
// Make newform 252.2.i.b in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_252_i();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_252_i_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_252_2_i_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_252_2_i_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 := [2187, 0, -1215, -243, 189, 270, 0, -117, 0, 30, 7, -3, -5, 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, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [46656, -63180, -23814, 9288, 8802, 2025, -3393, -1143, 1383, 290, -312, -157, -72, 35], [-8505, 5184, -2295, -1701, 297, -72, 225, 78, -127, 37, 5, -23, 2, -8], [-40581, -24381, 1701, 2916, 8370, -1152, -2151, 498, 546, 289, -135, -191, -15, 28], [-30618, -48438, 11124, 14985, -864, -2574, -2529, 846, 955, -301, -266, -70, 7, 26], [-293058, 22113, 61722, 35424, 2538, -26190, 1638, 8622, 165, -761, -1227, -62, 441, -56], [-237654, 87966, 116154, 7965, -20142, -17037, 7065, 7902, -2364, -2237, -354, 406, 423, -83], [454896, -59292, -89424, -51003, -1269, 42714, -2736, -10584, 1209, 2002, 1491, 1, -675, 97], [239841, -11178, -98901, -32022, -2295, 19845, 1413, -8694, -417, 1703, 804, -40, -360, -73], [-116397, 11583, 41445, 21789, -999, -11700, 1044, 3780, -247, -760, -463, 68, 203, 5], [267543, -113724, -46413, -4239, 15120, 20358, -13221, -6129, 3687, 1133, 93, -385, -324, 239], [-324405, -162081, 55485, 101142, 33021, -31266, -16785, 9288, 5184, -763, -2352, -769, 471, 179], [97686, -335097, 243, 89964, 40797, 5319, -27936, 0, 10167, -145, -1392, -1096, -81, 500]];
    Rf_basisdens := [1, 1, 43011, 4779, 14337, 14337, 43011, 43011, 43011, 43011, 14337, 43011, 43011, 43011];
    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_252_i();"
function MakeCharacter_252_i()
    N := 252;
    order := 3;
    char_gens := [127, 29, 73];
    v := [3, 2, 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_252_i_Hecke();"
function MakeCharacter_252_i_Hecke(Kf)
    N := 252;
    order := 3;
    char_gens := [127, 29, 73];
    char_values := [[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 1, 0, 0, 0, 0, 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, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -1, 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, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, -1, 0, 0], [0, 0, 0, 1, 1, 1, 0, -1, -1, -1, 0, 0, -1, 0], [-1, 1, 1, 0, 1, 0, -1, 1, 0, 1, 0, 0, 0, 1], [1, 0, -1, 0, 0, 1, 0, 0, 1, 0, 0, -1, 0, 0], [0, 0, 2, 1, -2, 0, 1, -1, 0, -1, 0, -1, 0, 0], [0, -1, 0, 0, -1, -1, 1, 0, 0, 0, 0, 1, -1, 1], [0, 0, 0, -1, 2, -1, -1, 3, -1, 2, 0, 1, 1, 1], [3, 3, -2, -3, 0, 0, 2, 0, -1, 1, 0, 1, -1, 0], [-3, 3, 4, -3, 0, 0, 2, 0, 0, 1, -1, 0, 0, 0], [1, -1, 1, 0, -1, 0, 0, -1, 0, -2, -1, 0, 0, -1], [1, 0, 0, 0, 0, 0, 2, -1, -1, 0, -1, 0, -2, 1], [0, 2, -1, -3, -1, -1, -1, 0, 0, 0, -2, -2, 2, 1], [2, 0, -1, 1, 1, -1, -1, 3, 1, 2, 1, 2, 2, -1], [0, 2, 1, -1, 1, -1, -1, 1, -1, 2, 1, 0, 0, 1], [-1, 2, 1, 0, -1, 0, 0, -1, 1, 0, 0, -1, -1, -1], [-1, 2, 1, -1, 1, -1, -2, 2, 0, 2, 1, 0, 1, 0], [1, -1, 2, -1, 1, 0, -1, 0, 0, 1, -1, 1, 0, -1], [3, 0, -1, 2, -1, 0, 3, -1, 1, 0, -1, 1, -2, -1], [2, -3, -7, -2, 1, -1, 2, 0, 0, 2, 1, 3, -1, -1], [-2, -2, 0, 2, 0, -1, -2, 0, 0, 0, 0, 2, 2, 0], [0, 2, -2, -3, 2, 0, -2, 1, 0, 0, -1, -1, 2, 0], [-4, -1, 3, 1, 0, -1, -1, 0, 0, 0, 3, 1, -2, 0], [1, 0, 0, 0, -3, 0, 0, -2, 0, -5, -2, -2, 1, -1], [4, -2, -4, 1, -1, 2, -1, 1, 0, 0, 2, 0, -1, 0], [-2, 1, 2, 5, 0, 2, 0, 0, 0, -4, 2, -1, -1, 0], [5, 2, -3, -2, 0, -2, 2, 0, -1, 0, -3, -1, 1, 0], [-3, 0, 0, -1, 2, -4, -1, 3, -1, 5, 0, 1, 1, 1], [2, 0, -7, -4, -1, -1, 2, -1, 0, 2, 0, -1, 2, -1], [-5, -2, 7, 2, 0, 0, 0, 0, -3, -2, -1, -1, -1, 0], [-2, -5, 1, 4, -1, -4, -3, 1, 3, -1, -2, -3, 4, 0], [-4, 1, 2, 9, 1, 3, 1, 1, 0, 1, 5, 0, -3, 1], [-3, -2, 1, 2, 0, 3, -2, 0, 3, 0, 2, -1, 0, 0], [5, -6, -3, 3, -3, 0, 0, -3, 3, -3, 0, 0, 3, -3], [0, 4, 0, -6, -2, -6, 2, 2, 2, 4, -1, 2, 3, 0], [-2, 4, 0, -2, 2, 0, 1, -2, 0, 1, 1, 3, -1, 0], [8, -2, 1, -3, 1, -2, -4, 4, 2, 1, -3, -2, 3, -2], [2, -1, -11, -1, 1, 1, -2, -1, 0, -3, -2, 0, 1, -3], [2, -4, 1, -3, -1, 0, -3, 0, 2, -3, -4, -4, 2, -2], [-10, 6, 2, -1, 2, 0, -1, 2, -2, 3, 3, 1, -1, 2], [-1, 6, 2, 0, 0, -2, 1, -1, -1, 4, 2, 0, -3, 1], [1, -4, -2, 2, -2, 2, 2, -3, 1, -4, -1, 0, 0, -1], [0, 1, -4, -3, 1, -1, -2, 1, 0, -2, -2, 0, 1, 1], [-5, -1, 2, 2, 1, -1, -2, -1, -1, 0, 0, 4, 2, 0], [2, 2, 1, -6, -4, -2, -1, -2, 0, 1, -5, -4, 4, 0], [4, 7, -1, -8, -1, 0, 5, 1, -2, 3, -1, 2, -1, 0], [-4, 1, 4, 7, 5, 3, -3, 3, 0, -1, 3, 2, -3, 1], [-3, 0, 1, 1, 1, 4, -1, -1, -1, 0, 4, 3, -3, 0], [-1, -1, -2, 0, -1, 3, -2, 1, 2, 2, 3, 3, 1, 0], [3, 7, 0, -6, 1, 0, 5, -1, -1, 1, -1, -1, -3, 0], [9, 0, -2, 1, 4, -6, 2, 5, -3, 9, 1, 5, 0, 3], [-1, 2, -6, -5, 3, -2, -1, 3, 0, 3, 0, 1, 1, 3], [-12, -3, 11, 4, 1, 0, -2, -1, 0, -2, -3, -1, 3, 0], [-2, 3, -8, 1, 1, 1, 1, 1, 0, -1, 3, -2, 1, 1], [1, 2, 1, 0, 2, 5, 2, -2, -2, -2, -3, -2, -1, 0], [0, -4, -2, 5, 1, 5, -3, -1, -2, -2, 3, 2, -2, 0], [4, -3, 2, -5, -2, -2, 0, -2, 0, -4, -4, 1, 1, -2], [-3, -4, 0, -2, 0, -1, 3, -2, -2, -1, -5, -2, -3, 2], [-11, 4, 13, -6, -2, -6, 3, 2, 1, 3, 1, 0, -1, 0], [5, -4, 1, -4, 1, -1, -6, 4, 2, -1, -3, -3, 5, -2], [2, 4, 2, 1, -4, 4, 4, -6, 2, -5, -2, -3, -6, -2], [2, 0, 2, -3, 0, 1, -5, 3, 3, -2, -2, -3, 3, -3], [-9, 10, 3, 0, 1, 5, -1, 0, -2, -1, 6, 1, -2, 2], [-1, 2, 0, 0, 2, 1, -5, 6, 2, 1, 3, 2, 5, -2], [5, -2, -3, 3, 1, 4, 0, -1, -2, -3, 1, -3, -4, 0], [10, 4, 4, -5, 2, -1, -3, 1, -3, 2, 0, -3, -1, 3], [0, -1, 3, 2, -2, 0, 2, -1, 0, -6, 1, 0, -1, 0], [-8, 1, 7, 0, 1, 7, 0, -1, 5, 0, 1, -4, -1, 0], [-1, -4, 1, 5, 1, -1, -2, -1, 3, -3, 1, -6, -2, 0], [9, 2, -7, 0, 2, 7, 2, -2, -3, -2, -2, -1, -2, 0], [0, 5, 2, -10, 7, -1, -3, 4, 0, 4, 0, -1, 5, 1], [7, -12, -7, 7, -5, 2, 5, -7, 3, -7, -1, 2, 2, -3], [1, -2, -4, 1, -4, -1, -1, -2, 0, -5, -4, 0, -1, 0], [-7, -7, 2, 6, -1, -2, -6, 1, 5, 0, -1, 0, 7, 0], [3, -4, -7, 5, 1, 0, -4, -1, -2, -1, 5, 5, -2, 0], [1, -5, 6, 7, -1, 2, 1, -2, 0, -5, 1, 3, -4, -3], [-6, 6, 2, 0, 0, 4, 3, -2, -2, -2, 1, 0, -5, 2], [-13, -4, 11, 1, -3, -5, -3, 3, -3, 2, 1, 7, 4, 0], [2, 0, 7, 2, -4, 2, -2, -4, 0, -6, -4, -4, 2, -4], [-2, 1, -11, -1, 3, -1, 0, 3, 0, 1, 2, 2, -1, 3], [3, -6, -5, 4, 1, -3, 7, -2, -4, 3, 0, 5, -2, 4], [9, 2, 2, -2, 0, 3, -2, -2, -2, -3, 2, -2, 0, 2], [-5, -10, -4, 3, -4, -1, -2, -4, 4, -4, 0, -1, 6, -4], [-13, -2, -2, 0, 4, -5, 2, 4, -4, 7, 0, 4, 0, 4], [3, -10, -6, 4, 0, -4, 3, 1, 1, 2, -2, 4, 3, -1], [3, -2, -15, 0, -2, 1, -4, -3, 0, -2, -6, -1, 1, -4], [-1, 7, 2, -5, 2, 3, 4, -2, -1, 1, 1, 1, -4, 0], [-10, 8, 12, -5, 3, 6, 5, -3, 1, 0, 3, -3, -8, 0], [6, 6, -6, -6, 0, -8, 3, 0, 2, 3, -3, 1, 3, 0], [0, 4, 6, -1, 6, 4, -3, 1, 0, 5, 2, -3, 4, -4], [-5, -5, 2, 7, 2, 1, -4, -2, -1, -3, 5, 1, -4, 0], [-5, -1, -2, 7, 3, 0, 0, 4, 0, 10, 4, 5, -6, 5], [15, -4, -1, 2, -5, 4, 5, -5, 5, -8, -7, -3, -4, -5], [3, -6, -13, 6, 0, 3, -2, -3, 0, -4, -2, 3, -3, -6], [4, -1, -8, -1, -8, 0, 2, -6, 0, 0, -4, -5, 3, -4], [-5, 1, 4, -3, -2, -5, 0, 2, 4, 3, 2, 0, 1, 0], [0, 2, 4, -5, -3, -3, 3, 3, 1, 2, -8, -3, 7, 0], [0, 3, -8, -8, 4, -2, -3, 3, 0, 14, -2, 0, 3, 2], [4, -3, -10, -4, -4, -2, 6, -3, 0, -1, 1, 0, 1, -2], [-4, 4, 1, -1, 3, -9, -4, 5, -1, 12, 4, 2, 3, 1], [0, -2, -3, 3, 1, -7, -1, 3, 1, 8, 3, 4, 4, -1], [1, -8, -3, 2, -3, 7, 3, -4, 2, -11, -4, -1, 0, -2], [1, -5, -7, 9, 1, 4, -3, -2, 0, -6, -1, 3, -4, -5], [2, 2, 1, 0, 2, 4, -5, -2, 0, -9, -3, -4, 4, -6], [-7, -4, 7, 3, -1, -8, -2, 1, -3, -1, 0, 2, 1, 0], [14, -2, 0, 0, -2, -4, -3, -2, 2, 2, 1, -2, 3, -2], [0, -1, -7, -5, 0, -4, 0, 2, 0, 1, -2, 3, -1, 4], [7, -10, 3, -2, -6, -3, 7, -5, 0, 1, -1, 5, -3, -4], [1, -2, -4, 2, -3, 0, -3, -2, 0, -2, -5, 0, -1, -1], [15, -3, -16, 5, 2, 8, -2, -2, -3, -3, -4, 1, 3, 0], [-1, -4, 7, 1, -3, -3, 1, 3, 2, -2, -6, -10, 3, 0], [7, 8, 2, -2, 6, -3, -3, 10, -2, 9, 3, 4, 1, 2], [-8, -2, 6, 3, 1, 1, -2, -1, 0, -1, 5, 1, -4, 0], [-8, -6, 2, 6, 0, 0, -6, 0, 3, 0, 0, 3, 6, 0], [2, -6, 9, -2, -2, -4, 7, 0, 0, -1, 3, 6, -4, 2], [3, -5, -12, -5, -4, -5, 8, -1, 0, 10, 2, 4, -2, 2], [0, -3, 2, 10, -5, 3, 4, -4, 0, -1, 3, -1, -3, -3], [-9, 12, 3, 1, 1, -2, 5, 1, -1, 7, 1, 2, -8, 1], [-4, 0, 5, 13, -8, 2, 6, -3, 0, -3, 5, -3, -4, 2], [7, 0, 2, -4, 2, 6, -6, 3, -1, -6, 1, -2, 4, 1], [-1, -2, -21, 1, -4, -3, 9, 0, 0, 10, 6, 2, -3, 4], [-13, -4, -3, 3, -1, -2, 1, -3, -1, 1, 3, 2, 2, 1], [2, -14, -7, 4, -1, -11, -5, 3, 5, 7, 1, 3, 12, -5], [-1, 0, 4, -3, -6, 2, -4, -5, 7, -7, -5, -9, 0, -7], [-4, 2, 4, -6, -4, -6, 1, 4, 2, 5, 3, 3, 1, 0], [-1, -3, -13, 2, 2, -1, 9, 2, 0, 9, 10, 5, -4, 2], [0, -8, 0, 13, -7, 1, 3, -4, 0, -1, 0, 4, -8, -1], [3, -2, 3, -5, -10, -5, 3, -4, 0, 2, -6, -2, 1, 2], [3, -6, -5, 9, 3, 1, -4, -3, -2, -5, 2, -1, -3, 0], [-3, 7, 11, 6, -5, 4, -2, -3, 0, -4, -1, -10, 4, -1], [-4, -2, -3, 1, 5, 3, -1, 9, -1, 0, 1, 6, 4, 1], [-3, 5, -18, -4, 4, -1, 4, 4, 0, 9, 7, -1, 2, 4], [-4, 7, 5, -5, 2, 3, 4, -2, 5, 1, 4, -5, -7, 0], [-6, 7, -4, 8, 9, 7, -9, 4, 0, -2, 2, -3, 1, -1], [8, -8, -3, 1, -1, -3, -4, 3, 5, 0, -2, 0, 7, -5], [-19, -7, 16, 12, 5, 12, -5, -5, -1, -7, 3, -3, -5, 0], [-1, -4, -9, 15, 2, 7, -4, -2, 0, -3, 1, 2, -5, -6], [-4, 2, -4, 6, 2, -1, 2, 3, -1, 5, 7, 8, 2, 1], [7, 0, 2, -3, 0, -8, 0, -1, -1, 7, -3, -3, -2, 1], [10, -2, 2, -4, 0, 2, 1, 0, 0, -4, -7, -4, -3, 0], [11, 10, 6, -3, -2, 5, 0, -2, 2, -4, -2, -5, -6, -2], [-8, 0, 15, 15, 6, 4, -6, 5, 0, 3, 3, 5, -8, 4], [2, 2, -13, -5, -6, -2, 3, -3, 0, 0, -2, -5, 4, 0], [3, -3, 5, 1, -5, 0, 5, -4, 0, 0, 1, -1, 0, -3], [-10, -7, 17, 4, -3, -5, 0, 3, 0, -4, -8, -11, 4, 0], [3, 4, -14, -16, 3, -4, -4, 2, 0, 1, -6, -2, 7, 1], [13, 1, -14, 2, 3, -1, 0, -3, -7, -2, -4, 6, 2, 0], [3, -4, -9, 2, -14, -3, 8, -7, 0, -6, -2, -3, -1, 0], [3, 10, 5, -4, 3, 2, -6, 1, -5, 2, 8, -1, 1, 5], [-9, 6, 1, 4, -5, 9, 7, -8, 2, -9, 0, -1, -8, -2], [-4, 3, 7, 2, -1, -1, 0, 2, 0, 8, 1, -1, -1, 5], [8, -1, -7, 0, -1, -10, 0, 1, -4, 0, -3, 3, 3, 0], [-6, -2, 10, 17, 0, 4, -5, 1, 0, 1, 0, 3, -8, 2], [0, -1, 3, -1, -2, 4, 1, 2, 4, 0, -2, -7, 1, 0], [18, -10, -4, 5, -8, -2, 11, -14, 2, -5, -7, -3, -7, -2], [-3, 6, 2, 0, 0, -8, -3, 1, 1, 10, 4, 0, 1, -1], [14, 2, 1, -3, 5, -6, -2, 10, 0, 9, -3, 2, 1, 0], [-12, -4, -1, -2, 3, -3, 8, -1, -7, 3, -6, 1, -7, 7], [-5, -14, 1, -7, -3, 6, -6, -3, 3, -15, -10, -10, 5, -3], [2, 5, -1, -8, -3, 0, 3, 3, -2, 5, -1, 6, 3, 0], [-17, -2, 13, 9, 7, 11, -3, -7, -3, -6, 1, 1, -4, 0], [-10, 18, 2, 5, 2, 12, 5, -4, -8, -3, 15, 7, -7, 8], [0, -6, -6, 5, -1, -2, -6, 1, 4, 1, 4, 3, 3, 0], [-1, 1, -20, -11, 11, -4, 0, 8, 0, 12, 4, 7, 0, 5], [12, 8, -8, -7, 1, 1, 6, -1, -6, 1, 3, 3, -8, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_252_i_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_252_2_i_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_252_2_i_b(:prec:=14)
    chi := MakeCharacter_252_i();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(997) - 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_252_2_i_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_252_2_i_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_252_i();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![6561, -8748, 18225, -12150, 21465, -14661, 13833, -4923, 2670, -357, 295, -16, 24, 2, 1]>],Snew);
    return Vf;
end function;