
// Make newform 1300.2.y.b in Magma, downloaded from the LMFDB on 06 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1300_y();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1300_y_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_1300_2_y_b();".
// 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_1300_2_y_b();".
// 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, -32, 20, 4, -11, 2, 5, -4, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [0, -8, 0, 1, -2, 1, 0, 1], [0, 14, -10, -1, 4, -1, -2, 1], [-48, 48, 2, -27, 8, 9, -6, 1], [-28, 40, -11, -15, 11, 3, -7, 3], [-64, 104, -42, -37, 32, 7, -18, 7], [-104, 140, -34, -55, 36, 13, -26, 9], [200, -256, 60, 95, -66, -25, 44, -17]];
    Rf_basisdens := [1, 8, 4, 8, 4, 8, 8, 8];
    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_1300_y();"
function MakeCharacter_1300_y()
    N := 1300;
    order := 6;
    char_gens := [651, 677, 301];
    v := [6, 6, 5];
    // 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_1300_y_Hecke();"
function MakeCharacter_1300_y_Hecke(Kf)
    N := 1300;
    order := 6;
    char_gens := [651, 677, 301];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, 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, -1, 0, 1, -1], [0, 0, 0, 0, 0, 0, 0, 0], [-1, -1, 0, 0, 1, 0, 0, 0], [1, 0, 0, 0, 1, 0, 0, 0], [0, 0, 0, -1, 1, 0, 1, -1], [-1, 1, 2, 0, -1, 0, 0, 0], [-1, 1, 0, 0, -1, 0, 2, -1], [1, -1, 0, 0, -2, -1, 1, -1], [0, 0, 1, 0, 0, -2, -1, 1], [-1, 2, 1, -1, 0, 2, 1, -2], [1, 0, -3, 0, -2, 0, 0, 0], [3, -1, -3, -2, 1, 1, -1, -1], [-1, 1, 2, 3, -5, -3, 0, 0], [-2, 0, 0, 0, 4, 0, 0, 0], [-3, 3, -2, 1, -3, 0, 2, 0], [-5, 1, 0, -1, 1, -1, 2, -1], [-1, 1, 2, 0, 1, 0, 0, 3], [3, 1, 2, 2, 4, -1, 1, 1], [-5, 1, 0, 2, 3, 2, -2, 1], [0, 1, 3, 2, -3, -4, -1, 2], [-5, -1, 2, 1, 1, 0, 4, 0], [-1, 2, 3, 1, 4, -2, -3, 6], [-3, 1, 3, 2, -3, -1, -1, -1], [5, -3, 0, -2, -1, -2, 0, 0], [-2, -1, 0, 0, 4, -1, -2, 2], [-2, -1, -1, -2, 1, 0, -5, 0], [-2, -1, -3, 0, 1, 5, 4, -4], [2, -1, -1, 0, 3, 0, -3, 6], [1, -1, -2, -2, -5, 2, 0, -4], [-9, -3, -2, 0, 10, 1, 1, -1], [0, 3, 1, 4, -3, 0, -1, 0], [-7, 5, 0, 1, -1, 1, 4, -2], [-2, 2, 4, 1, 3, -1, 0, -2], [5, -1, 0, 1, -5, 1, 6, -3], [0, -1, -1, 0, -1, 0, 1, -2], [0, -1, -3, -4, 1, 0, -1, 0], [4, 0, 0, 2, -3, 2, 2, -1], [4, -1, 3, -2, 7, 1, -2, -2], [-2, 2, 4, 0, 1, 0, 0, -3], [-5, -7, -3, 0, 7, -1, 1, -1], [0, 1, -3, -2, -1, 0, 7, 0], [-2, 2, 4, 1, 7, -1, 0, -4], [7, -2, -5, -4, 2, 2, -4, -4], [5, 2, 3, 4, 6, -2, 2, 2], [4, -4, -8, -6, 3, 6, 0, -6], [-2, 0, 1, 0, 0, -2, 1, -1], [2, 0, -2, 0, 3, 0, 3, 3], [7, 3, 0, 3, -3, 3, -4, 2], [0, 0, -4, -4, 8, 8, 0, 0], [14, 1, 1, 2, -1, 0, -5, 0], [8, -4, -6, -2, 4, 4, -6, 12], [12, -2, 0, -3, -7, -3, 4, -2], [-1, 1, 2, -3, 11, 3, 0, -3], [-20, -1, -1, 0, 19, 1, 2, -2], [2, -2, -4, 0, 5, 6, -3, 3], [-2, 2, 4, -2, 13, 2, 0, -4], [-6, 2, 3, 4, -4, -2, 3, 3], [0, 0, 0, 0, -7, 0, 0, -3], [-2, 2, 0, -2, -2, 4, 4, -8], [4, 7, 1, 0, -7, 5, 2, -2], [4, 0, 0, 0, -1, 0, -2, 1], [6, -1, -1, 0, -13, 0, 1, -2], [-8, 5, 3, 8, -5, 0, 1, 0], [7, -2, -1, -3, 2, 0, -3, 0], [0, 2, 0, -2, 6, 4, -2, 4], [2, 0, 0, -3, 3, -3, -8, 4], [-11, -5, 2, -3, 5, 0, 6, 0], [1, -1, -2, 2, 1, -2, 0, -8], [10, 2, -3, 4, 2, -2, -1, -1], [-12, 1, 3, 2, -11, -1, 0, 0], [6, 0, 0, 0, -4, 0, -4, 8], [12, 5, 1, 0, -13, 3, 0, 0], [0, 0, 0, 3, -17, -3, 0, 3], [12, 2, -1, 4, 8, -2, 1, 1], [-4, -12, 0, -3, 9, -3, -2, 1], [-5, -1, -4, -5, 1, 0, -10, 0], [-22, -6, 0, -2, 11, -2, 6, -3], [4, 1, 0, 2, 0, -1, -2, -2], [8, 2, 0, 0, -5, 0, 0, 0], [11, 7, 3, 0, -13, 1, -1, 1], [0, 2, 6, 4, 2, -8, -6, 12], [-9, 5, 15, 10, -3, -5, 1, 1], [2, -2, -4, -6, -3, 6, 0, -3], [-15, 5, 3, 0, 13, -1, -1, 1], [3, 3, 0, 3, -3, 0, 6, 0], [-11, -1, 0, 1, 3, 1, 6, -3], [10, 1, 3, 2, 13, -1, 2, 2], [1, -1, 0, 1, 3, 1, -6, 3], [3, -3, -6, -3, 7, 6, -2, 4], [-12, 0, 2, 2, 0, 0, -8, 0], [-1, -4, 0, -8, 5, 4, -2, -2], [19, 3, 0, 2, -11, 2, 0, 0], [-1, 10, 0, 0, -7, 10, 8, -8], [-1, 4, 7, 3, -6, -6, -1, 2], [-4, 4, 8, 4, -1, -4, 0, 5], [3, 4, 0, 8, 1, -4, 6, 6], [-6, -6, 2, -4, 6, 0, -14, 0], [-16, 8, 8, 0, 7, -8, 1, -1], [-1, -9, -10, -1, 5, 2, 4, -8], [-23, 3, -6, -3, -3, 0, 0, 0], [-4, 4, 6, 8, -9, -4, -3, -3], [3, -3, -6, -2, 21, 2, 0, -4], [-26, -10, -3, 0, 22, -4, 7, -7], [-7, 0, -1, -1, 0, 0, -1, 0], [18, -13, -17, -4, -13, 8, -1, 2], [21, -1, -6, -2, 14, 1, -3, -3], [-4, 1, 3, 2, -3, -4, 3, -6], [11, -5, -6, -11, 5, 0, -4, 0], [20, -12, -9, 0, -14, 6, 3, -3], [-9, 9, 18, 9, -1, -9, 0, 6], [-5, -2, 3, -4, 2, 2, 0, 0], [13, -7, 0, 1, 3, 1, -12, 6], [-4, 4, 10, 6, -12, -12, 2, -4], [0, -10, 0, -3, 11, -3, -12, 6], [-2, 2, 4, 4, 1, -4, 0, -10], [-17, -9, 0, 3, 9, 3, 8, -4], [-20, 2, 4, 0, 13, -6, 3, -3], [3, 3, 6, 0, -12, -9, 3, -3], [-7, 7, 14, 7, -1, -7, 0, 5], [-1, -1, 9, -2, 9, 1, -1, -1], [2, 3, 3, 0, -17, -3, 12, -12], [-12, 3, 5, 8, -3, 0, -5, 0], [-6, 6, 0, 0, 1, 0, -2, 1], [8, 1, 0, 2, 12, -1, 6, 6], [28, 2, -4, -2, -2, 0, -4, 0], [-30, 4, 0, 0, 15, 0, -4, 2], [7, -7, -14, 1, -5, -1, 0, 11], [15, -1, -8, -9, 1, 0, -6, 0], [-2, -8, -12, -4, 8, 8, 6, -12], [1, 3, -3, 6, -7, -3, 1, 1], [6, -7, -15, -14, 3, 7, -2, -2], [0, -9, -2, 0, -2, -5, 4, -4], [4, -2, 4, 0, 5, -10, -13, 13], [-3, 7, 0, 8, -7, 8, 10, -5], [12, 1, -2, 2, 2, -1, -6, -6], [-21, 11, 0, 1, 3, 1, 4, -2], [-16, 4, 0, -3, 9, -3, -6, 3], [1, -1, -2, -4, -29, 4, 0, -14], [11, -8, -4, 0, -13, 0, 6, -6], [10, -2, 2, 4, -14, -8, -6, 12], [-10, 2, 3, 4, -12, -2, -1, -1], [-4, 4, 8, 9, -11, -9, 0, 9], [0, -3, -9, -6, 13, 12, 1, -2], [3, -6, 2, 0, 11, -10, -16, 16], [5, -13, 0, -5, 5, -5, -2, 1], [-13, -1, 4, 5, 13, -10, 2, -4], [-15, 0, 7, 7, 0, 0, 5, 0], [-20, 3, 9, 12, -3, 0, 3, 0], [2, 5, -3, -8, 1, 16, 3, -6], [12, 4, 0, 4, -5, 4, -6, 3], [-11, -10, -6, 0, 19, 2, -2, 2], [-9, -4, 1, -3, 4, 0, 9, 0], [3, -6, -3, 0, -12, 0, 12, -12], [-15, -9, -2, 0, 4, -5, 13, -13], [-7, -2, -3, -5, 2, 0, -7, 0], [7, -7, -14, -14, 31, 14, 0, 1], [6, -4, 0, 1, 5, 1, -12, 6], [10, -5, -1, -6, 5, 0, -15, 0], [7, -1, -6, -5, 13, 10, -8, 16], [-12, 6, 12, 12, -5, -6, 7, 7], [7, -7, -14, 6, -5, -6, 0, 6], [9, -4, -9, -5, -14, 10, 5, -10], [-1, 1, 2, 3, 11, -3, 0, -3], [19, 3, 0, 6, 8, -3, -5, -5], [23, -3, -6, -3, -25, 6, -6, 12], [-24, 10, 5, 0, 10, 0, 9, -9], [2, -2, -4, 11, 3, -11, 0, 11], [5, -8, -13, -5, 10, 10, -1, 2], [32, 1, 3, 4, -1, 0, 5, 0], [12, -11, -15, -4, -19, 8, 7, -14], [13, 15, 0, 2, -15, 2, 2, -1], [-8, 8, 16, 11, -17, -11, 0, -2], [-5, 3, 0, 5, 7, 5, -12, 6], [-12, -4, 4, 0, 4, 0, 12, 0], [-2, 2, 4, 4, -11, -4, 0, -4], [-4, -2, -7, -4, 2, 2, 9, 9], [-12, -5, -3, 2, 11, -4, 7, -14], [29, -12, -9, 0, -8, 6, -12, 12], [6, 16, 3, 0, -22, 10, 13, -13], [-2, -6, -2, 4, 22, -8, -10, 20], [8, -14, 2, -12, 14, 0, -6, 0], [13, 0, -5, -5, 0, 0, -11, 0], [-3, 3, 6, 3, 5, -6, -4, 8], [-2, -4, 6, -8, 15, 4, 3, 3], [6, -6, -12, -5, 9, 5, 0, 2], [-11, 0, 3, 3, 0, 0, -3, 0], [-11, -2, 1, 3, -4, -6, 11, -22], [3, -4, -4, -8, 13, 4, 6, 6], [-8, 6, 0, -3, 3, -3, -4, 2], [4, -4, -8, -2, 1, 2, 0, 7], [-3, 5, 0, 7, 5, 7, -12, 6], [17, 5, 2, 7, -5, 0, 16, 0], [17, 1, -3, 2, 3, -1, -9, -9], [-5, 5, 10, 3, -29, -3, 0, -6], [-25, 1, -4, 0, 20, 9, 9, -9], [-19, 13, 18, 5, 23, -10, -4, 8], [4, -4, -8, -8, -3, 8, 0, -5], [15, -3, -6, -3, -37, 6, 8, -16], [8, -5, 5, 0, 5, 0, 9, 0], [-4, 10, 0, 5, 1, 5, -8, 4], [6, -6, -12, -6, -17, 6, 0, -6], [0, 5, 9, 10, -1, -5, 0, 0], [-10, 1, 7, 8, -1, 0, 5, 0], [8, 0, -3, 0, 4, 0, -1, -1], [0, -9, -13, 0, 15, 17, -2, 2], [9, -3, -4, -1, -17, 2, 2, -4], [10, -10, -20, -13, 25, 13, 0, -5], [12, 0, -3, 0, -4, 0, -13, -13], [9, -9, -18, -5, -9, 5, 0, -5], [-25, 6, 5, 0, 12, -4, 8, -8], [5, 7, -10, -3, -7, 0, 0, 0], [-23, 1, -2, 0, 14, 5, 11, -11], [-4, 0, 0, -3, 9, -3, -14, 7], [35, -11, 0, 2, -13, 2, 2, -1], [-16, -2, -6, -4, 30, 8, 6, -12], [1, -2, -7, -9, 2, 0, -15, 0], [-4, 4, 8, 9, 5, -9, 0, 18], [-1, 0, 3, 0, 8, 0, 6, 6], [-16, 13, 15, 2, -7, -4, 11, -22], [-11, -3, -3, 0, 11, 3, 3, -3], [12, -8, -14, -6, -8, 12, 2, -4], [-11, 3, 0, -3, 5, 6, 10, -20], [19, -1, 0, -8, -3, -8, -12, 6], [18, -6, -2, 0, -9, -2, -7, 7], [-20, -2, 8, 6, 2, 0, 0, 0], [9, 9, -2, 0, -18, 13, 11, -11], [10, -10, -20, -12, 19, 12, 0, 12], [-13, 13, 26, 17, -13, -17, 0, 2], [19, -7, -6, 1, -25, -2, -4, 8], [1, -5, -4, 0, 6, 3, -3, 3], [12, 0, 1, 0, -20, -2, 7, -7], [14, -4, -7, -8, 16, 4, 1, 1], [7, -7, 10, 3, 7, 0, 0, 0], [22, 3, -9, -6, -3, 0, -21, 0], [6, 5, 9, 4, -15, -8, -5, 10], [25, -13, 0, -11, -5, -11, -2, 1], [3, 5, 0, 10, -6, -5, 1, 1], [-3, 3, 6, 0, 3, 0, 0, 3], [6, 9, 10, 0, -12, -11, -4, 4], [-11, 10, 9, 20, -12, -10, 10, 10], [1, 16, 8, 0, -9, 0, 0, 0], [2, -2, -4, 9, 9, -9, 0, 15], [2, -7, -16, -14, 4, 7, 4, 4], [-30, -14, 0, -7, 27, -7, -10, 5], [12, 0, -4, -4, 0, 0, 4, 0], [21, -7, 0, 5, 1, 5, -16, 8], [2, -6, 3, -12, 20, 6, 3, 3], [4, 14, 0, -3, -9, -3, 0, 0], [21, -9, -4, 0, -6, -1, -11, 11], [1, -2, 3, -4, 14, 2, 6, 6], [-30, 0, 9, 0, -16, 0, 5, 5], [-20, 15, 21, 6, -9, -12, 11, -22], [-15, 5, 7, 0, -7, -9, 15, -15], [-24, -3, -5, -8, 3, 0, -19, 0], [-21, 5, 0, -2, 11, -2, -6, 3], [-5, 5, 10, 15, -39, -15, 0, -3], [1, 0, 9, 0, 14, 0, 4, 4], [-5, 5, -10, -5, -5, 0, 14, 0], [2, 22, 0, 5, -13, 5, 2, -1], [-32, 13, -1, 12, -13, 0, 3, 0], [16, -10, -4, 0, 1, -2, -13, 13], [-12, 0, 0, 0, 16, 0, 4, -8], [20, 18, 6, 0, -29, 6, 3, -3], [12, -8, 0, -7, -9, -7, 14, -7], [-3, 3, 6, 18, -25, -18, 0, 3], [-27, -5, 0, 2, 5, 2, 22, -11], [20, -20, -24, -4, -4, 8, -4, 8], [-44, -12, 0, -3, 29, -3, -2, 1], [-9, 9, 18, 5, 9, -5, 0, -2], [-11, 0, 9, 9, 0, 0, 9, 0], [-10, 5, 9, 4, 1, -8, 3, -6], [-22, 5, -3, 10, -33, -5, 2, 2], [-2, 2, 4, 0, -1, 0, 0, 0], [-1, 1, 2, -9, 21, 9, 0, 0], [12, -1, 5, -2, 17, 1, -2, -2], [-13, 7, 0, -1, 5, -1, -4, 2], [8, -10, -2, 8, -26, -16, 2, -4], [0, 10, 0, -3, 5, -3, -20, 10], [3, -3, -6, -7, 27, 7, 0, -11], [0, -12, -8, 0, 3, 4, 5, -5], [-22, 12, 21, 24, -20, -12, 5, 5], [2, -2, 2, 0, 2, 0, 12, 0], [19, -6, -20, -12, 7, 6, -4, -4], [-20, 17, 11, -6, 17, 12, 9, -18], [18, 6, 4, 10, -6, 0, 20, 0], [-28, -3, -9, -12, 3, 0, 21, 0], [-16, 1, 3, 2, 37, -4, -5, 10], [-26, 16, 0, 5, -5, 5, 20, -10], [1, -1, -2, 8, 1, -8, 0, -5], [32, 13, 8, 0, -34, -3, -6, 6], [3, -3, -6, -9, -9, 9, 0, 6], [-25, 2, 3, 4, -34, -2, -8, -8], [8, -10, 0, -5, 7, -5, -12, 6], [-8, 8, 16, 3, -21, -3, 0, -9], [-22, 4, 8, 12, -4, 0, -12, 0], [22, -3, 13, 10, 3, 0, -5, 0], [-26, -1, 7, 0, 19, -15, 0, 0], [8, -14, -1, 0, -4, -12, -3, 3], [1, -1, 6, -2, 6, 1, -3, -3], [-1, 1, 2, -1, -3, 1, 0, 2], [8, -10, 0, 10, -14, -20, -6, 12], [13, 0, 6, 0, -7, -12, -12, 12]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1300_y_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1300_2_y_b();".
// 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_1300_2_y_b(:prec:=8)
    chi := MakeCharacter_1300_y();
    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_1300_2_y_b();".
// 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_1300_2_y_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1300_y();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![1, 10, 106, -56, 55, -8, 10, -2, 1]>],Snew);
    return Vf;
end function;