
// Make newform 2304.3.e.n in Magma, downloaded from the LMFDB on 23 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_n();".
// 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_n();".
// 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 := [9, 0, 56, 0, 71, 0, 16, 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, 11, 0, 9, 0, 1, 0, 0], [0, 73, 0, 115, 0, 29, 0, 2], [14, 0, 32, 0, 4, 0, 0, 0], [0, -43, 0, -157, 0, -35, 0, -2], [16, 0, 68, 0, 24, 0, 2, 0], [0, 107, 0, 245, 0, 61, 0, 4], [48, 0, 76, 0, 24, 0, 2, 0]];
    Rf_basisdens := [1, 1, 3, 1, 3, 1, 3, 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_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, 1, 0, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0, 0, 0], [0, 0, 1, 0, 1, 0, 0, 0], [0, 0, 0, 0, 0, 1, 0, 0], [0, -1, 0, 0, 0, 0, -2, 0], [0, 0, 0, 0, 0, -1, 0, -1], [0, -3, 0, 0, 0, 0, -1, 0], [0, 0, -1, 0, 2, 0, 0, 0], [-16, 0, 0, -3, 0, 0, 0, 0], [0, 0, 0, 0, 0, -1, 0, -1], [0, 1, 0, 0, 0, 0, -2, 0], [0, 0, 0, 0, 0, 2, 0, -2], [0, -7, 0, 0, 0, 0, 3, 0], [0, 0, -5, 0, -2, 0, 0, 0], [0, 0, 4, 0, -4, 0, 0, 0], [0, 0, 0, 0, 0, -3, 0, 1], [0, 0, 0, 0, 0, 3, 0, -1], [0, -15, 0, 0, 0, 0, 3, 0], [20, 0, 0, 8, 0, 0, 0, 0], [-48, 0, 0, 11, 0, 0, 0, 0], [0, 0, -7, 0, 1, 0, 0, 0], [0, -7, 0, 0, 0, 0, 4, 0], [-24, 0, 0, -4, 0, 0, 0, 0], [0, 0, 13, 0, 6, 0, 0, 0], [-112, 0, 0, -1, 0, 0, 0, 0], [0, 0, 4, 0, 4, 0, 0, 0], [0, 0, 0, 0, 0, -1, 0, -4], [0, 13, 0, 0, 0, 0, 12, 0], [-96, 0, 0, -7, 0, 0, 0, 0], [0, 0, -22, 0, 10, 0, 0, 0], [0, 19, 0, 0, 0, 0, 6, 0], [0, 0, 0, 0, 0, 1, 0, 5], [0, 0, -29, 0, 4, 0, 0, 0], [-112, 0, 0, -7, 0, 0, 0, 0], [0, 0, 0, 0, 0, 7, 0, 3], [0, 0, 0, 0, 0, -8, 0, 4], [0, -44, 0, 0, 0, 0, 4, 0], [0, 0, 17, 0, 20, 0, 0, 0], [0, 0, 26, 0, -14, 0, 0, 0], [0, 0, 0, 0, 0, -5, 0, 4], [0, -44, 0, 0, 0, 0, 4, 0], [-2, 0, 0, 16, 0, 0, 0, 0], [0, 0, 37, 0, -22, 0, 0, 0], [-96, 0, 0, 19, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1, 0, -3], [-160, 0, 0, -15, 0, 0, 0, 0], [0, 0, -13, 0, -21, 0, 0, 0], [0, 0, 0, 0, 0, -11, 0, -4], [0, 59, 0, 0, 0, 0, 4, 0], [0, -62, 0, 0, 0, 0, -18, 0], [68, 0, 0, -36, 0, 0, 0, 0], [0, 0, 67, 0, 19, 0, 0, 0], [0, -5, 0, 0, 0, 0, 0, 0], [0, -54, 0, 0, 0, 0, -2, 0], [0, 0, -31, 0, 2, 0, 0, 0], [-32, 0, 0, 13, 0, 0, 0, 0], [0, 0, 0, 0, 0, -5, 0, 12], [0, -71, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -4, 0, -12], [0, 0, -29, 0, 22, 0, 0, 0], [0, 0, 0, 0, 0, 3, 0, -9], [0, 6, 0, 0, 0, 0, 50, 0], [-158, 0, 0, 0, 0, 0, 0, 0], [0, 0, 5, 0, 6, 0, 0, 0], [0, 0, 0, 0, 0, -31, 0, -11], [60, 0, 0, 32, 0, 0, 0, 0], [0, 0, 19, 0, -21, 0, 0, 0], [0, 0, 0, 0, 0, 1, 0, -7], [0, -43, 0, 0, 0, 0, 36, 0], [0, -17, 0, 0, 0, 0, -19, 0], [64, 0, 0, -35, 0, 0, 0, 0], [0, 0, 0, 0, 0, 25, 0, -19], [0, 0, 0, 0, 0, 13, 0, -11], [0, -14, 0, 0, 0, 0, 30, 0], [0, 0, 7, 0, -4, 0, 0, 0], [0, 0, 0, 0, 0, 17, 0, 17], [0, 67, 0, 0, 0, 0, 42, 0], [-384, 0, 0, 4, 0, 0, 0, 0], [0, 0, -43, 0, -19, 0, 0, 0], [0, 0, 0, 0, 0, -5, 0, 24], [0, 21, 0, 0, 0, 0, -57, 0], [170, 0, 0, 48, 0, 0, 0, 0], [224, 0, 0, 89, 0, 0, 0, 0], [0, 0, -71, 0, 1, 0, 0, 0], [0, -39, 0, 0, 0, 0, -30, 0], [184, 0, 0, -36, 0, 0, 0, 0], [0, 0, -109, 0, -40, 0, 0, 0], [-288, 0, 0, -25, 0, 0, 0, 0], [0, 0, 47, 0, -57, 0, 0, 0], [0, 75, 0, 0, 0, 0, 17, 0], [496, 0, 0, 51, 0, 0, 0, 0], [0, 0, -14, 0, 26, 0, 0, 0], [0, 0, 0, 0, 0, -24, 0, 8], [0, 39, 0, 0, 0, 0, 21, 0], [0, 0, -61, 0, 38, 0, 0, 0], [0, 99, 0, 0, 0, 0, -6, 0], [0, 0, 0, 0, 0, 39, 0, 15], [0, 0, 0, 0, 0, -29, 0, -8], [0, 0, 0, 0, 0, -34, 0, -6], [0, 0, -13, 0, -4, 0, 0, 0], [0, 0, -7, 0, -7, 0, 0, 0], [0, -137, 0, 0, 0, 0, -54, 0], [0, 0, 0, 0, 0, -38, 0, 2], [46, 0, 0, 16, 0, 0, 0, 0], [0, 0, -56, 0, 32, 0, 0, 0], [0, 51, 0, 0, 0, 0, 12, 0], [0, 93, 0, 0, 0, 0, -49, 0], [-494, 0, 0, 48, 0, 0, 0, 0], [-240, 0, 0, -55, 0, 0, 0, 0], [0, 0, 0, 0, 0, 25, 0, 25], [0, 249, 0, 0, 0, 0, 8, 0], [0, 0, 0, 0, 0, 6, 0, -10], [672, 0, 0, -65, 0, 0, 0, 0], [0, 43, 0, 0, 0, 0, 46, 0], [0, 0, 0, 0, 0, 4, 0, 0], [0, 159, 0, 0, 0, 0, 21, 0], [0, 0, 29, 0, 56, 0, 0, 0], [0, 0, 44, 0, 20, 0, 0, 0], [0, 0, 0, 0, 0, 59, 0, 15], [-610, 0, 0, 48, 0, 0, 0, 0], [0, 0, -107, 0, -20, 0, 0, 0], [0, 0, 109, 0, -43, 0, 0, 0], [0, 0, 0, 0, 0, 20, 0, 0], [0, 0, 119, 0, -66, 0, 0, 0], [0, 0, 0, 0, 0, 37, 0, 0], [0, 113, 0, 0, 0, 0, 19, 0], [656, 0, 0, -1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 37, 0, -24], [0, 0, 0, 0, 0, 46, 0, -26], [0, 40, 0, 0, 0, 0, -32, 0], [80, 0, 0, 117, 0, 0, 0, 0], [0, 0, 0, 0, 0, -27, 0, -36], [0, -111, 0, 0, 0, 0, 74, 0], [210, 0, 0, -64, 0, 0, 0, 0], [0, 0, 43, 0, 90, 0, 0, 0], [0, 0, 0, 0, 0, 21, 0, -43], [0, 0, -175, 0, 40, 0, 0, 0], [0, 237, 0, 0, 0, 0, -30, 0], [0, 0, 0, 0, 0, -43, 0, -19], [0, 0, -23, 0, 42, 0, 0, 0], [-656, 0, 0, 45, 0, 0, 0, 0], [0, 0, 94, 0, -2, 0, 0, 0], [0, 0, 0, 0, 0, 23, 0, -4], [0, 123, 0, 0, 0, 0, 49, 0], [0, 0, 0, 0, 0, -33, 0, 11], [0, -239, 0, 0, 0, 0, 54, 0], [0, 0, 0, 0, 0, 62, 0, -6], [0, 0, 0, 0, 0, 0, -72, 0], [0, 0, 0, 0, 0, 5, 0, -11], [0, 163, 0, 0, 0, 0, 80, 0], [0, 0, 0, 0, 0, -38, 0, -50], [0, -10, 0, 0, 0, 0, -126, 0], [0, 0, 0, 0, 0, 34, 0, -18], [0, 152, 0, 0, 0, 0, 96, 0], [992, 0, 0, -91, 0, 0, 0, 0], [0, -269, 0, 0, 0, 0, -62, 0], [742, 0, 0, -96, 0, 0, 0, 0], [0, 0, 227, 0, 32, 0, 0, 0], [0, 0, 152, 0, 80, 0, 0, 0], [0, -45, 0, 0, 0, 0, 66, 0], [-512, 0, 0, 99, 0, 0, 0, 0], [0, 0, -203, 0, -35, 0, 0, 0], [0, -51, 0, 0, 0, 0, -74, 0], [0, -66, 0, 0, 0, 0, -38, 0], [1136, 0, 0, 45, 0, 0, 0, 0], [0, 0, 0, 0, 0, 9, 0, -43], [400, 0, 0, 4, 0, 0, 0, 0], [0, 0, 145, 0, -84, 0, 0, 0], [0, 0, 58, 0, 18, 0, 0, 0], [0, 0, 0, 0, 0, 11, 0, 19], [0, 82, 0, 0, 0, 0, -50, 0], [24, 0, 0, 112, 0, 0, 0, 0], [-1056, 0, 0, 23, 0, 0, 0, 0], [0, -247, 0, 0, 0, 0, -24, 0], [0, 0, 0, 0, 0, 70, 0, -42], [0, 0, 19, 0, 20, 0, 0, 0], [240, 0, 0, 205, 0, 0, 0, 0], [0, 0, 0, 0, 0, 71, 0, -21], [432, 0, 0, -1, 0, 0, 0, 0], [0, 0, -109, 0, -37, 0, 0, 0], [0, 0, 0, 0, 0, -39, 0, -44], [0, 1, 0, 0, 0, 0, -16, 0], [0, -158, 0, 0, 0, 0, -66, 0], [0, 0, -71, 0, 100, 0, 0, 0], [0, 0, 0, 0, 0, -11, 0, -20], [0, 0, 0, 0, 0, 94, 0, -10], [1204, 0, 0, -60, 0, 0, 0, 0], [0, -42, 0, 0, 0, 0, -126, 0], [728, 0, 0, -24, 0, 0, 0, 0], [0, 0, 298, 0, -14, 0, 0, 0], [0, 0, 0, 0, 0, 62, 0, 38], [0, 0, 47, 0, 124, 0, 0, 0], [0, 0, -250, 0, 54, 0, 0, 0], [0, 287, 0, 0, 0, 0, 48, 0], [-322, 0, 0, -192, 0, 0, 0, 0], [0, 0, 0, 0, 0, -35, 0, -16], [0, 169, 0, 0, 0, 0, 16, 0], [0, -372, 0, 0, 0, 0, -12, 0], [0, 0, 173, 0, -124, 0, 0, 0], [608, 0, 0, 9, 0, 0, 0, 0], [0, 0, 0, 0, 0, 7, 0, 8], [-1532, 0, 0, -32, 0, 0, 0, 0], [0, 0, -197, 0, -37, 0, 0, 0], [0, 0, 113, 0, 38, 0, 0, 0], [-16, 0, 0, -45, 0, 0, 0, 0], [0, 0, -133, 0, 59, 0, 0, 0], [0, -53, 0, 0, 0, 0, 12, 0], [0, 0, 0, 0, 0, 46, 0, -70], [-366, 0, 0, -16, 0, 0, 0, 0], [0, 0, -155, 0, -92, 0, 0, 0], [-752, 0, 0, 153, 0, 0, 0, 0], [0, 0, -62, 0, 58, 0, 0, 0], [0, -29, 0, 0, 0, 0, 121, 0], [554, 0, 0, -16, 0, 0, 0, 0], [-1456, 0, 0, 51, 0, 0, 0, 0], [0, -11, 0, 0, 0, 0, 130, 0], [0, -46, 0, 0, 0, 0, 46, 0], [0, 0, -211, 0, -92, 0, 0, 0], [0, 0, 0, 0, 0, 31, 0, -64], [-1840, 0, 0, -55, 0, 0, 0, 0], [0, 311, 0, 0, 0, 0, 98, 0], [-96, 0, 0, -97, 0, 0, 0, 0], [0, 0, 95, 0, 151, 0, 0, 0], [0, 0, 0, 0, 0, -65, 0, 39], [0, -113, 0, 0, 0, 0, 156, 0], [0, -71, 0, 0, 0, 0, 155, 0], [144, 0, 0, 145, 0, 0, 0, 0], [0, 0, 100, 0, 76, 0, 0, 0], [0, 0, 0, 0, 0, -73, 0, 16], [0, 0, 0, 0, 0, -56, 0, -44], [-2112, 0, 0, 35, 0, 0, 0, 0], [0, -289, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 44, 0, 16], [0, -187, 0, 0, 0, 0, -225, 0], [-360, 0, 0, 308, 0, 0, 0, 0], [0, 0, 43, 0, -60, 0, 0, 0], [0, 0, 211, 0, 11, 0, 0, 0], [0, -221, 0, 0, 0, 0, -223, 0], [0, 0, 149, 0, -75, 0, 0, 0], [0, 0, 0, 0, 0, 101, 0, 41], [-2656, 0, 0, -13, 0, 0, 0, 0], [0, 0, 0, 0, 0, 27, 0, -57], [0, 31, 0, 0, 0, 0, 0, 0], [0, -399, 0, 0, 0, 0, 19, 0], [416, 0, 0, 135, 0, 0, 0, 0], [0, 0, 290, 0, -94, 0, 0, 0], [0, 0, 0, 0, 0, 19, 0, -73], [0, -438, 0, 0, 0, 0, -18, 0], [0, 0, 0, 0, 0, 63, 0, -28], [0, 75, 0, 0, 0, 0, -204, 0], [0, 71, 0, 0, 0, 0, 141, 0], [-1760, 0, 0, -72, 0, 0, 0, 0], [0, 0, -259, 0, 82, 0, 0, 0], [0, 0, -298, 0, -74, 0, 0, 0], [0, 0, 0, 0, 0, 15, 0, -84], [0, 0, 0, 0, 0, -104, 0, -24], [0, 0, 319, 0, 10, 0, 0, 0], [82, 0, 0, -48, 0, 0, 0, 0], [-144, 0, 0, -109, 0, 0, 0, 0], [0, 0, 350, 0, -106, 0, 0, 0], [0, 0, 0, 0, 0, 135, 0, 16], [0, 0, 0, 0, 0, 43, 0, 31], [0, -95, 0, 0, 0, 0, 150, 0], [0, 0, 0, 0, 0, -17, 0, 11], [0, 0, -367, 0, 240, 0, 0, 0], [0, 459, 0, 0, 0, 0, 52, 0], [0, 0, 0, 0, 0, 37, 0, -43], [0, 0, -341, 0, -36, 0, 0, 0], [0, 0, 0, 0, 0, 145, 0, 45], [0, 0, 0, 0, 0, 86, 0, 14], [192, 0, 0, -272, 0, 0, 0, 0], [-1424, 0, 0, 169, 0, 0, 0, 0], [-936, 0, 0, -16, 0, 0, 0, 0], [-432, 0, 0, -311, 0, 0, 0, 0], [0, 0, 109, 0, -99, 0, 0, 0], [0, 0, 0, 0, 0, -47, 0, -31], [-674, 0, 0, 96, 0, 0, 0, 0], [0, 0, 68, 0, -60, 0, 0, 0], [0, -503, 0, 0, 0, 0, -69, 0], [-704, 0, 0, -123, 0, 0, 0, 0], [0, -13, 0, 0, 0, 0, -79, 0], [0, 0, 0, 0, 0, 57, 0, 41], [0, 0, 0, 0, 0, -153, 0, 55], [0, 308, 0, 0, 0, 0, -20, 0], [-1356, 0, 0, 20, 0, 0, 0, 0], [0, 0, 271, 0, 168, 0, 0, 0], [512, 0, 0, -19, 0, 0, 0, 0], [0, -323, 0, 0, 0, 0, 140, 0], [0, 0, -355, 0, 120, 0, 0, 0], [0, 0, 92, 0, 228, 0, 0, 0], [0, 105, 0, 0, 0, 0, -244, 0], [0, 0, -350, 0, -102, 0, 0, 0], [0, 0, 0, 0, 0, -165, 0, -65], [0, 0, -325, 0, -120, 0, 0, 0], [224, 0, 0, -183, 0, 0, 0, 0], [0, 0, -467, 0, -146, 0, 0, 0], [0, 0, -131, 0, 253, 0, 0, 0], [0, 0, 0, 0, 0, -23, 0, 89], [-1682, 0, 0, 144, 0, 0, 0, 0], [0, 0, -217, 0, -164, 0, 0, 0], [2144, 0, 0, 135, 0, 0, 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_n();".
// 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_n(: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_n();".
// 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_n( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2304_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<5,R![68, 0, 60, 0, 1]>,<7,R![-52, 0, 1]>,<13,R![2448, 0, -472, 0, 1]>,<19,R![352512, 0, -1504, 0, 1]>,<31,R![-212, 32, 1]>],Snew);
    return Vf;
end function;