [-19, 2, 2, 0, -7, 5, -6, 3, -9, -3, 0, 7, -5, -2, 1, 0], [0, 10, -8, 0, -2, 12, -2, 6, -8, -5, -9, 4, -12, -10, 16, 9], [-4, -1, 0, 0, 13, 4, 7, 5, 2, -1, -7, 4, -4, -2, 10, 7], [0, 0, -5, -14, -5, -2, -6, 6, -10, 1, -2, 3, -2, -6, 14, 12], [4, -2, -4, 0, -4, -8, -4, -4, 0, 8, 0, -12, 8, -6, -16, 0], [-18, 4, 6, 0, -18, -2, -2, 4, -6, 5, 1, 4, 2, -2, -10, -1], [2, -1, -10, 0, 15, 0, 3, -5, 8, 2, -2, -10, 0, -3, 8, 2], [0, 1, 0, -16, 3, 0, 1, -1, 2, -9, 1, 0, 0, -4, -2, -1], [0, 8, -7, 16, 3, -4, 2, -2, -6, 10, -9, 3, -4, 3, 14, 5], [-2, -7, 3, 0, 1, -8, -2, -4, 2, 0, 5, -5, 8, 5, -14, -5], [0, 11, 6, 0, -7, -4, 5, -5, -6, -10, -12, -10, -4, 1, 14, 2], [0, 4, 1, -12, 1, -4, 8, -8, 14, 4, 1, -5, -4, 9, -6, -5], [0, -4, -8, -20, -4, 0, 0, 0, -4, -2, 6, 8, 0, -10, 4, 10], [0, -2, 2, 22, -6, 0, 4, -4, 4, 2, 0, -2, 0, 4, -4, -4], [0, 0, 1, -5, 17, 5, 0, 0, 3, 3, 9, 4, 5, -1, -13, -4], [-8, -7, -4, 0, 21, 0, 3, 5, -2, -13, 1, -4, 0, 4, -2, -1], [-12, -6, 2, 0, 4, -8, 2, -10, 12, 6, 12, -6, 8, 10, -4, -12], [0, -10, 2, 10, -14, 4, -4, 4, -8, 10, 4, 2, 4, 0, 0, 4], [0, 5, -4, 5, -1, -5, 3, -3, 3, -11, -2, -1, -5, -5, 7, 5], [0, -2, -4, 6, 7, 4, -5, 5, -10, -14, -2, 8, 4, -13, 2, 0], [0, 6, -3, 10, -11, -10, 0, 0, 4, -2, -11, -7, -10, 5, 16, 5], [0, 8, 1, -6, 12, 2, 7, -7, -2, 0, 7, 1, 2, -4, -2, 5], [0, -7, 0, 17, 3, 3, -7, 7, 1, 10, 5, 3, 3, 3, -7, -2], [0, 10, 9, 18, 11, -2, 4, -4, 10, -5, -2, -11, -2, 8, -6, -8], [-1, 2, -4, 0, 7, 3, 4, -1, 5, 3, -6, -1, -3, -4, 11, 6], [-6, 0, -1, 0, -5, -4, -2, -12, 10, 8, 7, -5, 4, 4, 2, -7], [0, -5, -4, -16, 2, 2, 0, 0, 2, 1, 3, 6, 2, -2, -6, -3], [0, -1, 5, -18, 5, 8, -2, 2, -14, -11, 2, 3, 8, -11, -2, 0], [9, -3, -4, 0, 10, 1, -1, 4, -5, -11, -4, -3, -1, -1, -3, 4], [0, -2, 3, -12, 14, 4, -13, 13, -4, -4, -5, 1, 4, 0, -4, -9], [0, 7, 15, -40, 22, 12, 1, -1, -12, -4, 11, -3, 12, -6, -12, -1], [0, 1, 10, 1, 13, 7, 3, -3, 3, -3, 8, -3, 7, 1, -17, -9], [0, -1, 2, -2, -5, 2, -1, 1, -8, 3, 5, 0, 2, -4, 4, 9], [0, 9, -10, -19, 1, -5, 3, -3, -9, 1, -4, 5, -5, -7, 19, 15], [-10, 4, -8, 0, 4, -2, 6, 0, 6, 9, -1, -10, 2, -6, 2, 1], [0, 0, 1, -36, -13, 0, 10, -10, -2, 6, 3, -1, 0, 1, 2, 5], [-18, 10, 0, 0, -14, 4, 2, -2, 4, 0, -4, 4, -4, 0, 12, 4], [-6, 4, 6, 0, -20, -12, 6, -10, 16, 10, 12, -6, 12, 12, -8, -12], [12, -1, -2, 0, 5, -6, 7, 5, 2, 6, 4, -8, 6, -1, -10, -4], [-4, -3, 5, 0, 2, -2, 3, 9, -6, 0, 1, 3, 2, 0, -10, -1], [0, -13, 4, 8, -23, 4, -1, 1, -6, 3, 1, 0, 4, 0, -2, -1], [24, -9, 7, 0, 4, -2, -5, -3, -2, 0, -1, 5, 2, 2, -6, 1], [0, 2, -3, 24, 2, 0, -3, 3, -8, -4, -7, 3, 0, -4, 8, 1], [-26, 12, 0, 0, -18, 10, -4, -2, -2, -8, -6, 10, -10, 0, 18, 6], [0, 12, 8, -13, -8, -5, 12, -12, 3, -13, -8, -13, -5, 3, 7, -1], [-20, -6, 8, 0, -4, -2, -6, 0, -6, 10, -4, 6, 2, -6, -10, 4], [3, 3, -2, 0, -6, -5, 3, -6, 9, 5, 8, -7, 5, 5, -1, -8], [7, 0, 9, 0, -11, -3, 1, -4, 5, 9, 7, 6, 3, 6, -1, -7], [0, 5, 10, -16, 23, 8, -5, 5, -6, 2, 8, -2, 8, -1, -10, -2], [10, -8, 3, 0, 3, -4, -6, -16, 10, 8, 3, -1, 4, 4, 2, -3], [12, 9, -6, 0, -5, 6, 1, 13, -12, -5, -11, 0, -6, -12, 0, 11], [-1, -5, -4, 0, 8, -1, -5, 2, -7, -11, 6, -5, 1, 3, -9, -6], [34, -7, 6, 0, -5, -4, -9, -5, -4, 2, -2, 2, 4, -1, -12, 2], [12, -9, 3, 0, 7, -14, 6, -10, 16, 10, 1, -11, 14, 5, -12, -1], [0, -2, -4, 2, 14, 4, -2, 2, 0, 4, 12, 8, 4, -6, -8, 4], [1, -6, 11, 0, -5, -1, -5, 4, -9, 3, 1, 10, 1, 0, -11, -1], [0, -2, -2, 14, -4, -4, 2, -2, 12, 4, -6, -2, -4, 10, -4, -10], [0, 5, 6, 0, -11, 6, -1, 5, -6, -10, -4, 12, -6, 3, 6, 4], [30, 2, -8, 0, 4, -2, 2, 4, -2, -10, -4, -10, 2, -2, -6, 4], [0, 1, 16, 25, -1, 7, 3, -3, -5, -1, 4, -9, 7, 3, -9, -5], [-31, -1, 6, 0, -10, 5, -11, -6, -5, 11, -8, 11, -5, -9, 5, 8], [0, -5, -2, 16, 13, 0, -11, 11, 10, 1, 9, 2, 0, 0, -10, -1]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1260_c_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1260_2_c_e();".
// 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_1260_2_c_e(:prec:=16)
    chi := MakeCharacter_1260_c();
    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_1260_2_c_e();".
// 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_1260_2_c_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1260_c();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<11,R![16384, 0, 172032, 0, 627712, 0, 947712, 0, 549248, 0, 75904, 0, 4292, 0, 108, 0, 1]>,<19,R![1408, 6784, 1696, -2400, -352, 288, 6, -12, 1]>],Snew);
    return Vf;
end function;