
// Make newform 2304.3.e.m in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2304_e();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_2304_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_2304_3_e_m();".
// 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_2304_3_e_m();".
// 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 := [256, 0, 0, 0, 49, 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, 65, 0, 0, 0, 1, 0], [0, -464, 0, -181, 0, -16, 0, -5], [0, 0, -33, 0, 0, 0, -1, 0], [98, 0, 0, 0, 4, 0, 0, 0], [0, 464, 0, -181, 0, 16, 0, -5], [0, -1616, 0, -701, 0, -16, 0, -13], [0, 1616, 0, -701, 0, 16, 0, -13]];
    Rf_basisdens := [1, 36, 576, 4, 9, 288, 576, 144];
    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_2304_e();"
function MakeCharacter_2304_e()
    N := 2304;
    order := 2;
    char_gens := [1279, 2053, 1793];
    v := [2, 2, 1];
    // 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_2304_e_Hecke();"
function MakeCharacter_2304_e_Hecke(Kf)
    N := 2304;
    order := 2;
    char_gens := [1279, 2053, 1793];
    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]];
    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 := 3;
    raw_aps := [[0, 0, 0, 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, 1, 0, 0], [0, 0, 0, -1, 0, 0, 0, 0], [0, 0, 0, 0, -1, 0, 0, 0], [0, 0, -5, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 1], [0, 5, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 5, 0], [0, 0, 0, 0, 0, -15, 0, 0], [0, 0, 0, 0, -6, 0, 0, 0], [0, 0, -19, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, -11, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 5, 0], [0, 0, 0, 4, 0, 0, 0, 0], [0, 0, 0, 0, -10, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -5], [0, 25, 0, 0, 0, 0, 0, 0], [-40, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -45, 0, 0], [0, 0, 0, -5, 0, 0, 0, 0], [0, 0, -81, 0, 0, 0, 0, 0], [-40, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -5, 0], [0, 0, 0, 0, 0, 59, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 25, 0, 0, 0], [0, 0, -145, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -51, 0, 0], [0, 0, 0, 6, 0, 0, 0, 0], [0, 0, -85, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 5], [0, 0, 0, 0, 0, 0, -25, 0], [0, 0, 0, 0, 0, 45, 0, 0], [0, 0, 0, 0, -6, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 10], [0, 40, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -55, 0], [0, 0, 0, 6, 0, 0, 0, 0], [0, 0, 0, 0, 25, 0, 0, 0], [0, 40, 0, 0, 0, 0, 0, 0], [-150, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 27, 0], [0, 0, 0, 0, 0, -25, 0, 0], [0, 0, 0, 0, 0, 0, 0, -15], [0, 0, 0, 0, 0, 1, 0, 0], [0, 0, 0, -15, 0, 0, 0, 0], [0, 0, 0, 0, -25, 0, 0, 0], [0, 0, -135, 0, 0, 0, 0, 0], [0, 10, 0, 0, 0, 0, 0, 0], [-240, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 21, 0, 0, 0, 0], [0, 0, -15, 0, 0, 0, 0, 0], [0, -90, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 75, 0], [0, 0, 0, 0, 0, 85, 0, 0], [0, 0, 0, 0, -39, 0, 0, 0], [0, 0, -161, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 20], [0, 0, 0, 0, 0, 0, 37, 0], [0, 0, 0, 0, 0, 0, 0, -5], [0, 10, 0, 0, 0, 0, 0, 0], [-250, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 3, 0], [0, 0, 0, 0, 0, 0, 0, 1], [-520, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -15, 0, 0, 0, 0], [0, 0, 0, 0, -10, 0, 0, 0], [0, 0, 255, 0, 0, 0, 0, 0], [0, -85, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 129, 0, 0], [0, 0, 0, 0, 6, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -5], [0, -46, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -65, 0], [0, 0, 0, 0, -66, 0, 0, 0], [0, 0, -189, 0, 0, 0, 0, 0], [-232, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -1, 0, 0, 0, 0], [0, 0, 0, 0, -15, 0, 0, 0], [0, -35, 0, 0, 0, 0, 0, 0], [-190, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -235, 0, 0], [0, 0, 0, -5, 0, 0, 0, 0], [0, 0, 309, 0, 0, 0, 0, 0], [600, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 15, 0], [0, 0, 0, 0, 0, 19, 0, 0], [0, 0, 0, 5, 0, 0, 0, 0], [0, 215, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 11, 0, 0], [0, 0, 0, 26, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 4], [0, 139, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -115, 0], [0, 0, 651, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 25], [0, 0, 0, 0, 105, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -40], [0, 0, 0, 0, 0, 0, 15, 0], [0, 0, 0, -65, 0, 0, 0, 0], [0, 0, 51, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -34], [890, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 60, 0, 0, 0, 0], [0, 0, -335, 0, 0, 0, 0, 0], [0, 185, 0, 0, 0, 0, 0, 0], [310, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -131, 0, 0], [0, 0, 0, 0, -34, 0, 0, 0], [0, 0, 415, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 30], [0, 0, 0, 0, 0, 135, 0, 0], [0, 0, 851, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 10], [0, 211, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 177, 0], [0, 0, 0, -16, 0, 0, 0, 0], [0, 0, 0, 0, -110, 0, 0, 0], [1070, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -7, 0], [0, 0, 0, -65, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 14], [0, 0, 0, 0, 0, 0, 85, 0], [0, 0, 0, 0, 15, 0, 0, 0], [0, -75, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 459, 0, 0], [0, 0, 0, 0, -9, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 6], [0, 316, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -235, 0, 0], [0, 0, 0, 0, -89, 0, 0, 0], [0, 0, -51, 0, 0, 0, 0, 0], [-10, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -43, 0], [0, 0, 0, 0, 0, 0, 0, 35], [0, 0, 0, 0, 0, 0, -55, 0], [0, 0, 389, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -5], [0, 0, 0, 0, 0, 0, 35, 0], [0, 0, 0, 0, 0, 509, 0, 0], [0, 0, 0, 30, 0, 0, 0, 0], [0, 0, 0, 0, -135, 0, 0, 0], [0, -205, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -134, 0, 0, 0], [0, 0, 525, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -40], [0, -340, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 46, 0, 0, 0], [0, 0, -79, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -60], [0, 210, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -40], [0, 140, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -275, 0, 0], [0, 0, 1059, 0, 0, 0, 0, 0], [510, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 95, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -645, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -101, 0, 0], [0, 0, 0, 39, 0, 0, 0, 0], [0, 0, -275, 0, 0, 0, 0, 0], [0, -246, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -315, 0, 0], [0, 0, 0, 0, -114, 0, 0, 0], [-200, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -255, 0], [0, 0, 0, -54, 0, 0, 0, 0], [0, 0, 0, 0, 10, 0, 0, 0], [0, 250, 0, 0, 0, 0, 0, 0], [1360, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 255, 0, 0], [0, 0, -1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 10], [0, 0, 0, 0, 0, 0, -185, 0], [0, 0, 0, 0, 0, 361, 0, 0], [0, 0, 0, 0, 210, 0, 0, 0], [0, 0, 0, 0, 0, 399, 0, 0], [0, 0, 0, -111, 0, 0, 0, 0], [0, 0, 0, 0, -81, 0, 0, 0], [0, 0, 255, 0, 0, 0, 0, 0], [0, -286, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 225, 0], [0, 0, 0, 0, -129, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 50], [-1440, 0, 0, 0, 0, 0, 0, 0], [0, -70, 0, 0, 0, 0, 0, 0], [-160, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -10, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 70], [0, 0, 0, 0, 0, 0, 55, 0], [0, 0, 0, 130, 0, 0, 0, 0], [0, 0, -1135, 0, 0, 0, 0, 0], [-1798, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -89, 0, 0, 0], [0, 0, -665, 0, 0, 0, 0, 0], [0, -76, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -95, 0], [0, 0, 0, 0, 0, -75, 0, 0], [0, 0, 0, 0, 41, 0, 0, 0], [680, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -51, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 107, 0], [0, 0, 0, 0, 0, 835, 0, 0], [0, 0, 0, 85, 0, 0, 0, 0], [0, 0, 489, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 64], [790, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -215, 0], [0, 0, 0, 0, 0, 21, 0, 0], [0, 0, 0, -130, 0, 0, 0, 0], [0, -405, 0, 0, 0, 0, 0, 0], [-142, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 111, 0, 0], [0, 0, -339, 0, 0, 0, 0, 0], [0, -54, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 385, 0], [0, 0, 0, 0, -135, 0, 0, 0], [0, 0, 0, 0, 0, -535, 0, 0], [0, 0, -661, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 311, 0, 0], [0, 0, 0, -75, 0, 0, 0, 0], [0, 0, 0, 0, -110, 0, 0, 0], [0, 0, -1455, 0, 0, 0, 0, 0], [0, 345, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -159, 0, 0], [0, 0, 0, -64, 0, 0, 0, 0], [0, 0, 0, 0, -159, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 70], [0, 0, 0, 0, 0, 455, 0, 0], [0, 0, 641, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -50], [0, 85, 0, 0, 0, 0, 0, 0], [-2872, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 263, 0], [0, 0, 0, 169, 0, 0, 0, 0], [0, 255, 0, 0, 0, 0, 0, 0], [0, 0, 0, -5, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 49], [0, 0, 0, 0, 0, 719, 0, 0], [0, 0, 0, 0, -30, 0, 0, 0], [0, 0, -855, 0, 0, 0, 0, 0], [0, -355, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -609, 0, 0], [0, 0, 0, -14, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 91], [0, -50, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -279, 0, 0, 0], [0, 0, -1711, 0, 0, 0, 0, 0], [0, -5, 0, 0, 0, 0, 0, 0], [1968, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -45, 0], [0, 0, 0, -26, 0, 0, 0, 0], [0, 0, 0, 0, -255, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 80], [0, 0, 0, 0, 0, 0, -275, 0], [-510, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -1001, 0, 0], [0, 0, 0, -30, 0, 0, 0, 0], [0, 0, 0, 0, -135, 0, 0, 0], [0, 0, 0, 0, 194, 0, 0, 0], [0, 0, 725, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 71], [0, 0, 0, 0, 0, 0, 425, 0], [0, 0, -231, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -45], [0, 0, 0, 0, 0, 0, 503, 0], [0, 0, 0, 0, 270, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 70], [-1280, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 965, 0, 0], [-1360, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 681, 0, 0], [0, 0, 0, -165, 0, 0, 0, 0], [0, 0, 0, 0, 190, 0, 0, 0], [1802, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 96, 0, 0, 0, 0], [0, -295, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1185, 0, 0], [0, 339, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 270, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -135], [0, 60, 0, 0, 0, 0, 0, 0], [160, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -193, 0], [0, 0, 0, 0, 0, -435, 0, 0], [0, 0, -209, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -95, 0], [0, 0, 0, 20, 0, 0, 0, 0], [0, 0, -1505, 0, 0, 0, 0, 0], [0, 0, 0, -74, 0, 0, 0, 0], [0, 0, 0, 0, -166, 0, 0, 0], [0, 0, 0, 0, 0, 0, -225, 0], [0, 0, 0, 0, 0, -495, 0, 0], [0, 0, 0, 0, 0, 0, 315, 0], [0, 0, 0, -61, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 15], [-2950, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -377, 0], [0, 0, 0, 0, 0, 375, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2304_e_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2304_3_e_m();".
// 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_2304_3_e_m(:prec:=8)
    chi := MakeCharacter_2304_e();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 3));
    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_2304_3_e_m();".
// 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_2304_3_e_m( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2304_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<5,R![34, 0, 1]>,<7,R![-8, 0, 1]>,<13,R![-68, 0, 1]>,<19,R![-544, 0, 1]>,<31,R![-1800, 0, 1]>],Snew);
    return Vf;
end function;