
// Make newform 1512.2.s.p in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1512_s();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1512_s_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_s_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_1512_2_s_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 := [2401, -1372, 245, 84, -76, 12, 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, 1, 0, 0, 0, 0, 0, 0], [343, -1911, 3290, -1282, -194, -26, -3, -1], [-42532, 2205, 574, -1520, 506, 128, 18, 13], [3773, -8967, -518, 334, -346, 74, 15, 19], [-1960, 1169, -468, -146, 48, -10, -4, 3], [27783, -5929, -2506, 762, -866, 114, 53, -43], [154693, -34398, 224, 5882, -2384, -746, 405, -124]];
    Rf_basisdens := [1, 1, 12348, 37044, 12348, 1764, 24696, 37044];
    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_s();"
function MakeCharacter_1512_s()
    N := 1512;
    order := 3;
    char_gens := [1135, 757, 785, 1081];
    v := [3, 3, 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_1512_s_Hecke();"
function MakeCharacter_1512_s_Hecke(Kf)
    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, 0, 0, 0], [0, 0, 0, 1, 0, 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, 0, 0, 0, 0], [0, 0, 0, -1, 0, -1, 1, 0], [0, -1, 0, -1, -1, 0, 0, 0], [1, -1, 0, 1, 0, 0, 0, 1], [-1, 1, -1, 1, 0, -2, 0, 1], [-1, 0, 0, -1, 0, 0, -3, 0], [-1, 1, 1, 1, 0, 2, -2, 0], [-1, 1, 1, 0, -1, -1, 1, 1], [-3, 2, 0, 2, 2, 1, 0, 2], [-2, 1, 0, -2, 0, 0, 2, -1], [2, -2, -2, 1, -1, -2, 2, 1], [1, 0, 2, 0, 2, 3, 0, 0], [-1, 0, 2, 0, 2, -2, 0, 0], [0, 0, 0, -5, -2, 1, -1, 2], [2, -1, 2, 0, -2, 0, 3, -1], [-5, -1, 0, -5, 0, 0, 2, 1], [1, -1, -1, 0, 1, 4, -4, -1], [0, 0, 1, -1, -1, 0, -4, -1], [-6, 0, 1, 0, 1, -2, 0, 0], [-3, 2, -1, -2, 1, 0, 0, -1], [-2, 2, 2, -3, 1, 0, 0, -1], [4, 2, 0, 2, 2, -2, 0, 2], [-2, 2, 2, -5, 0, -1, 1, 0], [1, 0, -2, 0, -2, 2, 0, 0], [3, -3, 0, 3, 0, 0, 0, 3], [0, 0, 0, 5, 1, -2, 2, -1], [-2, 2, 2, -4, 4, 4, -4, -4], [4, 1, -2, 6, 2, 0, -4, 1], [2, -2, 1, -2, -1, -4, 0, -2], [-7, 0, -1, 0, -1, 0, 0, 0], [-1, 1, 1, -2, 1, 3, -3, -1], [8, -1, -2, 10, 2, 0, -3, 3], [1, -1, 2, -1, 1, 4, 0, -1], [0, 0, 0, -5, 0, -1, 1, 0], [-2, 1, 0, -2, 0, 0, -2, -1], [-4, -2, 0, -4, 0, 0, 6, 2], [-2, 2, 2, -11, 0, 0, 0, 0], [8, -2, -2, -2, -4, -2, 0, -2], [1, -1, -1, -18, -1, 1, -1, 1], [13, 0, 2, 11, -2, 0, -1, -2], [2, 3, -2, 3, 1, -2, 0, 3], [2, -2, -2, 3, -2, 5, -5, 2], [-4, -2, -1, -3, 1, 0, 0, 3], [-8, -4, 2, -4, -2, 4, 0, -4], [12, 1, 2, 10, -2, 0, -8, -3], [-5, -2, 0, -2, -2, -6, 0, -2], [-2, 2, 0, 2, 2, -2, 0, 2], [2, -1, -2, 4, 2, 0, -1, 3], [1, -1, -1, -10, -3, -2, 2, 3], [-1, 1, 1, 0, 5, 1, -1, -5], [-1, 0, 2, 0, 2, 9, 0, 0], [-3, 2, -6, 3, 6, 0, -2, 4], [3, -2, 1, -2, -1, -1, 0, -2], [2, -2, -2, 6, 0, 2, -2, 0], [9, 3, -2, 11, 2, 0, 4, -1], [2, 4, 0, 2, 0, 0, -6, -4], [3, -3, -3, 2, 3, 4, -4, -3], [2, 3, 0, 2, 0, 0, 0, -3], [-14, 2, -3, 2, -1, -8, 0, 2], [1, 0, 0, 1, 0, 0, 2, 0], [8, -2, 1, -2, -1, -2, 0, -2], [-3, 4, 0, 4, 4, -8, 0, 4], [-15, 3, 0, -15, 0, 0, -4, -3], [1, -1, -1, -3, 2, -6, 6, -2], [1, -1, -1, -1, 1, -8, 8, -1], [-1, 1, 1, -9, -4, -4, 4, 4], [12, 5, 0, 5, 5, -2, 0, 5], [24, 0, 0, 24, 0, 0, 4, 0], [-1, -6, 1, -6, -5, -2, 0, -6], [0, 6, -4, 4, 4, 0, 2, -2], [2, -2, -2, 4, -4, -4, 4, 4], [13, 0, 1, 12, -1, 0, 4, -1], [-3, 3, 3, -1, 0, 14, -14, 0], [-15, 3, 2, 3, 5, 8, 0, 3], [-2, 2, 2, 8, 4, -4, 4, -4], [16, -5, 0, 16, 0, 0, 5, 5], [-3, 3, 3, -12, 1, -4, 4, -1], [-3, 3, 3, -15, 3, 0, 0, -3], [-6, 0, 3, -9, -3, 0, -8, -3], [9, 2, 0, 2, 2, 5, 0, 2], [4, -3, 4, -3, 1, 4, 0, -3], [13, 5, -2, 15, 2, 0, -2, -3], [-1, -4, 2, -4, -2, 2, 0, -4], [4, -4, -4, 2, -2, -10, 10, 2], [-3, 3, 3, 12, -1, 7, -7, 1], [-7, 2, 0, 2, 2, -7, 0, 2], [4, -4, -4, 17, -4, 2, -2, 4], [-4, -2, -3, -2, -5, 6, 0, -2], [-12, -1, -4, -1, -5, -4, 0, -1], [-3, 3, 3, -9, -3, 4, -4, 3], [3, 2, -4, 7, 4, 0, -1, 2], [-1, 6, -3, 2, 3, 0, -6, -3], [8, -2, -2, -2, -4, -10, 0, -2], [0, 0, 0, 11, -4, 0, 0, 4], [3, -2, 3, -2, 1, 15, 0, -2], [0, 0, 0, 6, -6, -8, 8, 6], [-6, -6, 2, -8, -2, 0, 8, 4], [-3, 3, 3, 0, -4, -6, 6, 4], [-3, 3, 3, 11, 2, 4, -4, -2], [-17, 6, -2, 6, 4, 4, 0, 6], [7, -2, 0, 7, 0, 0, 11, 2], [-16, -3, 2, -18, -2, 0, -1, 1], [3, -3, -3, -5, -1, -2, 2, 1], [3, 4, -3, 6, 3, 0, 0, -1], [-18, 0, 1, -19, -1, 0, -12, -1], [3, -2, 2, -2, 0, -9, 0, -2], [3, -3, -3, -11, -1, 4, -4, 1], [24, -1, 0, 24, 0, 0, -7, 1], [-21, 4, 4, 4, 8, 6, 0, 4], [3, -3, -3, 15, 4, -2, 2, -4], [16, -5, 0, 16, 0, 0, 0, 5], [18, 0, 3, 0, 3, 6, 0, 0], [0, 0, -1, 1, 1, 0, -12, 1], [-3, -4, 1, -4, -3, 4, 0, -4], [9, 2, 4, 5, -4, 0, -11, -6], [19, 0, 2, 0, 2, 0, 0, 0], [2, -2, 0, 2, 0, 0, -4, 2], [-1, 1, 1, 2, 3, 13, -13, -3], [-25, 6, -2, 6, 4, -1, 0, 6], [-3, -8, 1, -4, -1, 0, 2, 7], [30, 1, -2, 1, -1, 8, 0, 1], [-6, 6, 6, -26, 2, -12, 12, -2], [3, -3, -2, 5, 2, 0, -12, 5], [4, -4, -4, 7, -2, -2, 2, 2], [-15, 4, -3, 4, 1, -1, 0, 4], [-1, 1, 1, 4, 3, -8, 8, -3], [3, -3, -3, -23, -7, 6, -6, 7], [13, -2, -1, -2, -3, 0, 0, -2], [-1, 1, 1, -20, 3, 4, -4, -3], [-9, 2, -2, -7, 2, 0, 0, 0], [9, 2, 2, 2, 4, -3, 0, 2], [3, -3, -3, 1, 4, 12, -12, -4], [-1, 8, -3, 8, 5, -12, 0, 8], [0, 0, 0, 20, 4, 8, -8, -4], [-20, 3, -2, 3, 1, -8, 0, 3], [2, -3, 4, -2, -4, 0, 5, -1], [11, -6, 6, 5, -6, 0, 8, 0], [10, 2, 1, 2, 3, 4, 0, 2], [8, -2, -6, 14, 6, 0, 2, 8], [-16, 4, -4, 4, 0, -2, 0, 4], [-10, 10, 10, -8, 8, 12, -12, -8], [22, -3, 0, 22, 0, 0, -8, 3], [-10, 4, 2, 4, 6, 18, 0, 4], [-39, -2, -1, -38, 1, 0, 6, 3], [-8, 2, -4, 2, -2, 6, 0, 2], [-4, -1, 0, -1, -1, -8, 0, -1], [-9, 3, 6, -15, -6, 0, 10, -9], [8, -8, -8, 9, -8, -8, 8, 8], [-1, 1, 1, -6, 5, 7, -7, -5], [-1, 1, 1, -4, -3, -14, 14, 3], [1, 6, -3, 6, 3, -11, 0, 6], [2, 5, -6, 5, -1, -14, 0, 5], [4, -4, -4, 22, -10, -16, 16, 10], [18, -2, -1, 19, 1, 0, 18, 3], [-33, -2, 2, -2, 0, -5, 0, -2], [-1, 1, 1, -16, -1, -22, 22, 1], [2, -2, -2, -19, -4, 13, -13, 4], [-11, -7, 2, -7, -5, 0, 0, -7], [-9, -9, 6, -15, -6, 0, 6, 3], [-2, 2, 2, 1, 6, -5, 5, -6], [-11, 6, -1, 6, 5, -7, 0, 6], [-1, -5, -3, -5, -8, 2, 0, -5], [3, -3, -3, -4, 5, 1, -1, -5], [18, 6, -6, 24, 6, 0, 4, 0], [17, 1, 4, 13, -4, 0, -22, -5], [-1, 7, -9, 8, 9, 0, -14, 2], [-12, -1, -4, -8, 4, 0, -8, 5], [31, -1, -2, -1, -3, -4, 0, -1], [-7, 7, 7, 10, 9, 3, -3, -9], [-1, 0, -2, 1, 2, 0, -7, 2], [0, -6, 6, -6, 0, -10, 0, -6], [-4, 4, 4, -6, 0, -14, 14, 0], [-15, 2, -7, -8, 7, 0, -8, 5], [-17, -1, 3, -20, -3, 0, -10, -2], [-1, 6, -1, 6, 5, -5, 0, 6], [-1, -6, 6, -6, 0, -6, 0, -6], [5, -5, 4, 1, -4, 0, -16, 1], [-9, -2, 13, -2, 11, 2, 0, -2], [-13, 13, 13, -8, 9, 4, -4, -9], [-1, 1, 1, -17, 6, -6, 6, -6], [-11, -8, 3, -8, -5, 7, 0, -8], [15, 4, -9, 4, -5, 0, 0, 4], [-8, 8, 8, 7, 0, 11, -11, 0], [-12, 6, 0, -12, 0, 0, -14, -6], [26, 4, -8, 34, 8, 0, -2, 4], [26, -1, 0, 26, 0, 0, 10, 1], [3, -2, 3, 0, -3, 0, 8, -1], [-3, 3, 3, -4, 10, 12, -12, -10], [-7, 7, -8, 1, 8, 0, 4, 1], [-5, 5, 5, 5, 12, 2, -2, -12], [6, -18, 9, -18, -9, 4, 0, -18], [0, 0, 0, -9, 2, 14, -14, -2], [1, -1, -1, -9, 9, 8, -8, -9], [-5, 4, -4, -1, 4, 0, 13, 0], [-22, 10, 0, -22, 0, 0, 0, -10], [-23, 4, -11, -12, 11, 0, 0, 7], [1, -1, -1, 2, 9, -2, 2, -9], [-5, -12, 9, -12, -3, -3, 0, -12], [-1, 1, 1, 6, -5, -15, 15, 5], [12, -5, 8, 4, -8, 0, -7, -3], [-1, -5, -3, -5, -8, 2, 0, -5], [5, -5, -5, -18, -7, -10, 10, 7], [13, -14, -2, 15, 2, 0, 10, 16], [-12, 6, -3, 6, 3, 2, 0, 6], [-15, 3, -6, -9, 6, 0, -4, 3], [-1, 1, 1, -12, 5, 0, 0, -5], [-4, 4, 4, 10, 8, -2, 2, -8], [-1, 0, -13, 0, -13, -15, 0, 0], [3, 2, -4, 7, 4, 0, -8, 2], [7, -7, -7, 32, -2, 12, -12, 2], [-2, -10, 2, -10, -8, 8, 0, -10], [-21, -2, 7, -2, 5, 10, 0, -2], [-11, 11, 11, -23, 5, -6, 6, -5], [10, -3, -2, 12, 2, 0, -15, 5], [11, -11, -11, 8, -14, -16, 16, 14], [12, 0, 10, 2, -10, 0, -18, -10], [1, 0, 6, -5, -6, 0, 7, -6], [-3, 3, 3, 3, 3, 10, -10, -3], [31, -6, 8, -6, 2, 7, 0, -6], [10, -10, -10, -15, -9, -10, 10, 9], [27, 10, -3, 10, 7, 2, 0, 10], [3, -3, -3, -2, 1, 3, -3, -1], [4, -4, -4, -24, -12, 2, -2, 12], [-5, 0, 2, 0, 2, -11, 0, 0], [1, 4, 1, 4, 5, 14, 0, 4], [2, -2, -2, -8, 8, 4, -4, -8], [-15, -2, -6, -9, 6, 0, 21, 8], [-1, 1, 1, -21, 4, -10, 10, -4], [11, -11, -11, 23, 1, 12, -12, -1], [15, 8, -3, 18, 3, 0, 8, -5], [16, 12, -2, 18, 2, 0, 0, -10], [-19, 2, 9, 2, 11, -2, 0, 2], [-3, 2, -8, 5, 8, 0, -19, 6], [-9, 1, 0, 1, 1, -6, 0, 1], [36, -6, 2, 34, -2, 0, 10, 4], [13, -13, -13, 25, -8, 6, -6, 8], [-5, 5, 5, 21, 9, -10, 10, -9], [25, 6, 3, 6, 9, -3, 0, 6], [-22, -2, -9, -2, -11, -4, 0, -2], [-39, -2, 6, -45, -6, 0, 5, -4], [-1, 1, 1, 15, 2, -20, 20, -2], [-31, -10, 3, -34, -3, 0, 0, 7], [11, -11, -11, 32, -7, 0, 0, 7], [8, -8, -5, -8, -13, 0, 0, -8], [12, -12, -12, 1, -6, -17, 17, 6], [-27, 1, -5, 1, -4, -12, 0, 1], [-32, 9, 8, -40, -8, 0, 7, -17], [24, -12, 8, 16, -8, 0, 6, 4], [5, 0, -4, 0, -4, -29, 0, 0], [3, -4, -3, -4, -7, 18, 0, -4], [7, -7, -7, 12, -1, -5, 5, 1], [10, 1, -2, 12, 2, 0, -3, 1], [-11, 5, -6, 5, -1, 0, 0, 5], [32, 3, -2, 34, 2, 0, 1, -1], [12, -12, -12, 12, -6, 10, -10, 6], [33, -5, -5, 38, 5, 0, 0, 10], [-9, 4, -4, -5, 4, 0, 2, 0], [-34, -2, 4, -2, 2, 12, 0, -2], [-4, 4, 4, 24, 6, -14, 14, -6], [5, -4, -3, 8, 3, 0, 8, 7], [-8, -4, 5, -4, 1, 0, 0, -4], [21, 3, 5, 16, -5, 0, -10, -8], [7, -3, -1, -3, -4, 12, 0, -3], [20, -2, 2, 18, -2, 0, 6, 0], [-2, 2, 2, 13, 10, -6, 6, -10], [9, -2, 7, -2, 5, 21, 0, -2], [-17, 6, 0, 6, 6, 7, 0, 6], [-32, -4, -2, -4, -6, 10, 0, -4], [-8, -4, 4, -12, -4, 0, 20, 0], [-1, 1, 1, -37, 8, 8, -8, -8], [-5, 4, -12, 7, 12, 0, 20, 8], [35, -4, 6, 29, -6, 0, 4, -2], [-10, 10, 10, -27, -3, -2, 2, 3], [-18, 0, 4, 0, 4, 2, 0, 0], [-5, 5, 5, -32, -5, -4, 4, 5], [14, -14, -14, 17, 0, 1, -1, 0], [6, 5, 4, 2, -4, 0, 4, -9], [-2, 2, 2, 23, 12, -12, 12, -12], [-4, 4, 4, 8, -2, 18, -18, 2], [-1, -11, 14, -15, -14, 0, -4, -3], [6, -5, 2, 4, -2, 0, 24, 3], [-34, 10, 9, 10, 19, 16, 0, 10], [-5, 5, -5, 5, 0, 2, 0, 5], [0, 0, 0, -19, 0, 14, -14, 0], [-8, 8, 8, 18, 0, 14, -14, 0], [-14, 8, -9, -5, 9, 0, -12, 1], [25, 6, -4, 6, 2, -3, 0, 6], [35, -5, -7, 42, 7, 0, -14, 12], [-42, 0, -2, 0, -2, 6, 0, 0], [-31, -11, 2, -33, -2, 0, 6, 9], [21, -4, 6, 15, -6, 0, 3, -2], [-10, 10, 10, -15, 8, -3, 3, -8], [-41, 6, -1, 6, 5, -9, 0, 6], [-31, -12, 7, -12, -5, 2, 0, -12], [-24, -1, -2, -22, 2, 0, 1, 3], [4, -4, -4, -3, -3, -6, 6, 3], [-15, 8, -9, 8, -1, -33, 0, 8], [-1, 1, 1, 19, -7, 4, -4, 7], [23, -3, -12, -3, -15, -18, 0, -3], [4, -4, -4, 13, 10, -12, 12, -10], [0, 0, 0, -16, 4, -24, 24, -4], [12, 0, -2, 14, 2, 0, -22, 2]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1512_s_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1512_2_s_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_1512_2_s_p(:prec:=8)
    chi := MakeCharacter_1512_s();
    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_s_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_1512_2_s_p( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1512_s();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![4, 4, 6, -2, 1]>,<11,R![20736, 3456, 4320, -48, 580, 4, 30, -2, 1]>,<13,R![-264, -204, -35, 4, 1]>],Snew);
    return Vf;
end function;