
// Make newform 1840.2.e.d in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1840_e();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1840_e_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_1840_2_e_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_1840_2_e_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 := [1, 4, 8, -10, 7, 2, 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], [-740, 354, -205, 398, -124, 80, -46, 25], [-882, -401, 128, -187, -221, 7, 31, -27], [-726, 731, 512, -748, -417, 28, 124, -108], [-280, -1469, 2005, -897, -211, -361, 336, -142], [418, 2650, -3210, 1209, 325, 649, -595, 232], [-537, -3286, 2486, -1742, -403, -674, 551, -269], [560, 2471, -3543, 1794, 422, 722, -672, 284]];
    Rf_basisdens := [1, 467, 467, 467, 467, 467, 467, 467];
    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_1840_e();"
function MakeCharacter_1840_e()
    N := 1840;
    order := 2;
    char_gens := [1151, 1381, 737, 1201];
    v := [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_1840_e_Hecke();"
function MakeCharacter_1840_e_Hecke(Kf)
    N := 1840;
    order := 2;
    char_gens := [1151, 1381, 737, 1201];
    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]];
    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, -1, 1, 0, 1, 0, 0, 0], [-1, 0, 0, 0, 0, 0, -1, 0], [0, 0, 0, 0, -1, 0, -1, 1], [-1, -1, 0, 1, 0, 1, 0, -1], [0, -1, 1, 0, -3, 0, 0, 0], [0, 0, 0, 0, 4, 1, 0, 1], [-1, -1, 0, 1, 0, 0, 0, 0], [0, 0, 0, 0, -1, 0, 0, 0], [-1, 1, -1, -2, 0, -1, 0, 1], [1, 1, 3, 2, 0, 1, 0, -1], [0, -2, 2, 0, 1, -2, -1, -1], [-3, 0, 2, 2, 0, 4, 0, -4], [0, 4, -4, 0, -1, -1, -1, 0], [0, 0, 0, 0, 1, 3, 0, 3], [0, 0, 0, 0, -2, 2, 4, -2], [0, -1, -1, 0, 0, -1, 0, 1], [-3, -3, 0, 3, 0, 2, 0, -2], [0, 0, 0, 0, 1, 0, -1, 1], [7, 2, 4, 2, 0, 2, 0, -2], [0, 2, -2, 0, -5, -3, -4, 1], [5, -1, -2, -1, 0, 0, 0, 0], [0, -6, 6, 0, 0, 1, 4, -3], [1, -3, -2, 1, 0, 0, 0, 0], [0, 2, -2, 0, 9, 0, -3, 3], [-2, 2, 2, 0, 0, 0, 0, 0], [0, 0, 0, 0, 2, 1, -2, 3], [0, 2, -2, 0, 1, -3, -1, -2], [-1, -1, -2, -1, 0, -2, 0, 2], [0, 0, 0, 0, 1, 0, 1, -1], [0, 3, -3, 0, -11, 4, 4, 0], [-1, 2, -4, -6, 0, 2, 0, -2], [0, 4, -4, 0, -8, -2, -4, 2], [-11, 2, 2, 0, 0, 4, 0, -4], [2, 0, 4, 4, 0, 2, 0, -2], [-1, -3, -1, 2, 0, -1, 0, 1], [0, 4, -4, 0, 7, 3, -1, 4], [0, 1, -1, 0, -7, -2, -2, 0], [0, -3, 3, 0, 10, -3, -2, -1], [0, -1, 1, 0, 2, 1, -4, 5], [-7, 1, -3, -4, 0, -5, 0, 5], [-5, 3, 0, -3, 0, 3, 0, -3], [-2, 2, -2, -4, 0, -7, 0, 7], [0, 2, -2, 0, -19, 1, 2, -1], [0, -2, 2, 0, -5, -5, -2, -3], [5, -1, 6, 7, 0, 5, 0, -5], [-2, -2, -2, 0, 0, 0, 0, 0], [0, 4, -4, 0, -4, -4, -6, 2], [0, -2, 2, 0, -3, -4, -7, 3], [-13, 5, 4, -1, 0, 2, 0, -2], [0, -4, 4, 0, -5, -1, 0, -1], [-7, -3, 1, 4, 0, 7, 0, -7], [-10, 0, 0, 0, 0, 5, 0, -5], [-2, -4, -6, -2, 0, 3, 0, -3], [0, 3, -3, 0, -3, 0, 4, -4], [0, -4, 4, 0, 9, -1, -5, 4], [-5, 4, -2, -6, 0, 0, 0, 0], [-10, -6, 0, 6, 0, 8, 0, -8], [0, 3, -3, 0, 11, -4, -4, 0], [1, -1, -4, -3, 0, 1, 0, -1], [0, -4, 4, 0, -2, -1, 0, -1], [0, -2, 2, 0, 4, 0, 4, -4], [0, 5, -5, 0, 4, 3, 8, -5], [5, 5, -1, -6, 0, -1, 0, 1], [0, 2, -2, 0, -22, 2, 2, 0], [0, 1, -1, 0, -14, 1, 0, 1], [7, 8, 6, -2, 0, 0, 0, 0], [0, -2, 2, 0, 9, 3, 1, 2], [0, 2, -2, 0, 4, -2, -6, 4], [3, 4, -4, -8, 0, 4, 0, -4], [0, -4, 4, 0, -5, 3, 0, 3], [-7, 3, -6, -9, 0, 0, 0, 0], [0, -4, 4, 0, 6, -5, -10, 5], [0, -4, 4, 0, -12, -7, 6, -13], [-8, 0, 8, 8, 0, 0, 0, 0], [0, -2, 2, 0, -19, -3, 1, -4], [-8, 0, -4, -4, 0, 0, 0, 0], [0, 5, -5, 0, 11, 2, -10, 12], [4, 0, -6, -6, 0, -5, 0, 5], [7, -6, -2, 4, 0, 0, 0, 0], [1, 5, -6, -11, 0, 0, 0, 0], [15, 3, -2, -5, 0, -1, 0, 1], [-7, 3, -4, -7, 0, -4, 0, 4], [0, -8, 8, 0, 4, 5, 2, 3], [-13, 4, -4, -8, 0, 0, 0, 0], [0, 0, 0, 0, -3, 7, 0, 7], [-18, 5, -5, -10, 0, 1, 0, -1], [0, -6, 6, 0, 0, -1, 0, -1], [-5, -1, -3, -2, 0, -13, 0, 13], [0, -6, 6, 0, -2, 4, 4, 0], [0, 0, 0, 0, 4, 6, 6, 0], [18, -6, -4, 2, 0, -5, 0, 5], [0, -2, 2, 0, 1, -11, -8, -3], [-7, 0, 2, 2, 0, 4, 0, -4], [7, 7, -1, -8, 0, -3, 0, 3], [0, 0, 0, 0, 0, -6, -6, 0], [25, 3, 1, -2, 0, 3, 0, -3], [12, 0, -8, -8, 0, 4, 0, -4], [0, 2, -2, 0, 1, 10, 9, 1], [11, 10, 6, -4, 0, -2, 0, 2], [0, -14, 14, 0, 7, 5, 8, -3], [0, -4, 4, 0, -9, 7, -1, 8], [0, 8, -8, 0, -12, 6, 8, -2], [-18, 4, 0, -4, 0, 11, 0, -11], [-18, 4, -6, -10, 0, -1, 0, 1], [0, 11, -11, 0, 9, 2, -2, 4], [0, 1, -1, 0, 5, -4, -8, 4], [0, 0, 0, 0, 10, 6, 8, -2], [-4, 5, 5, 0, 0, -1, 0, 1], [-3, 3, 13, 10, 0, -5, 0, 5], [0, -5, 5, 0, 0, 1, 2, -1], [0, -2, 2, 0, 6, 7, -6, 13], [0, 0, 0, 0, 9, 7, 7, 0], [-10, -14, -4, 10, 0, 0, 0, 0], [19, 3, -2, -5, 0, -14, 0, 14], [3, 7, -2, -9, 0, -8, 0, 8], [0, 10, -10, 0, 2, -5, -16, 11], [0, 9, -9, 0, -15, 0, 6, -6], [0, 3, -3, 0, -1, -8, -12, 4], [10, 4, -10, -14, 0, 1, 0, -1], [12, -2, 4, 6, 0, -10, 0, 10], [0, 4, -4, 0, -21, -11, -16, 5], [0, -6, 6, 0, 20, 4, -2, 6], [0, -11, 11, 0, -7, 0, 2, -2], [-2, -2, -4, -2, 0, -12, 0, 12], [-21, -1, -2, -1, 0, -5, 0, 5], [-11, 5, -6, -11, 0, -3, 0, 3], [6, -4, 10, 14, 0, 10, 0, -10], [0, 12, -12, 0, -4, -5, -10, 5], [0, 6, -6, 0, -12, -9, -4, -5], [-1, 10, 12, 2, 0, 0, 0, 0], [0, -4, 4, 0, 4, -1, 4, -5], [9, -1, -4, -3, 0, -10, 0, 10], [0, 2, -2, 0, -7, -8, -9, 1], [1, 0, -2, -2, 0, 4, 0, -4], [-6, -10, -10, 0, 0, 3, 0, -3], [0, 10, -10, 0, -5, 3, -1, 4], [0, -2, 2, 0, 14, 6, -2, 8], [0, 0, 0, 0, 7, -3, 1, -4], [6, 0, 8, 8, 0, -12, 0, 12], [-13, -11, 5, 16, 0, 5, 0, -5], [14, -6, 0, 6, 0, -4, 0, 4], [0, 9, -9, 0, -7, 2, 12, -10], [0, 4, -4, 0, -32, 2, -4, 6], [20, -8, -2, 6, 0, -4, 0, 4], [-13, 9, -4, -13, 0, 11, 0, -11], [0, -1, 1, 0, -18, -1, -6, 5], [0, -16, 16, 0, 19, -5, -4, -1], [-17, -7, -11, -4, 0, 3, 0, -3], [0, 11, -11, 0, 25, 2, -8, 10], [0, -13, 13, 0, 4, 1, 2, -1], [8, 8, 12, 4, 0, 1, 0, -1], [0, 1, -1, 0, -2, -9, -12, 3], [0, -4, 4, 0, -5, 3, -2, 5], [0, 6, -6, 0, 29, -5, -1, -4], [-29, 5, 6, 1, 0, 12, 0, -12], [27, 5, 4, -1, 0, -1, 0, 1], [-9, -4, 2, 6, 0, 6, 0, -6], [0, 4, -4, 0, 11, -6, 5, -11], [-8, -4, 4, 8, 0, 19, 0, -19], [0, -9, 9, 0, 9, 2, 2, 0], [0, -4, 4, 0, 12, 0, -2, 2], [0, -13, 13, 0, -1, -10, 0, -10], [-20, -10, -12, -2, 0, -6, 0, 6], [0, 12, -12, 0, 11, -2, 9, -11], [0, 6, -6, 0, 8, 8, -2, 10], [-4, 0, -6, -6, 0, -6, 0, 6], [0, -5, 5, 0, 8, 7, 4, 3], [35, 1, 2, 1, 0, -8, 0, 8], [0, 8, -8, 0, 17, -3, -2, -1], [-9, -13, -12, 1, 0, 7, 0, -7], [15, 6, 2, -4, 0, 4, 0, -4], [0, -4, -4, 0, 0, 3, 0, -3], [0, -4, 4, 0, -4, 8, -8, 16], [5, -4, -8, -4, 0, -2, 0, 2], [3, -1, -6, -5, 0, -1, 0, 1], [13, 4, 16, 12, 0, 10, 0, -10], [-15, -1, -3, -2, 0, -3, 0, 3], [0, 10, -10, 0, -7, 9, 15, -6], [-16, 4, 4, 0, 0, 17, 0, -17], [0, 13, -13, 0, 3, -2, -10, 8], [-3, -1, -4, -3, 0, 2, 0, -2], [0, 18, -18, 0, -7, -3, -12, 9], [0, -5, 5, 0, 36, 3, 4, -1], [0, 8, -8, 0, 6, 7, 16, -9], [12, 12, 16, 4, 0, 3, 0, -3], [0, 12, -12, 0, -26, 8, 12, -4], [0, -8, 8, 0, 13, 15, 11, 4], [-3, 10, 8, -2, 0, 12, 0, -12], [32, -2, 8, 10, 0, -4, 0, 4], [0, -5, 5, 0, -21, 2, 6, -4], [0, -5, 5, 0, 16, 1, -6, 7], [-18, -4, -4, 0, 0, 1, 0, -1], [-18, 0, 6, 6, 0, 6, 0, -6], [0, 8, -8, 0, -26, 3, 8, -5], [0, 6, -6, 0, 29, 5, -7, 12], [-7, -17, -2, 15, 0, 1, 0, -1], [0, 10, -10, 0, 26, 10, -10, 20], [0, -6, 6, 0, 8, -4, -8, 4], [0, -2, 2, 0, -27, -7, -8, 1], [-2, -2, 16, 18, 0, 0, 0, 0], [4, -1, -5, -4, 0, 1, 0, -1], [0, -1, 1, 0, -9, 4, 8, -4], [-9, 3, -4, -7, 0, 21, 0, -21], [12, 4, 16, 12, 0, 4, 0, -4], [0, 14, -14, 0, -21, -3, -4, 1], [-1, 5, 6, 1, 0, -11, 0, 11], [0, 10, -10, 0, -5, 5, 8, -3], [13, 8, -12, -20, 0, -2, 0, 2], [37, -3, 11, 14, 0, 1, 0, -1], [0, -9, 9, 0, 19, 10, 10, 0], [19, 10, 6, -4, 0, -14, 0, 14], [0, 4, -4, 0, -3, -7, -15, 8], [0, -6, 6, 0, -32, 0, 6, -6], [26, 1, 9, 8, 0, -9, 0, 9], [1, 13, -10, -23, 0, -10, 0, 10], [0, 6, -6, 0, 5, 3, -4, 7], [-19, 9, -5, -14, 0, -7, 0, 7], [0, -14, 14, 0, 8, -1, -6, 5], [0, -13, 13, 0, 25, -6, -8, 2], [43, -1, -5, -4, 0, -15, 0, 15], [5, 3, -2, -5, 0, 11, 0, -11], [-7, 6, -10, -16, 0, 12, 0, -12], [0, -4, 4, 0, -1, -6, -9, 3], [0, 0, 0, 0, 23, -7, 6, -13], [-13, -9, -13, -4, 0, -3, 0, 3], [0, -2, 2, 0, -9, 18, 1, 17], [-1, -1, 7, 8, 0, -1, 0, 1], [0, 12, -12, 0, -2, 14, 0, 14], [-40, -3, -5, -2, 0, -3, 0, 3], [0, -14, 14, 0, 40, -14, -8, -6], [-1, 3, -2, -5, 0, -9, 0, 9], [12, 0, 6, 6, 0, 15, 0, -15], [-6, 1, -15, -16, 0, 7, 0, -7], [0, 2, -2, 0, 26, -4, 6, -10], [0, 16, -16, 0, -1, 8, -1, 9], [16, 4, 4, 0, 0, -14, 0, 14], [0, 8, -8, 0, 41, 4, -1, 5], [-7, 23, 19, -4, 0, 5, 0, -5], [-25, 8, -6, -14, 0, 12, 0, -12], [0, -2, 2, 0, 30, 2, 6, -4], [1, -7, -7, 0, 0, -7, 0, 7], [0, 4, -4, 0, 25, 7, -10, 17], [11, -6, 12, 18, 0, 18, 0, -18], [0, 7, -7, 0, 4, -5, 6, -11], [19, 11, 9, -2, 0, 11, 0, -11], [0, 12, -12, 0, -21, 3, -10, 13], [12, 18, 20, 2, 0, 8, 0, -8], [-24, -12, 2, 14, 0, -2, 0, 2], [0, -4, 4, 0, 22, -11, 4, -15], [0, -6, 6, 0, 0, 9, -12, 21], [32, 10, 4, -6, 0, -3, 0, 3], [0, 2, -2, 0, 18, 10, -4, 14], [19, -5, -2, 3, 0, 7, 0, -7], [0, -17, 17, 0, 8, -5, 10, -15], [-13, 8, -8, -16, 0, -8, 0, 8], [-29, -5, 8, 13, 0, 13, 0, -13], [0, -10, 10, 0, -4, 5, 22, -17], [0, 7, -7, 0, 29, 2, -12, 14], [0, 1, -1, 0, 5, -6, 0, -6], [0, -16, 16, 0, 26, 6, 14, -8], [0, -4, 4, 0, 15, 19, 15, 4], [16, 7, 15, 8, 0, 9, 0, -9], [0, 8, -8, 0, -19, 0, 5, -5], [0, -18, 18, 0, 10, 0, 12, -12], [3, -7, -2, 5, 0, 24, 0, -24], [29, -11, -12, -1, 0, -19, 0, 19], [-23, -9, -24, -15, 0, -6, 0, 6], [0, -12, 12, 0, -19, -17, 3, -20], [0, -5, 5, 0, -15, 6, 12, -6], [-5, -4, -10, -6, 0, 24, 0, -24], [0, -8, 8, 0, 24, -14, -26, 12], [0, -4, 4, 0, 15, 7, 11, -4], [11, -5, 2, 7, 0, -19, 0, 19], [0, -2, 2, 0, 25, -7, -8, 1], [0, -6, 6, 0, -13, -11, 6, -17], [0, -4, 4, 0, -8, 14, 32, -18], [4, 20, 10, -10, 0, 2, 0, -2], [8, 8, 2, -6, 0, -15, 0, 15], [-30, -2, 8, 10, 0, 14, 0, -14], [0, 18, -18, 0, -29, -17, -26, 9], [4, 6, 16, 10, 0, -5, 0, 5], [0, -18, 18, 0, -11, 8, 19, -11], [8, 12, -18, -30, 0, -14, 0, 14], [0, 4, -4, 0, 8, -2, -8, 6], [20, 2, -10, -12, 0, 12, 0, -12], [0, 26, -26, 0, -26, -12, -16, 4], [0, -8, 8, 0, -37, 5, 5, 0], [10, 10, -10, -20, 0, -20, 0, 20], [-16, 0, 6, 6, 0, 28, 0, -28], [-20, -14, 12, 26, 0, 12, 0, -12], [0, -12, 12, 0, 6, -7, -22, 15], [0, -2, 2, 0, -8, -14, -20, 6], [15, -3, -6, -3, 0, -26, 0, 26], [0, -14, 14, 0, 29, 5, -2, 7], [-38, -6, -6, 0, 0, -4, 0, 4], [-38, 2, 0, -2, 0, 21, 0, -21], [0, 21, -21, 0, -21, 4, 12, -8], [-11, -10, -8, 2, 0, 16, 0, -16], [0, 8, -8, 0, 1, 5, -14, 19], [0, 32, -32, 0, -15, -8, -11, 3], [0, -8, 8, 0, -49, -1, 21, -22], [-18, 16, 18, 2, 0, 11, 0, -11]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1840_e_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1840_2_e_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_1840_2_e_d(:prec:=8)
    chi := MakeCharacter_1840_e();
    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_1840_2_e_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_1840_2_e_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1840_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![25, 0, 96, 0, 70, 0, 16, 0, 1]>,<7,R![16, 0, 176, 0, 152, 0, 28, 0, 1]>],Snew);
    return Vf;
end function;