
// Make newform 1274.2.g.p in Magma, downloaded from the LMFDB on 16 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_1274_g();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1274_g_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_1274_2_g_p();".
// 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_1274_2_g_p();".
// 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 := [243, -162, -81, 117, 3, -39, 1, 13, -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], [405, -189, -162, 111, 75, -41, -17, 12, 1, -2], [-243, 837, -36, -354, 21, 131, -7, -42, 8, 5], [-648, 675, 342, -393, -99, 136, 28, -45, -2, 7], [-729, -459, 243, 273, -96, -92, 31, 30, -5, -8], [-171, 33, 78, -13, -26, 5, 8, -2, -1, 0], [-972, -108, 783, -87, -276, 37, 88, -15, -20, 4], [1215, -891, -567, 480, 174, -164, -59, 57, 4, -8], [2592, -27, -1602, 201, 549, -83, -173, 36, 31, -5], [-972, 1107, 1098, -885, -312, 320, 101, -105, -19, 23]];
    Rf_basisdens := [1, 81, 81, 81, 81, 9, 81, 81, 81, 81];
    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_1274_g();"
function MakeCharacter_1274_g()
    N := 1274;
    order := 3;
    char_gens := [885, 197];
    v := [3, 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_1274_g_Hecke();"
function MakeCharacter_1274_g_Hecke( : Kf := false)
    N := 1274;
    order := 3;
    char_gens := [885, 197];
    char_values := [[1, 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 := 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 := [[-1, 0, 0, -1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 1, 0, -1, 0, 0, 0, 1, 0, -1], [-1, 0, 0, 0, 0, -1, 0, -1, -1, 0], [0, 0, 2, 1, 0, 0, 0, 0, 0, 1], [0, 0, 0, 1, 0, 0, 0, 0, 0, 2], [1, 0, 0, 1, 0, 0, 1, 0, 0, 0], [1, 1, 0, 1, -1, 0, 2, 0, 0, -1], [5, 1, -1, 0, 0, 1, 1, -1, 1, 0], [3, 0, 0, 3, -2, 0, 2, 1, 0, 0], [-2, 0, 0, -2, 0, 0, 1, 2, 0, 0], [0, 0, 1, -1, 0, -3, 0, 0, 0, 0], [2, -1, -1, 0, 0, -2, -2, -1, -1, 0], [2, -1, 0, 0, 0, 1, 1, 0, -3, 0], [0, 0, -1, -1, 2, -1, 0, 0, 2, 1], [0, 0, 0, 2, 3, -1, 0, 0, 3, -1], [2, 1, 0, 2, -4, 0, 2, 0, 0, -1], [0, 0, 1, 0, -1, 4, 0, 0, -1, 1], [2, -1, 0, 0, 0, 2, 2, 0, 2, 0], [-1, -1, 3, 0, 0, -1, -1, 3, -1, 0], [9, 0, 0, 0, 0, -3, -3, 0, 0, 0], [-3, 1, 0, -3, 2, 0, -2, 0, 0, -1], [0, 0, -2, 4, -2, 1, 0, 0, -2, 0], [-4, 3, 0, -4, -1, 0, 2, -1, 0, -3], [-4, 1, -1, 0, 0, 4, 4, -1, 1, 0], [-6, -1, 0, -6, 0, 0, -1, 0, 0, 1], [3, 3, 1, 0, 0, 1, 1, 1, 3, 0], [0, 0, 0, 1, 4, 1, 0, 0, 4, -1], [3, 1, 0, 3, -1, 0, 0, 2, 0, -1], [3, 0, 2, 0, 0, 0, 0, 2, 0, 0], [0, 0, 1, 0, 0, 1, 0, 0, 0, 4], [0, 0, -2, -3, -4, 3, 0, 0, -4, 1], [0, 0, 2, -8, 1, 7, 0, 0, 1, -1], [-1, -2, 2, 0, 0, 1, 1, 2, -3, 0], [2, -4, 0, 0, 0, -4, -4, 0, -1, 0], [0, 0, -5, 4, -4, 2, 0, 0, -4, -2], [6, -3, 0, 6, -3, 0, 3, 0, 0, 3], [0, 0, -3, -2, 1, 1, 0, 0, 1, 2], [-5, 0, 0, -5, 2, 0, 1, -5, 0, 0], [-9, 0, -1, 0, 0, -8, -8, -1, -2, 0], [0, 0, -2, 6, 3, 5, 0, 0, 3, -1], [7, -3, 0, 7, 0, 0, -3, 0, 0, 3], [4, -1, 0, 4, 1, 0, 0, -1, 0, 1], [0, 0, -3, -4, -2, 4, 0, 0, -2, 1], [4, -3, 0, 4, 4, 0, -3, 1, 0, 3], [-2, -2, 0, -2, -1, 0, 4, 2, 0, 2], [0, 0, -7, -4, -2, 2, 0, 0, -2, 1], [2, 2, 3, 0, 0, -1, -1, 3, -3, 0], [5, -3, -4, 0, 0, -7, -7, -4, 0, 0], [-6, -2, -7, 0, 0, 1, 1, -7, -4, 0], [0, 0, -1, 4, 2, 1, 0, 0, 2, 0], [0, 0, 2, -2, 0, -3, 0, 0, 0, -2], [-5, -5, 0, -5, 2, 0, -1, -1, 0, 5], [2, 1, 0, 2, 3, 0, -6, 5, 0, -1], [0, 0, -1, 7, 2, 2, 0, 0, 2, 0], [2, 0, 0, 2, -2, 0, -2, -2, 0, 0], [0, 0, 2, 10, -6, -7, 0, 0, -6, 1], [3, 2, 0, 0, 0, -1, -1, 0, 0, 0], [-6, -4, 0, -6, 4, 0, -2, -1, 0, 4], [0, 0, 2, 7, -3, 1, 0, 0, -3, 2], [2, 3, 0, 0, 0, 3, 3, 0, 0, 0], [-10, -4, -5, 0, 0, 1, 1, -5, -1, 0], [5, -2, -1, 0, 0, -5, -5, -1, 2, 0], [-4, 4, 3, 0, 0, 3, 3, 3, -1, 0], [0, 0, -5, -8, -6, 6, 0, 0, -6, 2], [-6, -4, 7, 0, 0, -3, -3, 7, 0, 0], [0, 0, 1, -17, 0, 5, 0, 0, 0, 3], [11, 2, 0, 11, 5, 0, 6, 1, 0, -2], [5, -1, 0, 5, 0, 0, -2, -1, 0, 1], [-4, 0, -5, 0, 0, 5, 5, -5, 3, 0], [-4, 1, 0, -4, 3, 0, -1, -6, 0, -1], [0, 0, 1, 15, -1, -2, 0, 0, -1, 2], [-8, 5, 0, -8, 4, 0, -1, 7, 0, -5], [0, 0, -2, 22, -3, 2, 0, 0, -3, 0], [4, -1, -5, 0, 0, 6, 6, -5, -3, 0], [0, 0, -1, -5, 7, -1, 0, 0, 7, 1], [4, -2, 0, 4, 6, 0, 5, -3, 0, 2], [0, 0, 4, 6, 2, -9, 0, 0, 2, 1], [5, -2, 0, 5, 0, 0, 3, -7, 0, 2], [1, -1, 2, 0, 0, 4, 4, 2, 6, 0], [2, -3, 0, 2, 3, 0, 2, -5, 0, 3], [0, 0, -11, -6, -2, 5, 0, 0, -2, 3], [-3, 3, 0, -3, -8, 0, -4, 2, 0, -3], [-6, -4, -3, 0, 0, 8, 8, -3, -4, 0], [0, 0, 1, -14, 8, 2, 0, 0, 8, 0], [14, 7, 0, 14, -2, 0, 2, -1, 0, -7], [0, 0, -2, 24, 1, 2, 0, 0, 1, -4], [-8, 3, -1, 0, 0, -4, -4, -1, -2, 0], [-14, -4, -6, 0, 0, 3, 3, -6, 0, 0], [13, 4, 0, 13, 1, 0, -1, -2, 0, -4], [0, 0, -4, 9, -2, 1, 0, 0, -2, 2], [18, 0, 0, 18, -2, 0, 3, -6, 0, 0], [-1, -1, -2, 0, 0, -7, -7, -2, 6, 0], [0, 0, 6, -18, 5, 0, 0, 0, 5, 0], [10, -3, 0, 10, 2, 0, -5, 0, 0, 3], [13, 1, -4, 0, 0, 5, 5, -4, -2, 0], [10, -1, 0, 10, 6, 0, 5, -2, 0, 1], [-3, -5, 1, 0, 0, -7, -7, 1, -1, 0], [-2, -5, 0, 0, 0, 3, 3, 0, 4, 0], [3, 4, 0, 3, 1, 0, 1, -4, 0, -4], [0, 0, 3, 3, 0, 2, 0, 0, 0, 2], [2, 7, 0, 2, -2, 0, 6, -1, 0, -7], [-7, 6, 4, 0, 0, 6, 6, 4, 12, 0], [22, 3, 3, 0, 0, -1, -1, 3, 8, 0], [2, -1, 0, 2, -2, 0, -5, -1, 0, 1], [3, -1, -11, 0, 0, 2, 2, -11, -6, 0], [14, 6, -5, 0, 0, 5, 5, -5, 1, 0], [1, -1, 0, 1, 0, 0, -7, 6, 0, 1], [0, 0, 7, -8, -1, -3, 0, 0, -1, 2], [-19, 0, 0, -19, 9, 0, -8, 5, 0, 0], [0, 0, 2, -14, 2, -6, 0, 0, 2, 1], [4, -6, -7, 0, 0, -4, -4, -7, -2, 0], [0, 0, -1, -21, -3, 8, 0, 0, -3, 0], [0, 0, 3, -5, -2, -12, 0, 0, -2, -8], [0, 0, 0, 27, -4, -6, 0, 0, -4, -2], [2, 1, 0, 2, -3, 0, 0, 7, 0, -1], [-3, 4, 0, -3, -8, 0, -5, 2, 0, -4], [0, 0, 6, 7, 0, -2, 0, 0, 0, 2], [-2, -4, 0, -2, 5, 0, 2, -2, 0, 4], [16, 0, 0, 16, 2, 0, 9, 0, 0, 0], [15, 3, -1, 0, 0, -3, -3, -1, 7, 0], [0, 0, -3, 11, 2, 14, 0, 0, 2, -2], [5, -1, 0, 5, 2, 0, 6, -8, 0, 1], [5, 7, 0, 0, 0, 6, 6, 0, 8, 0], [0, 0, 4, 9, 3, -4, 0, 0, 3, 3], [0, 0, -6, 9, -5, -9, 0, 0, -5, 0], [-9, -9, -3, 0, 0, 1, 1, -3, -8, 0], [13, 0, -1, 0, 0, 8, 8, -1, -13, 0], [7, -8, 0, 7, -2, 0, -5, -5, 0, 8], [16, -3, 0, 16, 1, 0, -8, -1, 0, 3], [20, -5, 0, 20, 2, 0, 2, -5, 0, 5], [15, 2, 0, 15, -3, 0, 4, -7, 0, -2], [0, 0, 0, -15, 2, 5, 0, 0, 2, -1], [-16, -1, 0, -16, 2, 0, -9, -7, 0, 1], [0, 0, 9, 18, -4, -5, 0, 0, -4, -2], [0, 0, 3, 8, -6, -3, 0, 0, -6, -6], [0, 0, 6, 11, 1, 9, 0, 0, 1, 2], [-4, -11, 0, -4, -2, 0, -6, 0, 0, 11], [16, -6, -3, 0, 0, -13, -13, -3, -2, 0], [-12, -5, 0, -12, -4, 0, -14, 7, 0, 5], [0, 0, -11, 11, -8, 0, 0, 0, -8, 0], [-8, -4, 1, 0, 0, -3, -3, 1, 6, 0], [7, 4, 0, 7, -1, 0, 1, -2, 0, -4], [0, 0, 1, -15, 13, 10, 0, 0, 13, 4], [11, 9, 7, 0, 0, -9, -9, 7, 4, 0], [-17, -3, 0, 0, 0, -5, -5, 0, 0, 0], [-24, 5, 4, 0, 0, 6, 6, 4, 0, 0], [-12, 8, -1, 0, 0, -7, -7, -1, -5, 0], [0, 0, 4, 0, -3, -9, 0, 0, -3, -5], [-19, 0, 0, -19, -2, 0, -7, 5, 0, 0], [1, -3, -6, 0, 0, -4, -4, -6, 0, 0], [-8, -3, 0, -8, 12, 0, -11, 4, 0, 3], [-23, 2, 0, -23, -4, 0, 5, -3, 0, -2], [-20, -1, 5, 0, 0, -12, -12, 5, -4, 0], [0, 0, -11, -5, 1, -4, 0, 0, 1, -2], [0, 0, -3, 27, 2, -1, 0, 0, 2, 2], [-2, -2, 0, 0, 0, 6, 6, 0, -12, 0], [-3, 6, -4, 0, 0, -3, -3, -4, -7, 0], [14, -8, 0, 14, 4, 0, -6, -6, 0, 8], [0, 0, -7, 13, -4, -2, 0, 0, -4, -1], [-5, 1, 10, 0, 0, 0, 0, 10, -2, 0], [0, 0, -1, -21, 0, 5, 0, 0, 0, 2], [-13, 0, 0, -13, 2, 0, -16, 0, 0, 0], [4, -4, 1, 0, 0, -3, -3, 1, 5, 0], [26, 5, 0, 26, 1, 0, 6, 1, 0, -5], [0, 0, 4, 7, -6, 6, 0, 0, -6, -1], [-11, -2, -3, 0, 0, 0, 0, -3, -10, 0], [-24, -1, 4, 0, 0, -13, -13, 4, -9, 0], [19, -1, 8, 0, 0, -3, -3, 8, 2, 0], [0, 0, -2, 5, -5, -12, 0, 0, -5, -9], [0, 0, 0, 5, -3, 9, 0, 0, -3, -4], [0, 0, -4, -7, -2, -7, 0, 0, -2, -4], [-4, 1, 6, 0, 0, -17, -17, 6, -7, 0], [0, 0, 4, 2, -6, -26, 0, 0, -6, -3], [-2, 8, 0, -2, 2, 0, 8, -11, 0, -8], [-11, 3, 2, 0, 0, -2, -2, 2, -5, 0], [-13, 3, 0, -13, 7, 0, -5, 1, 0, -3], [24, 5, 0, 24, -6, 0, 1, -6, 0, -5], [9, -4, 5, 0, 0, 0, 0, 5, -3, 0], [4, 13, -6, 0, 0, 17, 17, -6, 4, 0], [-19, -1, -5, 0, 0, -1, -1, -5, -6, 0], [19, -5, -4, 0, 0, 5, 5, -4, 6, 0], [-19, -2, 0, -19, 9, 0, -18, 7, 0, 2], [0, 0, 4, 13, 8, 2, 0, 0, 8, 3], [5, -7, 0, 0, 0, 2, 2, 0, 1, 0], [13, -6, -8, 0, 0, -4, -4, -8, -2, 0], [-11, 0, 0, -11, -10, 0, 0, 11, 0, 0], [0, 0, -4, -19, 6, 4, 0, 0, 6, 9], [0, 0, 2, 10, 12, 9, 0, 0, 12, 8], [0, 0, -7, -9, -8, 6, 0, 0, -8, 9], [-7, 6, -2, 0, 0, -12, -12, -2, -2, 0], [12, -2, 0, 12, -10, 0, -5, -5, 0, 2], [0, 0, 6, -3, 14, 3, 0, 0, 14, 6], [19, 7, 0, 19, -4, 0, 14, -1, 0, -7], [8, 0, -1, 0, 0, 18, 18, -1, 10, 0], [0, 0, 3, -12, 3, -9, 0, 0, 3, -8], [2, -8, 0, 0, 0, -12, -12, 0, -2, 0], [7, -7, -11, 0, 0, -9, -9, -11, -11, 0], [0, 0, -14, -2, 2, -8, 0, 0, 2, -4], [0, 0, 3, -15, -1, 3, 0, 0, -1, 7], [-32, 5, 0, -32, -2, 0, 11, 0, 0, -5], [-16, 5, -6, 0, 0, -4, -4, -6, -8, 0], [-24, 6, 0, -24, 2, 0, -9, 4, 0, -6], [10, -6, 0, 10, -5, 0, 7, 3, 0, 6], [14, 4, 6, 0, 0, 4, 4, 6, -3, 0], [0, 0, 5, 5, -4, -3, 0, 0, -4, 8], [22, 7, 0, 22, -6, 0, -13, 9, 0, -7], [0, 0, -2, -7, -4, 13, 0, 0, -4, 4], [0, 5, 0, 0, 10, 0, 1, -2, 0, -5], [33, 1, -2, 0, 0, 4, 4, -2, 0, 0], [10, -7, 0, 10, 3, 0, -4, 5, 0, 7], [0, 0, 3, -16, -8, 1, 0, 0, -8, -3], [0, 0, 3, 16, -8, 8, 0, 0, -8, -3], [6, 1, 15, 0, 0, 11, 11, 15, 2, 0], [-9, -5, 12, 0, 0, -16, -16, 12, 5, 0], [0, 0, -6, -12, 6, 11, 0, 0, 6, -1], [-17, -2, 0, -17, -12, 0, -13, -4, 0, 2], [2, -3, -6, 0, 0, 5, 5, -6, -14, 0], [0, 2, 0, 0, -4, 0, 4, -12, 0, -2], [6, -7, -5, 0, 0, 6, 6, -5, 6, 0], [-16, -11, -3, 0, 0, 9, 9, -3, -10, 0], [0, 0, -9, -7, 1, -3, 0, 0, 1, 3], [-6, 4, 0, -6, -10, 0, -11, 6, 0, -4], [-3, 3, 0, 0, 0, 10, 10, 0, -4, 0], [-18, -6, 0, -18, 0, 0, -12, -2, 0, 6], [0, 0, -3, -14, 10, 8, 0, 0, 10, -9], [0, 0, -10, -26, 3, 0, 0, 0, 3, 2], [28, 3, -1, 0, 0, 7, 7, -1, 6, 0], [9, 0, 0, 9, 7, 0, 11, -6, 0, 0], [34, -8, 0, 34, 2, 0, -2, -5, 0, 8], [-19, -3, 0, -19, -11, 0, -5, 5, 0, 3], [8, 7, 14, 0, 0, -6, -6, 14, -2, 0], [-24, -3, -1, 0, 0, -2, -2, -1, 12, 0], [-6, 1, 7, 0, 0, 4, 4, 7, -7, 0], [0, 0, 10, -8, 0, -10, 0, 0, 0, -11], [-4, 7, 0, -4, 12, 0, 3, 10, 0, -7], [0, 0, 17, 11, 12, -5, 0, 0, 12, 0], [12, -11, 0, 12, 3, 0, 7, 2, 0, 11], [-6, 5, 0, -6, -2, 0, -10, 6, 0, -5], [11, -6, 0, 11, -8, 0, 10, -11, 0, 6], [0, 0, -3, -15, -10, 4, 0, 0, -10, -10], [20, -9, 0, 20, 0, 0, -14, -7, 0, 9], [0, 0, -8, 30, 0, 9, 0, 0, 0, 0], [-39, 0, 11, 0, 0, -6, -6, 11, -5, 0], [0, 0, -2, 13, 5, 8, 0, 0, 5, -5], [2, 0, 0, 2, -10, 0, 17, -4, 0, 0], [0, 0, 18, -16, 2, -10, 0, 0, 2, 8], [-7, 3, 0, -7, 1, 0, 17, 5, 0, -3], [-27, 14, 0, -27, -7, 0, 16, 11, 0, -14], [18, 9, 0, 18, -12, 0, 0, 7, 0, -9], [-34, 3, 11, 0, 0, 2, 2, 11, -3, 0], [8, 8, 0, 8, -2, 0, 9, 2, 0, -8], [-5, -5, -1, 0, 0, 11, 11, -1, -1, 0], [0, 0, 4, -23, -2, 4, 0, 0, -2, -4], [0, 0, 11, 13, 14, -2, 0, 0, 14, 6], [4, 0, 0, 4, -12, 0, 0, 3, 0, 0], [-5, -6, 9, 0, 0, -14, -14, 9, 8, 0], [0, 0, -7, 5, 0, -5, 0, 0, 0, -14], [-23, 0, -1, 0, 0, -16, -16, -1, -5, 0], [17, 5, 0, 17, -22, 0, 10, -1, 0, -5], [-19, 9, 13, 0, 0, -3, -3, 13, -7, 0], [0, -2, 0, 0, 1, 0, -9, -11, 0, 2], [0, 0, 8, -22, 4, 11, 0, 0, 4, -11], [0, 0, 1, 21, 6, 1, 0, 0, 6, -4], [0, 0, -1, 7, -11, 11, 0, 0, -11, -9], [-31, -5, -10, 0, 0, 0, 0, -10, -4, 0], [0, 0, 6, -7, -14, -4, 0, 0, -14, 8], [0, 0, -12, -7, -6, 5, 0, 0, -6, 4], [19, -9, -17, 0, 0, -13, -13, -17, -1, 0], [-1, 2, 2, 0, 0, 2, 2, 2, 14, 0], [-31, 0, 0, -31, -10, 0, -17, -3, 0, 0], [0, 0, 15, 2, -3, -3, 0, 0, -3, 6], [0, 0, 9, 11, 14, -7, 0, 0, 14, 2], [-5, 0, 0, -5, -8, 0, -9, 12, 0, 0], [0, 0, 9, 12, 0, -16, 0, 0, 0, -10], [7, 0, -2, 0, 0, 5, 5, -2, -11, 0], [0, 0, 1, 4, -2, 6, 0, 0, -2, 2], [0, 0, 11, -5, -2, -11, 0, 0, -2, 14], [12, 1, 0, 12, -23, 0, 13, 0, 0, -1], [32, -4, 0, 32, 1, 0, 6, -8, 0, 4], [-13, -5, -17, 0, 0, 15, 15, -17, 0, 0], [-4, -10, 0, -4, -3, 0, 9, -11, 0, 10], [-17, -14, -1, 0, 0, -3, -3, -1, -2, 0], [6, -12, -2, 0, 0, 6, 6, -2, 3, 0], [5, 10, -9, 0, 0, 7, 7, -9, -8, 0], [28, 9, 2, 0, 0, 1, 1, 2, -10, 0], [0, 0, 5, -14, -8, -13, 0, 0, -8, -2], [0, 0, -8, 6, -12, 3, 0, 0, -12, -6], [-5, -1, 0, -5, -17, 0, -12, 8, 0, 1], [0, 0, 14, 0, -4, -12, 0, 0, -4, 12], [7, 10, 0, 7, -10, 0, -1, 2, 0, -10], [0, 0, -2, 3, -14, 4, 0, 0, -14, 4], [0, 0, -2, 20, -1, 11, 0, 0, -1, -1], [-22, 7, -10, 0, 0, 21, 21, -10, 3, 0], [-37, 1, 1, 0, 0, -5, -5, 1, -17, 0], [23, -4, 0, 23, 7, 0, 4, -13, 0, 4], [-4, 3, 0, -4, 6, 0, -4, -15, 0, -3], [-28, -1, 0, -28, 4, 0, -18, 2, 0, 1], [0, 0, -11, -19, -6, -5, 0, 0, -6, -2], [-15, -5, 13, 0, 0, -11, -11, 13, -7, 0], [9, -5, 0, 9, 3, 0, -3, 7, 0, 5]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1274_g_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1274_2_g_p();".
// 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_1274_2_g_p(:prec:=10)
    chi := MakeCharacter_1274_g();
    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_1274_2_g_p();".
// 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_1274_2_g_p( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1274_g();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![9, 54, 342, -54, 195, 21, 69, 3, 10, 1, 1]>,<5,R![-21, 7, 25, -12, -2, 1]>,<11,R![81, -981, 11296, -7517, 6877, 679, 727, 34, 40, 4, 1]>],Snew);
    return Vf;
end function;