
// Make newform 3150.2.m.k in Magma, downloaded from the LMFDB on 20 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3150_m();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_3150_m_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_3150_2_m_k();".
// 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_3150_2_m_k();".
// 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 := [1, 0, 0, 0, -1, 0, 0, 0, 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, 0, 1, 0, 0, 0, 0], [0, 1, 0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 0, 0, 1, 0], [-1, 0, 0, 0, 2, 0, 0, 0], [0, 1, 0, 0, 0, -1, 0, 0], [0, 0, 2, 0, 0, 0, -1, 0], [0, 0, 0, -1, 0, 0, 0, 2]];
    Rf_basisdens := [1, 1, 1, 1, 1, 1, 1, 1];
    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_3150_m();"
function MakeCharacter_3150_m()
    N := 3150;
    order := 4;
    char_gens := [2801, 127, 451];
    v := [2, 1, 4];
    // 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_3150_m_Hecke();"
function MakeCharacter_3150_m_Hecke(Kf)
    N := 3150;
    order := 4;
    char_gens := [2801, 127, 451];
    char_values := [[-1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 1, 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 := 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, 1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0, 0, 0], [0, 1, 0, 2, 0, -1, 0, 0], [1, 0, 0, -1, 1, -1, -1, 0], [0, 0, 0, 0, -1, -2, 1, -1], [0, -1, 0, 0, 2, 1, 0, 0], [-2, 1, 2, -2, 0, 0, 0, 0], [0, -1, -2, 0, 0, -1, -1, 2], [0, -4, -1, 0, 0, -4, 1, 1], [-2, -2, 0, -2, 2, 0, 2, 0], [0, -1, 2, 1, 2, 1, 0, 2], [-1, 0, 0, 1, 2, -2, -2, -1], [1, 0, 0, -1, 2, 2, -2, -2], [-4, -1, 0, -4, 1, 0, 1, 0], [-1, 3, -1, 0, 0, 3, 2, 1], [-4, -5, 1, 0, 0, -5, 1, -1], [2, 4, -2, 2, 2, 0, 2, 0], [0, 0, 0, 4, -4, 0, 0, 0], [0, 0, 0, 0, 1, -6, -1, 0], [0, 2, 3, -2, 2, -2, 0, 3], [1, -2, 3, 1, -1, 0, -1, 0], [2, -2, -2, 0, 0, -2, 0, 2], [2, 0, 2, 2, 3, 0, 3, 0], [0, 2, 1, 8, 2, -2, 0, 1], [-1, 0, 0, 1, 2, 1, -2, 4], [4, 0, 0, -4, -2, -2, 2, -4], [0, 4, -3, 0, 4, -4, 0, -3], [-4, 2, 2, -4, 1, 0, 1, 0], [4, -8, 2, 4, 4, 0, 4, 0], [0, -6, -2, 6, 0, 6, 0, -2], [0, 0, 0, 0, -2, 2, 2, -6], [0, 0, -4, -2, -2, 0, 0, -4], [0, -1, -4, 0, 0, -1, 3, 4], [4, -4, 0, 0, 0, -4, -2, 0], [-8, -4, -4, -8, 4, 0, 4, 0], [-5, 0, 0, 5, 0, -2, 0, -3], [3, 0, 0, -3, 0, -2, 0, -4], [-5, 0, 6, -5, -2, 0, -2, 0], [10, 2, -2, 0, 0, 2, -4, 2], [-4, -10, -2, 0, 0, -10, 0, 2], [0, 4, -4, 6, 1, -4, 0, -4], [6, 0, 0, -6, 2, 0, -2, -2], [6, 0, 0, -6, 5, 11, -5, -4], [0, 4, 6, 0, 4, -4, 0, 6], [-9, -6, 5, 0, 0, -6, -2, -5], [7, 0, 0, -7, 0, 1, 0, 4], [7, 0, 0, -7, -1, -10, 1, -1], [0, 2, 4, 0, -2, -2, 0, 4], [4, 4, -6, 4, 3, 0, 3, 0], [-8, -6, 0, 0, 0, -6, 0, 0], [4, 4, -6, 0, 0, 4, -2, 6], [0, -3, -7, -1, 2, 3, 0, -7], [6, 0, 0, -6, 7, -10, -7, 3], [8, -3, 2, 8, 0, 0, 0, 0], [8, 2, -2, 0, 0, 2, 6, 2], [-8, 2, -2, 0, 0, 2, -10, 2], [7, -14, 0, 7, 2, 0, 2, 0], [0, -10, 0, 8, 0, 10, 0, 0], [0, 0, 0, 0, 1, 18, -1, 0], [-2, -12, 0, -2, -4, 0, -4, 0], [2, -12, -6, 2, -2, 0, -2, 0], [0, 0, -6, -8, 6, 0, 0, -6], [-8, 0, 0, 8, 1, -10, -1, 0], [-2, 0, 0, 2, -3, -9, 3, 0], [5, -4, -1, 0, 0, -4, -6, 1], [8, 12, 7, 8, -4, 0, -4, 0], [6, 0, 0, -6, -7, 14, 7, 8], [0, -3, -3, -8, 11, 3, 0, -3], [-6, 8, 8, -6, 4, 0, 4, 0], [10, 0, 4, 0, 0, 0, 7, -4], [-19, -3, 0, -19, -2, 0, -2, 0], [-7, 0, 0, 7, 0, 0, 0, -8], [0, -2, -7, -3, -2, 2, 0, -7], [14, 6, 2, 14, -2, 0, -2, 0], [-24, 8, -4, 0, 0, 8, -6, 4], [-3, -5, 8, -3, -5, 0, -5, 0], [0, 10, 4, -6, -2, -10, 0, 4], [0, -1, 12, -8, -2, 1, 0, 12], [-5, 9, -1, 0, 0, 9, -10, 1], [2, 8, 5, 0, 0, 8, 4, -5], [0, 6, -4, -2, -3, -6, 0, -4], [2, 0, 0, -2, 0, 6, 0, 8], [0, -8, 3, 12, 3, 8, 0, 3], [2, 0, 8, 2, 7, 0, 7, 0], [14, -3, 0, 0, 0, -3, -8, 0], [-2, 0, -9, -2, -6, 0, -6, 0], [0, -2, -3, 8, -10, 2, 0, -3], [-4, 0, 0, 4, -4, 6, 4, -2], [-5, 0, 0, 5, 7, -2, -7, -3], [-8, -4, -1, 0, 0, -4, 16, 1], [7, -12, -12, 7, -6, 0, -6, 0], [0, 6, -8, -12, -6, -6, 0, -8], [0, 6, 3, 15, -14, -6, 0, 3], [-1, -4, -10, -1, -6, 0, -6, 0], [-6, -10, -5, 0, 0, -10, 12, 5], [0, 9, 2, -7, 10, -9, 0, 2], [8, 0, 0, -8, -1, 20, 1, 2], [6, 18, -1, 0, 0, 18, 2, 1], [19, -6, -1, 19, -4, 0, -4, 0], [12, 0, 0, -12, -6, 4, 6, 0], [3, 2, -5, 3, 1, 0, 1, 0], [-10, -5, -4, 0, 0, -5, 6, 4], [13, 12, 1, 0, 0, 12, -2, -1], [-22, -2, 0, -22, -4, 0, -4, 0], [5, 0, 0, -5, -1, 18, 1, 11], [-8, 4, -20, -8, -2, 0, -2, 0], [6, 8, 2, 0, 0, 8, -13, -2], [-24, 9, 6, 0, 0, 9, 8, -6], [-6, 4, 16, -6, 0, 0, 0, 0], [2, 0, 0, -2, -4, -16, 4, 10], [-20, 0, 0, 20, 2, 4, -2, 10], [0, -4, 14, -4, 2, 4, 0, 14], [-6, 0, 4, 0, 0, 0, -12, -4], [0, -2, 8, -22, -4, 2, 0, 8], [2, 0, 0, -2, -5, -4, 5, 4], [-3, 0, 0, 3, 2, -36, -2, -2], [-10, 10, 0, -10, 6, 0, 6, 0], [-4, -1, 2, 0, 0, -1, -14, -2], [16, 6, -10, 0, 0, 6, 6, 10], [6, 0, 0, -6, -8, -20, 8, -5], [5, 0, 0, -5, 6, 0, -6, 6], [6, -4, -10, 6, 2, 0, 2, 0], [-2, 15, -2, 0, 0, 15, 10, 2], [0, -9, 8, -24, -9, 9, 0, 8], [0, 10, 6, -6, 14, -10, 0, 6], [22, -6, 3, 0, 0, -6, -4, -3], [15, 3, 0, 15, -10, 0, -10, 0], [1, 0, 0, -1, -9, -9, 9, 0], [0, -10, -1, -29, 6, 10, 0, -1], [-2, -5, 6, -2, 6, 0, 6, 0], [-2, 6, -7, 0, 0, 6, 4, 7], [13, -14, 0, 13, -2, 0, -2, 0], [0, -10, 12, 4, 4, 10, 0, 12], [0, 5, -4, -2, -14, -5, 0, -4], [-7, -6, -16, -7, 2, 0, 2, 0], [-18, 14, -4, -18, -8, 0, -8, 0], [-7, 0, 0, 7, -4, 8, 4, 0], [28, 3, 0, 0, 0, 3, 2, 0], [8, 9, 6, 0, 0, 9, -8, -6], [0, 10, 6, -4, 12, -10, 0, 6], [-1, 0, 0, 1, -2, 34, 2, 2], [2, 0, 0, -2, 0, 16, 0, -14], [0, -7, 15, 16, 5, 7, 0, 15], [-30, 14, 8, 0, 0, 14, 10, -8], [-3, 0, 0, 3, -1, -29, 1, 12], [-12, 0, 0, 12, -8, -20, 8, 8], [0, 6, 4, 4, -8, -6, 0, 4], [-2, 2, -16, -2, 6, 0, 6, 0], [-14, 0, -4, -14, -8, 0, -8, 0], [0, 3, 16, 5, -6, -3, 0, 16], [5, 0, 0, -5, -4, 10, 4, -3], [4, 0, 0, -4, 8, -10, -8, 8], [7, 18, 1, 7, 0, 0, 0, 0], [0, 4, -2, 2, 3, -4, 0, -2], [0, -10, 7, 16, 2, 10, 0, 7], [-21, 1, 4, 0, 0, 1, -2, -4], [12, 8, -2, 12, 8, 0, 8, 0], [0, 4, 8, -20, 8, -4, 0, 8], [-4, 0, 0, 4, 3, -22, -3, 6], [-8, -8, -12, -8, -2, 0, -2, 0], [9, 18, 12, 9, -2, 0, -2, 0], [0, -2, -6, -12, -4, 2, 0, -6], [-16, 0, 0, 16, 0, 16, 0, -6], [26, 20, 12, 26, -8, 0, -8, 0], [-6, 10, 15, 0, 0, 10, 0, -15], [-26, 4, -8, -26, 0, 0, 0, 0], [0, 8, -8, -1, -8, -8, 0, -8], [-16, 18, 6, -16, 0, 0, 0, 0], [-16, -6, 12, 0, 0, -6, 2, -12], [-8, -5, 9, 0, 0, -5, 9, -9], [0, -10, -6, 30, -9, 10, 0, -6], [-6, 0, 0, 6, 6, -4, -6, 7], [0, -12, 12, -16, -18, 12, 0, 12], [-5, 19, 8, 0, 0, 19, -2, -8], [10, -8, 8, 0, 0, -8, 8, -8], [0, -9, 8, 0, 7, 9, 0, 8], [-9, 0, 0, 9, 2, 15, -2, 4], [0, 7, 3, 4, 15, -7, 0, 3], [-12, -2, 0, -12, 4, 0, 4, 0], [0, 3, -1, -13, -22, -3, 0, -1], [8, 0, 0, -8, 10, 14, -10, 0], [-18, 0, 0, 18, -3, 6, 3, -13], [-2, -2, 8, -2, -4, 0, -4, 0], [6, 20, -3, 0, 0, 20, 14, 3], [24, 14, -2, 24, -2, 0, -2, 0], [-2, 0, 0, 2, 1, -10, -1, 18], [0, -18, 4, -7, -4, 18, 0, 4], [0, -22, -2, -2, 14, 22, 0, -2], [12, 0, 0, -12, 5, 10, -5, -10], [8, -18, 10, 8, 5, 0, 5, 0], [0, -12, -10, 0, 0, -12, -12, 10], [0, 14, -4, 14, 4, -14, 0, -4], [-6, 0, 0, 6, -4, 2, 4, -14], [-18, -2, 19, -18, 1, 0, 1, 0], [1, -10, 6, 0, 0, -10, -4, -6], [7, 0, 0, -7, 0, -16, 0, -6], [-28, 0, 0, 28, -6, 4, 6, 0], [16, -14, 2, 16, 10, 0, 10, 0], [-28, 0, -4, 0, 0, 0, -14, 4], [12, -18, 7, 0, 0, -18, 7, -7], [3, 9, 0, 3, 5, 0, 5, 0], [0, 5, 12, -28, 2, -5, 0, 12], [43, 11, 7, 0, 0, 11, -6, -7], [20, 0, 0, -20, -8, -18, 8, -10], [0, 6, 1, 20, -13, -6, 0, 1], [34, 0, 6, 34, 2, 0, 2, 0], [-22, -22, 8, 0, 0, -22, -12, -8], [-22, 2, -6, 0, 0, 2, -12, 6], [-20, 28, 10, -20, 6, 0, 6, 0], [0, -10, -2, 28, 10, 10, 0, -2], [13, 0, 0, -13, 6, 2, -6, -18], [-7, 0, 0, 7, -7, -18, 7, 3], [-12, 4, 13, 0, 0, 4, 2, -13], [20, 0, -4, 0, 0, 0, 16, 4], [14, -10, -14, 14, 4, 0, 4, 0], [0, 18, 6, 30, -4, -18, 0, 6], [-14, 0, 0, 14, 4, -5, -4, -18], [12, 21, -4, 12, 9, 0, 9, 0], [12, -12, -6, 0, 0, -12, -16, 6], [0, 8, 3, -32, -11, -8, 0, 3], [8, -6, 16, 0, 0, -6, 6, -16], [-5, 0, 0, 5, 10, 18, -10, -4], [8, 0, 0, -8, -8, -4, 8, -2], [0, -12, -9, 10, 8, 12, 0, -9], [20, -20, 12, 20, 12, 0, 12, 0], [12, 32, 8, 0, 0, 32, -10, -8], [4, 26, -8, 4, 4, 0, 4, 0], [0, 0, -4, -4, 22, 0, 0, -4], [1, 0, 0, -1, 2, -18, -2, -2], [0, 29, 4, 30, -8, -29, 0, 4], [18, 2, -1, 0, 0, 2, -8, 1], [0, -7, 4, 26, -10, 7, 0, 4], [-12, 0, 0, 12, 1, 30, -1, 4], [-5, 0, 0, 5, 4, -16, -4, -12], [0, -22, 2, 12, -10, 22, 0, 2], [-24, 2, 0, -24, 2, 0, 2, 0], [14, 3, 8, 0, 0, 3, 8, -8], [0, 10, -14, 6, -4, -10, 0, -14], [18, 22, 10, 18, -4, 0, -4, 0], [16, -36, -2, 0, 0, -36, -6, 2], [-11, 0, 0, 11, 2, 13, -2, 24], [0, 18, 9, 40, -18, -18, 0, 9], [-8, -30, 19, -8, -3, 0, -3, 0], [8, -8, 5, 0, 0, -8, -8, -5], [-3, 17, -16, -3, 12, 0, 12, 0], [0, -20, -4, 26, 4, 20, 0, -4], [0, -16, 0, 6, -12, 16, 0, 0], [18, 6, -8, 18, 16, 0, 16, 0], [16, -18, 2, 16, 10, 0, 10, 0], [0, 13, 10, 9, -2, -13, 0, 10], [-4, 0, 0, 4, -2, 27, 2, -6], [0, 29, 2, -6, 8, -29, 0, 2], [-3, 8, -6, -3, -2, 0, -2, 0], [32, -12, 4, 0, 0, -12, 14, -4], [-28, -14, 3, 0, 0, -14, 6, -3], [-14, -10, 2, -14, 15, 0, 15, 0], [14, 0, 0, -14, -2, 20, 2, -18], [-2, 12, -10, -2, 3, 0, 3, 0], [13, 0, 0, -13, -2, -24, 2, -2], [-22, 0, 0, 22, -6, -6, 6, 6], [0, 21, 9, -12, 17, -21, 0, 9], [10, 0, 0, -10, -2, -8, 2, 20], [-28, 0, 0, 28, -7, -26, 7, 1], [0, 6, 6, -34, -8, -6, 0, 6], [12, -19, -2, 0, 0, -19, -3, 2], [0, -27, 10, -25, -22, 27, 0, 10], [9, 0, 0, -9, 4, 30, -4, -9], [12, -7, 4, 12, -13, 0, -13, 0], [12, 9, 5, 0, 0, 9, -15, -5], [-1, -14, 7, -1, 12, 0, 12, 0], [-14, 0, 0, 14, 8, 2, -8, 14], [0, 34, -1, -12, -2, -34, 0, -1], [4, -24, -4, 4, 6, 0, 6, 0], [-4, 0, 0, 4, -2, 16, 2, -16], [-12, 0, 0, 12, 14, -8, -14, -20], [0, -12, 7, -2, 12, 12, 0, 7], [-7, -8, 0, 0, 0, -8, 24, 0], [0, -5, 1, 21, 10, 5, 0, 1], [-6, -30, 2, -6, -10, 0, -10, 0], [6, -36, 2, 0, 0, -36, -2, -2], [13, 0, 0, -13, -10, 34, 10, -2], [0, -9, -5, 0, 0, -9, -7, 5], [-16, -12, 2, -16, -1, 0, -1, 0], [0, -6, 12, -14, -7, 6, 0, 12], [-18, 0, 0, 18, -2, 8, 2, 11], [16, 0, 0, -16, 2, -22, -2, -4], [0, -16, 7, -24, -7, 16, 0, 7], [14, 24, 4, 0, 0, 24, -16, -4], [0, -16, -4, 28, -20, 16, 0, -4], [-4, 0, 0, 4, 16, 4, -16, -10], [-4, 20, 8, -4, 6, 0, 6, 0], [0, 20, -4, 22, 20, -20, 0, -4], [-20, 0, 0, 20, 4, -28, -4, -14], [32, -2, -17, 0, 0, -2, 2, 17], [16, -20, 5, 0, 0, -20, 1, -5], [-8, 38, 8, -8, -10, 0, -10, 0], [-25, 3, 9, 0, 0, 3, 2, -9], [-2, 0, -10, -2, 2, 0, 2, 0], [-8, 0, 0, 8, 0, -4, 0, 4], [2, 0, 0, -2, 9, -27, -9, 0], [0, 14, -7, -20, 12, -14, 0, -7]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3150_m_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3150_2_m_k();".
// 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_3150_2_m_k(:prec:=8)
    chi := MakeCharacter_3150_m();
    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_3150_2_m_k();".
// 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_3150_2_m_k( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3150_m();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<11,R![4, 0, 12, 0, 1]>,<13,R![1, 40, 800, -72, 2, -24, 32, -8, 1]>,<17,R![2209, -4512, 4608, -2496, 770, -96, 0, 0, 1]>],Snew);
    return Vf;
end function;