
// Make newform 1560.2.l.d in Magma, downloaded from the LMFDB on 28 December 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_1560_l();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1560_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_1560_2_l_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_1560_2_l_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 := [16, -72, 162, -170, 89, 0, 2, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [-2736, 9448, 63783, 6804, 32, 76, 747, 72], [67844, -234110, 244926, -167761, -836, -2894, 2970, -1881], [28512, -60588, 136611, -144063, -716, -792, 1683, -1611], [2749, 11647, -30358, 32014, 361, 176, -374, 358], [6221, 84217, -169443, 178781, 1893, 1089, -2087, 1988], [-59636, 213034, -289098, 147349, 740, 2666, -3394, 1665], [328276, -1096418, 1516102, -811589, -4052, -14166, 17838, -9117]];
    Rf_basisdens := [1, 18170, 36340, 18170, 1817, 9085, 7268, 36340];
    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_1560_l();"
function MakeCharacter_1560_l()
    N := 1560;
    order := 2;
    char_gens := [391, 781, 521, 937, 1081];
    v := [2, 2, 2, 1, 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_1560_l_Hecke();"
function MakeCharacter_1560_l_Hecke( : Kf := false)
    N := 1560;
    order := 2;
    char_gens := [391, 781, 521, 937, 1081];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 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, -1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -1], [0, -1, 2, 0, 0, 0, 0, 0], [0, 0, 0, 1, 1, 1, 0, 0], [0, 0, 1, 0, 0, 0, 0, 0], [0, -1, 0, 0, 1, -1, -1, 1], [0, 0, 0, 0, 0, 0, 0, 0], [0, -1, -2, 0, 1, -1, -1, 1], [-2, 0, 0, 0, 1, 1, -1, -1], [0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 1, -1, -1, 1], [2, 0, 0, -1, 0, 0, -1, -1], [0, 0, -2, 0, 1, -1, 1, -1], [0, 0, 2, 0, 0, 0, -1, 1], [0, 1, 0, 0, 1, -1, -1, 1], [-2, 0, 0, 0, 2, 2, 1, 1], [2, 0, 0, 1, 2, 2, 0, 0], [0, 0, 4, 0, 0, 0, 2, -2], [4, 0, 0, 3, 0, 0, 1, 1], [0, -2, 2, 0, 1, -1, -1, 1], [0, 0, 0, -1, -2, -2, 2, 2], [0, 4, -6, 0, 0, 0, 1, -1], [-2, 0, 0, 3, 2, 2, -1, -1], [0, 5, 0, 0, 0, 0, 2, -2], [2, 0, 0, -2, -1, -1, -3, -3], [0, 0, 2, 0, -2, 2, 4, -4], [0, 1, 6, 0, 1, -1, 3, -3], [-6, 0, 0, 2, 2, 2, 0, 0], [0, -4, 2, 0, 2, -2, -2, 2], [0, 0, 8, 0, -1, 1, 1, -1], [4, 0, 0, -6, -3, -3, -1, -1], [0, 2, 4, 0, -2, 2, 3, -3], [2, 0, 0, 1, 3, 3, 1, 1], [-6, 0, 0, 1, -3, -3, -2, -2], [-4, 0, 0, 2, -2, -2, 2, 2], [0, -2, 4, 0, 3, -3, 1, -1], [0, -5, -2, 0, 1, -1, 1, -1], [0, 0, 6, 0, 2, -2, -1, 1], [0, 0, -6, 0, 0, 0, -4, 4], [0, 0, 0, 0, 3, 3, -1, -1], [0, 0, 0, 1, -1, -1, -3, -3], [0, 0, 0, -2, 2, 2, -2, -2], [0, -3, 4, 0, 2, -2, -2, 2], [0, 2, 4, 0, -2, 2, -5, 5], [6, 0, 0, 4, 1, 1, 3, 3], [-14, 0, 0, -4, 1, 1, 3, 3], [0, -2, 0, 0, -3, 3, -1, 1], [0, -2, 10, 0, -2, 2, 1, -1], [-2, 0, 0, 4, -2, -2, 0, 0], [0, 3, -4, 0, -5, 5, -1, 1], [4, 0, 0, 1, 0, 0, -5, -5], [-6, 0, 0, -2, -2, -2, 4, 4], [0, 0, 0, 8, 3, 3, 1, 1], [0, 0, 14, 0, -2, 2, -2, 2], [0, 0, 0, 0, 0, 0, 6, -6], [-6, 0, 0, -2, -1, -1, -3, -3], [-12, 0, 0, -2, -2, -2, -4, -4], [0, 8, 4, 0, -3, 3, 3, -3], [-4, 0, 0, -6, -1, -1, -2, -2], [0, 6, -8, 0, 0, 0, 0, 0], [0, 4, 4, 0, -6, 6, 5, -5], [0, 5, 2, 0, -5, 5, 1, -1], [-4, 0, 0, 4, 4, 4, 2, 2], [0, 4, 2, 0, 1, -1, 7, -7], [0, -4, 24, 0, -2, 2, -1, 1], [-4, 0, 0, 0, -6, -6, 0, 0], [0, -8, 2, 0, 1, -1, -1, 1], [0, 1, -2, 0, 3, -3, -5, 5], [-2, 0, 0, 0, 0, 0, -4, -4], [0, 2, -4, 0, 0, 0, 3, -3], [6, 0, 0, 6, 3, 3, 2, 2], [0, 4, 18, 0, -4, 4, 6, -6], [0, -4, 2, 0, -2, 2, 2, -2], [16, 0, 0, 4, -2, -2, -6, -6], [0, -12, 2, 0, 0, 0, -5, 5], [-6, 0, 0, -8, -3, -3, -3, -3], [0, -3, 8, 0, 1, -1, 3, -3], [-8, 0, 0, 6, 5, 5, 0, 0], [-6, 0, 0, -2, 8, 8, 6, 6], [8, 0, 0, 2, 1, 1, 7, 7], [14, 0, 0, 0, -4, -4, -2, -2], [14, 0, 0, 2, -1, -1, -6, -6], [0, 10, -14, 0, 1, -1, 3, -3], [14, 0, 0, -3, -5, -5, -3, -3], [0, -1, -10, 0, -3, 3, -1, 1], [2, 0, 0, 5, -4, -4, 7, 7], [0, -3, 4, 0, 2, -2, 0, 0], [6, 0, 0, -3, -1, -1, -2, -2], [0, 1, -2, 0, -6, 6, 6, -6], [0, 5, -10, 0, 1, -1, 3, -3], [16, 0, 0, -5, -2, -2, -3, -3], [0, 1, 10, 0, -4, 4, -6, 6], [8, 0, 0, 0, -1, -1, -7, -7], [8, 0, 0, -6, -4, -4, -2, -2], [0, -4, 16, 0, 6, -6, 0, 0], [-26, 0, 0, 1, 5, 5, 2, 2], [2, 0, 0, 4, 6, 6, 8, 8], [0, 2, -14, 0, 1, -1, -3, 3], [-2, 0, 0, -6, -6, -6, -4, -4], [0, 0, -2, 0, -3, 3, 9, -9], [0, 4, 12, 0, 2, -2, -1, 1], [0, 7, -10, 0, -1, 1, 3, -3], [10, 0, 0, 10, 0, 0, 2, 2], [-24, 0, 0, 1, 2, 2, 2, 2], [0, 1, 16, 0, 2, -2, 2, -2], [0, 0, -6, 0, -4, 4, -3, 3], [0, -8, 4, 0, 6, -6, 1, -1], [-4, 0, 0, -2, 2, 2, 0, 0], [-18, 0, 0, 3, 0, 0, 6, 6], [0, 10, 18, 0, -2, 2, 4, -4], [0, 1, 0, 0, 7, -7, -7, 7], [0, 8, 8, 0, 0, 0, 7, -7], [0, 0, 0, -10, -2, -2, -4, -4], [4, 0, 0, 4, 2, 2, 2, 2], [-2, 0, 0, -2, -2, -2, 6, 6], [0, 7, 10, 0, -3, 3, 7, -7], [0, 7, -10, 0, 5, -5, 5, -5], [0, 4, -6, 0, 4, -4, 8, -8], [8, 0, 0, -6, 1, 1, -3, -3], [-6, 0, 0, -4, 0, 0, -4, -4], [0, 6, -12, 0, 0, 0, -4, 4], [0, -5, 4, 0, 7, -7, 1, -1], [0, 0, 14, 0, 0, 0, 7, -7], [-12, 0, 0, 10, 10, 10, 0, 0], [18, 0, 0, 4, 3, 3, -1, -1], [2, 0, 0, -6, -4, -4, 2, 2], [12, 0, 0, -10, -2, -2, -4, -4], [0, -2, 4, 0, -1, 1, 1, -1], [0, 7, -28, 0, 3, -3, -1, 1], [24, 0, 0, 4, 4, 4, -6, -6], [0, -2, 10, 0, -2, 2, -5, 5], [-14, 0, 0, 7, -3, -3, -1, -1], [0, 4, 6, 0, 0, 0, 0, 0], [-4, 0, 0, -10, -9, -9, 6, 6], [-2, 0, 0, 8, 0, 0, 0, 0], [0, -14, 8, 0, 0, 0, -3, 3], [0, -4, 0, 0, 10, -10, 4, -4], [0, 5, 12, 0, -9, 9, 9, -9], [-6, 0, 0, -2, 2, 2, 4, 4], [-12, 0, 0, -8, 0, 0, -12, -12], [34, 0, 0, 1, 3, 3, 0, 0], [0, 8, -2, 0, 2, -2, 0, 0], [0, -8, -26, 0, 8, -8, -13, 13], [12, 0, 0, 8, 5, 5, -7, -7], [12, 0, 0, 5, 2, 2, 9, 9], [0, 5, 12, 0, -3, 3, -5, 5], [0, 9, 16, 0, -1, 1, 11, -11], [8, 0, 0, 9, 4, 4, 8, 8], [0, 2, -6, 0, 4, -4, -3, 3], [0, -4, -10, 0, 4, -4, -2, 2], [-18, 0, 0, -4, -4, -4, -2, -2], [0, -6, 8, 0, 4, -4, -4, 4], [0, -1, -2, 0, 7, -7, 7, -7], [0, 2, -8, 0, 6, -6, -10, 10], [-8, 0, 0, -6, -8, -8, 0, 0], [0, 0, 0, 7, -2, -2, -6, -6], [18, 0, 0, 3, 8, 8, 3, 3], [0, 2, -22, 0, -1, 1, -7, 7], [30, 0, 0, -1, 3, 3, -2, -2], [0, -4, 2, 0, -8, 8, 1, -1], [0, -7, 8, 0, -5, 5, 3, -3], [0, 2, 16, 0, -5, 5, 1, -1], [16, 0, 0, -12, 7, 7, -1, -1], [0, -14, 0, 0, 2, -2, -1, 1], [0, 6, 18, 0, 6, -6, 11, -11], [-30, 0, 0, 7, 1, 1, -1, -1], [0, -4, -20, 0, -1, 1, -3, 3], [-10, 0, 0, -8, -2, -2, -8, -8], [0, -15, 0, 0, 9, -9, 1, -1], [6, 0, 0, 12, 2, 2, 1, 1], [6, 0, 0, -2, -4, -4, 4, 4], [-4, 0, 0, -12, -2, -2, 4, 4], [0, 7, 8, 0, 6, -6, 12, -12], [30, 0, 0, 9, -1, -1, 1, 1], [-22, 0, 0, 4, 4, 4, 4, 4], [-24, 0, 0, 6, 6, 6, -4, -4], [6, 0, 0, -9, 3, 3, 2, 2], [0, 6, -14, 0, 0, 0, 2, -2], [0, 0, 0, 12, 1, 1, 9, 9], [0, -9, 2, 0, -6, 6, 6, -6], [-32, 0, 0, -8, 3, 3, 7, 7], [0, -4, 18, 0, 4, -4, -4, 4], [0, 0, 20, 0, 0, 0, -5, 5], [0, 10, 6, 0, -2, 2, 1, -1], [-18, 0, 0, -2, -9, -9, 7, 7], [0, -6, -20, 0, -1, 1, -11, 11], [0, 9, 2, 0, 3, -3, 7, -7], [10, 0, 0, -8, -10, -10, 0, 0], [8, 0, 0, 5, 6, 6, 11, 11], [0, 4, 16, 0, -10, 10, 6, -6], [0, 0, -10, 0, -12, 12, 3, -3], [-6, 0, 0, 1, -9, -9, 5, 5], [2, 0, 0, 9, 1, 1, 0, 0], [0, -4, -40, 0, -2, 2, 2, -2], [0, 13, -20, 0, -7, 7, 1, -1], [6, 0, 0, 6, 0, 0, 4, 4], [0, -2, -14, 0, -10, 10, -2, 2], [0, 6, 0, 0, 10, -10, 5, -5], [0, 7, 18, 0, -9, 9, 11, -11], [-16, 0, 0, 4, -4, -4, 3, 3], [6, 0, 0, -15, -9, -9, -7, -7], [0, 15, -20, 0, 3, -3, 3, -3], [-12, 0, 0, -9, -7, -7, -1, -1], [4, 0, 0, -7, -7, -7, -2, -2], [0, 13, 4, 0, 1, -1, 3, -3], [-4, 0, 0, 2, 2, 2, -6, -6], [0, -16, 12, 0, 2, -2, -4, 4], [-18, 0, 0, -1, -6, -6, -3, -3], [-16, 0, 0, -5, 10, 10, 10, 10], [0, 4, -6, 0, -11, 11, -5, 5], [22, 0, 0, 6, -5, -5, -1, -1], [0, -12, 0, 0, -3, 3, -5, 5], [0, 4, 2, 0, -8, 8, 9, -9], [-30, 0, 0, -2, -5, -5, 8, 8], [-14, 0, 0, 6, 6, 6, -2, -2], [0, 4, -24, 0, -9, 9, 5, -5], [22, 0, 0, 4, 8, 8, 0, 0], [0, 12, -22, 0, -12, 12, 3, -3], [0, 14, 8, 0, 0, 0, 7, -7], [-12, 0, 0, 3, -1, -1, 9, 9], [-12, 0, 0, -8, -6, -6, 8, 8], [0, 0, 0, -14, -11, -11, -12, -12], [0, 2, 0, 0, -9, 9, -11, 11], [0, -7, -34, 0, 1, -1, -1, 1], [32, 0, 0, 15, 3, 3, -3, -3], [0, -1, -32, 0, 5, -5, -9, 9], [-4, 0, 0, 10, 14, 14, 0, 0], [0, -6, -36, 0, 7, -7, -7, 7], [4, 0, 0, 5, -5, -5, -8, -8], [0, -10, 4, 0, -3, 3, -1, 1], [18, 0, 0, -3, 3, 3, -3, -3], [24, 0, 0, -14, -16, -16, -2, -2], [-6, 0, 0, 4, 2, 2, 2, 2], [0, 2, 34, 0, -7, 7, 9, -9], [0, -14, -18, 0, 4, -4, -9, 9], [-22, 0, 0, 4, 8, 8, 10, 10], [0, -12, 12, 0, -8, 8, -7, 7], [-28, 0, 0, 2, -7, -7, 1, 1], [-32, 0, 0, -16, -2, -2, 6, 6], [0, -2, 14, 0, -10, 10, 3, -3], [8, 0, 0, 7, 10, 10, -10, -10], [0, -6, -12, 0, -3, 3, 11, -11], [10, 0, 0, 8, 2, 2, -6, -6], [0, 4, 4, 0, 2, -2, -13, 13], [-4, 0, 0, 4, 8, 8, -2, -2], [0, 6, 24, 0, 1, -1, 9, -9], [-4, 0, 0, -11, -13, -13, 0, 0], [8, 0, 0, 12, 0, 0, -4, -4], [0, -8, 40, 0, 0, 0, -6, 6], [0, 4, -10, 0, -12, 12, 12, -12], [28, 0, 0, 2, -7, -7, -4, -4], [0, 2, -26, 0, 8, -8, 7, -7], [22, 0, 0, -9, -16, -16, -10, -10], [0, 11, -28, 0, 9, -9, 5, -5], [-12, 0, 0, -3, -5, -5, 4, 4], [28, 0, 0, 8, 3, 3, -1, -1], [0, 13, -18, 0, -3, 3, 9, -9], [0, 0, -14, 0, -2, 2, 8, -8], [0, -3, -12, 0, 10, -10, 2, -2], [0, 4, -8, 0, -3, 3, 3, -3], [0, -9, -14, 0, 1, -1, -11, 11], [-38, 0, 0, -14, -4, -4, 2, 2], [0, 0, 20, 0, -13, 13, 9, -9], [0, -14, 0, 0, -4, 4, -7, 7], [-28, 0, 0, -5, 8, 8, 8, 8], [14, 0, 0, -15, -5, -5, -10, -10], [14, 0, 0, 13, 10, 10, -5, -5], [0, 1, -2, 0, 15, -15, 1, -1], [0, 4, 14, 0, -14, 14, 14, -14], [-16, 0, 0, -8, -13, -13, 3, 3], [0, -1, -22, 0, 11, -11, -7, 7], [0, -1, 12, 0, 8, -8, -6, 6], [24, 0, 0, 16, 4, 4, 4, 4], [0, 8, -30, 0, -15, 15, -1, 1], [0, 15, -2, 0, -4, 4, 4, -4], [0, 4, 2, 0, -12, 12, -3, 3], [-2, 0, 0, 4, -8, -8, 4, 4], [-10, 0, 0, -6, -4, -4, -8, -8], [-20, 0, 0, 4, -3, -3, -1, -1], [0, -9, 14, 0, 11, -11, -1, 1], [-20, 0, 0, 18, 6, 6, 8, 8], [0, 0, -20, 0, -10, 10, 2, -2], [22, 0, 0, -12, 4, 4, 0, 0], [0, -8, -16, 0, 4, -4, -2, 2], [-4, 0, 0, 0, -4, -4, -12, -12], [0, 4, 42, 0, -5, 5, -11, 11], [0, 10, 8, 0, 2, -2, -5, 5], [-8, 0, 0, -10, 0, 0, 4, 4], [-2, 0, 0, 6, 0, 0, -2, -2], [-6, 0, 0, -2, 13, 13, 5, 5], [0, 8, -16, 0, 14, -14, 6, -6], [0, -8, -20, 0, -4, 4, -13, 13], [34, 0, 0, 4, -2, -2, -5, -5], [0, 4, -24, 0, -7, 7, -9, 9], [-10, 0, 0, -8, 11, 11, 3, 3], [-32, 0, 0, 16, 4, 4, 8, 8], [0, -13, 12, 0, 9, -9, -1, 1], [16, 0, 0, -12, 7, 7, -9, -9], [0, 1, 2, 0, 11, -11, 1, -1], [0, -6, -32, 0, 10, -10, -10, 10], [0, 0, 8, 0, 2, -2, 11, -11], [-6, 0, 0, 7, 5, 5, -1, -1]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1560_l_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1560_2_l_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_1560_2_l_d(:prec:=8)
    chi := MakeCharacter_1560_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_1560_2_l_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_1560_2_l_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1560_l();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![256, 0, 768, 0, 340, 0, 37, 0, 1]>],Snew);
    return Vf;
end function;