
// Make newform 1512.2.r.d in Magma, downloaded from the LMFDB on 03 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_1512_r();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1512_r_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_1512_2_r_d();".
// 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_1512_2_r_d();".
// 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 := [19, 3, -22, 3, 21, -15, 5, -3, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [-2806, -3419, 5037, 2226, 36, -270, -317, -43], [-10991, 2696, 12564, -10143, 1878, -1023, 620, 193], [-3125, 2048, 6795, -828, -1389, -312, -235, 130], [3799, -2596, 1371, 1086, -1944, 585, -298, 145], [-67, 730, -438, -2019, 1944, -585, 298, -145], [929, -6752, 1812, 3138, -3045, 912, -581, 242], [-2363, 1697, 1221, -6162, 4422, -1443, 818, -254]];
    Rf_basisdens := [1, 2799, 5598, 2799, 2799, 1866, 2799, 2799];
    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_1512_r();"
function MakeCharacter_1512_r()
    N := 1512;
    order := 3;
    char_gens := [1135, 757, 785, 1081];
    v := [3, 3, 2, 3];
    // 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_1512_r_Hecke();"
function MakeCharacter_1512_r_Hecke( : Kf := false)
    N := 1512;
    order := 3;
    char_gens := [1135, 757, 785, 1081];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, -1, 0, 0], [1, 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, 1, 0, 0, 1, -1, 0, 1], [0, 0, 0, 0, 0, 1, 0, 0], [0, 0, 0, -1, 0, -2, 0, 1], [-1, -1, -1, 0, 1, -1, -1, 1], [1, -1, 0, 1, 2, 0, 1, 0], [0, 1, 0, -1, 0, 0, -1, 0], [1, 1, 2, 0, 0, 1, 2, 0], [0, 0, 2, -1, 0, 0, 0, -2], [3, 0, -1, 0, 0, 3, -1, 0], [7, 0, 0, 0, 0, 0, -1, 0], [-2, -1, 2, 0, 3, -2, 2, 3], [0, 0, -1, 2, 0, 3, 0, 2], [0, 0, 2, 2, 0, 2, 0, -1], [1, 0, 0, 0, 3, 0, -3, 0], [-2, 1, -2, 0, 2, -2, -2, 2], [0, 0, 2, 1, 0, 1, 0, 2], [-3, 1, -1, 0, 2, -3, -1, 2], [4, -2, 0, 2, -3, 0, -3, 0], [4, -2, 0, 2, -3, 0, -2, 0], [0, 0, 5, -1, 0, 4, 0, 0], [0, 0, -3, -3, 0, -1, 0, -2], [4, 3, 0, -3, 1, 0, 0, 0], [0, 0, 0, 1, 0, 2, 0, -6], [0, 0, 1, -3, 0, 4, 0, -2], [-3, -2, -3, 0, 1, -3, -3, 1], [-3, 1, 0, -1, 1, 0, 3, 0], [-2, -2, 0, 2, 7, 0, 0, 0], [0, -1, 3, 0, -1, 0, 3, -1], [2, 2, 0, -2, 6, 0, -1, 0], [2, -4, -1, 0, -3, 2, -1, -3], [0, 0, 1, 2, 0, -3, 0, 8], [-2, -3, -1, 0, 6, -2, -1, 6], [-2, -1, 4, 0, -5, -2, 4, -5], [0, 0, -5, 4, 0, 0, 0, 1], [-8, -5, -1, 0, -2, -8, -1, -2], [11, 1, 0, -1, -2, 0, -1, 0], [3, 1, -3, 0, 1, 3, -3, 1], [0, 0, -3, 2, 0, 9, 0, 6], [-6, 4, 0, -4, 1, 0, 4, 0], [-7, 5, 0, -5, 7, 0, 4, 0], [0, 0, -2, 1, 0, -17, 0, 1], [-4, 2, -5, 0, -1, -4, -5, -1], [12, 1, 0, -1, -4, 0, 2, 0], [7, -2, 0, 2, -5, 0, -5, 0], [-5, -3, -3, 0, -3, -5, -3, -3], [0, 0, 7, -1, 0, -3, 0, 1], [0, 0, -7, 0, 0, 8, 0, 3], [-7, 2, 2, 0, -4, -7, 2, -4], [-3, 1, 0, -1, 9, 0, 2, 0], [5, 0, 0, 0, 0, 5, 0, 0], [0, 0, -5, -2, 0, 9, 0, -5], [-11, -2, 0, 2, -3, 0, 2, 0], [-2, -4, -4, 0, 2, -2, -4, 2], [0, 0, -3, 5, 0, -4, 0, 6], [-2, -4, 0, 4, 1, 0, -3, 0], [4, -5, 0, 5, -5, 0, 4, 0], [0, 0, -5, 1, 0, 1, 0, 0], [0, 0, -2, 0, 0, 9, 0, -1], [1, 1, -4, 0, -7, 1, -4, -7], [-20, 0, 3, 0, -1, -20, 3, -1], [5, -6, 0, 6, -7, 0, 0, 0], [12, 3, -4, 0, 7, 12, -4, 7], [0, 0, -4, 3, 0, 10, 0, 8], [0, 0, -2, -1, 0, 12, 0, -5], [0, 0, -5, 1, 0, 3, 0, -10], [-20, -3, -2, 0, 2, -20, -2, 2], [-6, -2, 2, 0, -7, -6, 2, -7], [0, 0, -4, -2, 0, -14, 0, 0], [0, 0, -5, 4, 0, -1, 0, 1], [-1, -2, 0, 2, 7, 0, 4, 0], [0, 0, 1, 0, 0, -13, 0, 8], [-7, -10, 1, 0, 0, -7, 1, 0], [4, 5, 0, -5, 6, 0, -2, 0], [13, 2, -7, 0, 8, 13, -7, 8], [0, 0, -2, 4, 0, 10, 0, 7], [-9, 6, 0, -6, 6, 0, -5, 0], [-8, -2, -1, 0, -2, -8, -1, -2], [2, 6, -2, 0, 0, 2, -2, 0], [11, 7, 2, 0, -10, 11, 2, -10], [0, 0, 0, 8, 0, 6, 0, 3], [4, 5, 0, -5, 2, 0, -4, 0], [-11, 1, 0, -1, -7, 0, 5, 0], [0, 0, -4, -3, 0, 0, 0, -8], [0, 0, 6, -6, 0, -4, 0, -4], [5, -2, 0, 2, 7, 0, 6, 0], [0, 0, 9, -3, 0, -4, 0, -4], [0, 0, 0, 2, 0, 7, 0, -5], [-13, -1, 2, 0, 8, -13, 2, 8], [-13, 2, 0, -2, 0, 0, -3, 0], [0, 0, 0, -6, 0, 10, 0, 1], [-10, 4, 0, -4, 1, 0, 11, 0], [5, -2, 1, 0, 4, 5, 1, 4], [-14, 4, 6, 0, 8, -14, 6, 8], [6, -1, 0, 1, 1, 0, -6, 0], [-11, 4, -3, 0, 3, -11, -3, 3], [-3, 3, 0, -3, -13, 0, -5, 0], [-8, -6, 0, 6, -7, 0, 5, 0], [9, 1, 0, -1, 7, 0, -11, 0], [0, 0, 1, 4, 0, -1, 0, 5], [0, 5, 0, -5, 0, 0, -8, 0], [0, -6, -11, 0, 0, 0, -11, 0], [0, 0, -1, -5, 0, -25, 0, -3], [-8, 5, -4, 0, -12, -8, -4, -12], [-23, -4, 0, 4, -5, 0, 3, 0], [0, 0, -3, 8, 0, -12, 0, -2], [-18, -8, 0, 8, 1, 0, -6, 0], [0, -2, 8, 0, 0, 0, 8, 0], [0, 0, 8, 5, 0, -14, 0, 3], [1, 1, 5, 0, -3, 1, 5, -3], [7, 5, 0, -5, -15, 0, -7, 0], [-8, 1, 2, 0, 1, -8, 2, 1], [0, 0, -4, 7, 0, 7, 0, -8], [8, 8, 0, -8, 4, 0, 1, 0], [0, 0, -4, 3, 0, 5, 0, 10], [38, -1, 2, 0, 0, 38, 2, 0], [-4, 0, 0, 0, 7, 0, -12, 0], [-5, 7, 1, 0, -2, -5, 1, -2], [0, 0, -2, 2, 0, -8, 0, -11], [-2, 6, -5, 0, -1, -2, -5, -1], [0, 0, 1, -8, 0, 6, 0, -8], [0, 0, 8, 5, 0, 6, 0, 6], [-2, 4, 0, -4, 10, 0, 8, 0], [0, 0, -3, -8, 0, 0, 0, 0], [-4, 2, 0, -2, 4, 0, -2, 0], [0, 0, 8, 6, 0, 20, 0, 0], [16, 1, 0, -1, 4, 0, 8, 0], [0, 0, 7, -4, 0, -23, 0, -4], [25, -6, -8, 0, -6, 25, -8, -6], [-2, -15, 0, 15, -6, 0, 0, 0], [17, 3, 3, 0, 6, 17, 3, 6], [-4, 1, -1, 0, -2, -4, -1, -2], [-9, -1, 0, 1, -4, 0, -5, 0], [8, -4, -18, 0, -6, 8, -18, -6], [17, -2, 5, 0, -7, 17, 5, -7], [31, 2, 0, -2, 4, 0, 2, 0], [1, 2, -1, 0, 10, 1, -1, 10], [-15, -6, -4, 0, 7, -15, -4, 7], [-19, 1, 0, -1, -2, 0, 1, 0], [-23, 6, 0, -6, -1, 0, -9, 0], [0, 0, -6, 10, 0, 2, 0, 11], [8, -8, -8, 0, -4, 8, -8, -4], [6, 9, 0, -9, 6, 0, 6, 0], [-2, -6, 0, 6, 11, 0, 8, 0], [0, 0, 3, -3, 0, -27, 0, 9], [0, 0, -11, -13, 0, -20, 0, 4], [0, 0, -1, 0, 0, -9, 0, 13], [-5, 7, 5, 0, -3, -5, 5, -3], [-1, 12, 0, -12, 5, 0, 0, 0], [9, 13, -7, 0, 12, 9, -7, 12], [5, 16, 0, -16, 8, 0, -1, 0], [17, -4, 0, 4, -16, 0, -7, 0], [9, -4, 7, 0, 1, 9, 7, 1], [0, 0, -7, -3, 0, -17, 0, 5], [0, 0, 3, 8, 0, 2, 0, 11], [34, -4, 0, 4, 4, 0, 1, 0], [0, 0, 4, 3, 0, 4, 0, -6], [-15, -1, 0, 1, 6, 0, 5, 0], [-13, -3, -4, 0, 5, -13, -4, 5], [0, 0, -3, 4, 0, -1, 0, -11], [-14, 4, 0, -4, 0, 0, -4, 0], [5, 10, 2, 0, -3, 5, 2, -3], [-29, -2, 0, 2, 3, 0, 2, 0], [19, 1, -9, 0, 3, 19, -9, 3], [0, 0, -7, 12, 0, 13, 0, 9], [-40, 4, 0, -4, 7, 0, 4, 0], [0, 0, 3, -3, 0, -12, 0, -17], [1, 11, 0, -11, 6, 0, 1, 0], [-10, 1, -1, 0, -2, -10, -1, -2], [0, 0, -6, -7, 0, -4, 0, -10], [10, -2, -11, 0, 14, 10, -11, 14], [11, 6, 6, 0, -8, 11, 6, -8], [0, 0, 4, 8, 0, -7, 0, 0], [19, -7, 1, 0, 3, 19, 1, 3], [-6, -1, 2, 0, -7, -6, 2, -7], [0, 0, -3, -6, 0, -7, 0, 18], [5, -4, 0, 4, -2, 0, -1, 0], [-27, -12, 0, 12, 1, 0, -3, 0], [0, 0, 10, -7, 0, 5, 0, -15], [0, 0, 2, 11, 0, -23, 0, 12], [0, 0, -4, -6, 0, 8, 0, 3], [-18, -2, 8, 0, -20, -18, 8, -20], [24, -4, 0, 4, -9, 0, -14, 0], [-29, 3, -4, 0, -6, -29, -4, -6], [0, 0, 4, 2, 0, -30, 0, -11], [-7, 8, 0, -8, 8, 0, -3, 0], [0, 0, -12, 4, 0, -19, 0, 7], [-12, -2, 13, 0, -6, -12, 13, -6], [4, -6, 0, 6, 11, 0, 0, 0], [-6, 7, 0, -7, -4, 0, -2, 0], [0, 0, -5, 4, 0, 11, 0, 4], [-4, 3, 0, -3, -14, 0, 8, 0], [0, 0, -1, 0, 0, 52, 0, -6], [1, -16, 0, 16, -6, 0, 9, 0], [15, -4, -7, 0, -3, 15, -7, -3], [14, -3, 9, 0, -6, 14, 9, -6], [0, 0, -10, 8, 0, 22, 0, 13], [0, 0, -5, 5, 0, -29, 0, -4], [-27, 3, 0, -3, -5, 0, -5, 0], [31, 9, -5, 0, 9, 31, -5, 9], [0, 0, -9, -5, 0, 7, 0, -21], [-12, 8, 5, 0, -5, -12, 5, -5], [0, 0, 6, -3, 0, -6, 0, 1], [-2, -10, 0, 10, 2, 0, 5, 0], [-6, 2, 0, -2, 14, 0, 3, 0], [8, 8, 0, -8, -6, 0, 2, 0], [3, 0, -23, 0, 5, 3, -23, 5], [0, 0, 8, 10, 0, 2, 0, -8], [6, -6, -9, 0, 2, 6, -9, 2], [-16, 4, 0, -4, 2, 0, 15, 0], [0, -2, -10, 0, -13, 0, -10, -13], [0, 0, -12, -10, 0, 0, 0, 6], [0, 0, -1, -8, 0, -20, 0, -2], [-35, -3, 11, 0, -16, -35, 11, -16], [0, 0, 10, 5, 0, -12, 0, 10], [11, 3, 8, 0, -16, 11, 8, -16], [0, 0, -5, -12, 0, -3, 0, -6], [-3, 1, 0, -1, -2, 0, 3, 0], [-24, -3, 4, 0, 0, -24, 4, 0], [20, 1, -8, 0, -20, 20, -8, -20], [10, 16, -4, 0, 0, 10, -4, 0], [10, 13, 3, 0, 11, 10, 3, 11], [-9, 4, 0, -4, -20, 0, -3, 0], [-4, 2, -4, 0, 8, -4, -4, 8], [0, 0, -13, -7, 0, 1, 0, -14], [0, 0, -4, 5, 0, 4, 0, 11], [35, 5, 0, -5, -14, 0, 0, 0], [0, 0, -4, -9, 0, 4, 0, -11], [0, 0, -13, 4, 0, 1, 0, 16], [-21, 0, 5, 0, 6, -21, 5, 6], [30, 5, 0, -5, 1, 0, -3, 0], [6, 1, -3, 0, -9, 6, -3, -9], [16, 8, -2, 0, 7, 16, -2, 7], [0, 0, -17, 5, 0, 6, 0, 17], [0, 0, -17, -2, 0, 7, 0, -4], [10, -16, 2, 0, -12, 10, 2, -12], [-29, -6, 0, 6, 3, 0, -1, 0], [-5, -12, -3, 0, 18, -5, -3, 18], [2, 0, 0, 0, -15, 0, 4, 0], [0, 0, 0, 13, 0, 44, 0, -3], [-4, -9, 0, 9, 2, 0, 4, 0], [4, 9, -12, 0, -3, 4, -12, -3], [12, -1, 0, 1, -10, 0, -14, 0], [27, -10, 11, 0, -3, 27, 11, -3], [0, 0, 3, 9, 0, 9, 0, -16], [-26, -5, 0, 5, -6, 0, 4, 0], [-8, -16, 5, 0, 10, -8, 5, 10], [0, -2, 16, 0, 9, 0, 16, 9], [15, -4, 0, 4, -26, 0, -10, 0], [4, 7, -8, 0, 3, 4, -8, 3], [-4, 4, 0, -4, -10, 0, 28, 0], [-7, -1, 12, 0, -8, -7, 12, -8], [0, 0, 11, -7, 0, -3, 0, -12], [0, 0, -2, 12, 0, 4, 0, -15], [17, 3, 0, -3, 12, 0, 7, 0], [-12, -2, 0, 2, 13, 0, 2, 0], [0, 0, 13, 11, 0, -4, 0, -8], [21, 1, 0, -1, -15, 0, 4, 0], [9, 13, 0, -13, 17, 0, -13, 0], [0, 0, -3, 16, 0, 11, 0, -5], [0, 0, 5, -3, 0, 35, 0, 2], [-42, 1, 8, 0, 5, -42, 8, 5], [-7, 8, 0, -8, -20, 0, 6, 0], [-12, 4, 4, 0, 0, -12, 4, 0], [0, 0, 8, 4, 0, 12, 0, -14], [24, 12, 0, -12, -23, 0, -2, 0], [0, 0, 5, 6, 0, 39, 0, 0], [-16, 7, 4, 0, 13, -16, 4, 13], [3, 2, 1, 0, -5, 3, 1, -5], [1, 3, 21, 0, -8, 1, 21, -8], [6, -12, 0, 12, -27, 0, -2, 0], [0, 0, 5, 15, 0, 3, 0, 9], [-31, -7, -9, 0, -6, -31, -9, -6], [-2, -5, 16, 0, -10, -2, 16, -10], [29, -1, 0, 1, 1, 0, -17, 0], [23, 0, -11, 0, -13, 23, -11, -13], [0, 0, 22, 8, 0, 8, 0, -3], [24, -6, 0, 6, 16, 0, -2, 0], [0, 0, -4, 5, 0, 9, 0, -27], [-23, 1, -5, 0, 11, -23, -5, 11], [-4, 19, -4, 0, 7, -4, -4, 7], [0, 0, 5, 12, 0, 13, 0, 9], [0, 0, 9, 14, 0, -30, 0, 23], [-1, -14, 0, 0, -9, -1, 0, -9], [-8, -13, 0, 13, 0, 0, 7, 0], [-8, 5, 0, -5, 4, 0, 21, 0], [8, 1, -16, 0, -8, 8, -16, -8], [0, 0, 22, -1, 0, 28, 0, 0], [-37, -3, 0, 3, 8, 0, -1, 0], [0, 0, 3, 0, 0, -3, 0, 9], [8, 21, 0, -21, 15, 0, 3, 0], [18, -3, -5, 0, 10, 18, -5, 10], [24, 10, 0, 0, -19, 24, 0, -19], [0, 0, -7, 16, 0, 5, 0, 11], [25, -17, 8, 0, -13, 25, 8, -13], [0, 0, 4, 9, 0, -26, 0, 22], [0, 0, 9, -3, 0, 22, 0, -3], [3, -6, 0, 6, 10, 0, 5, 0], [0, 0, 13, -1, 0, -5, 0, -7], [-17, 3, -3, 0, 3, -17, -3, 3], [-19, 6, 0, -6, -19, 0, -1, 0], [4, 10, 0, -10, 5, 0, -5, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1512_r_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1512_2_r_d();".
// 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_1512_2_r_d(:prec:=8)
    chi := MakeCharacter_1512_r();
    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_1512_2_r_d();".
// 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_1512_2_r_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1512_r();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![961, -899, 1120, 13, 166, 22, 25, 4, 1]>],Snew);
    return Vf;
end function;