
// Make newform 2304.3.h.j in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2304_h();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_2304_h_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_h_j();".
// 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_h_j();".
// 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 := [81, 0, 0, 0, 7, 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, 32, 0, 0, 0, 2, 0], [0, -9, 0, -13, 0, -9, 0, 2], [0, -18, 0, 26, 0, -18, 0, -4], [28, 0, 0, 0, 8, 0, 0, 0], [0, -9, 60, 13, 0, -9, -30, -2], [0, 558, 0, 166, 0, 18, 0, 16], [0, 1116, 0, -332, 0, 36, 0, -32]];
    Rf_basisdens := [1, 45, 135, 135, 5, 135, 135, 135];
    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_h();"
function MakeCharacter_2304_h()
    N := 2304;
    order := 2;
    char_gens := [1279, 2053, 1793];
    v := [2, 1, 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_h_Hecke();"
function MakeCharacter_2304_h_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, 1, 0, -1, 0, 0], [-2, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -4, 0, 2, 0, 0], [0, 4, 0, 0, 0, 0, -1, 0], [0, 0, -1, 0, 1, 0, 0, 0], [0, 4, 0, 0, 0, 0, 2, 0], [0, 0, -6, 0, -2, 0, 0, 0], [0, 0, 0, 6, 0, 1, 0, 0], [2, 0, 0, 0, 0, 0, 0, 2], [0, -13, 0, 0, 0, 0, 0, 0], [0, 0, 11, 0, -1, 0, 0, 0], [0, -14, 0, 0, 0, 0, 4, 0], [0, 0, 38, 0, -2, 0, 0, 0], [0, 0, 0, -12, 0, 3, 0, 0], [0, 0, 0, -26, 0, 0, 0, 0], [0, 15, 0, 0, 0, 0, 4, 0], [0, 42, 0, 0, 0, 0, 2, 0], [0, 0, 22, 0, 4, 0, 0, 0], [28, 0, 0, 0, 0, 0, 0, 6], [38, 0, 0, 0, 0, 0, 0, -2], [0, 0, 0, -6, 0, 6, 0, 0], [0, 0, 47, 0, 6, 0, 0, 0], [-88, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, -42, 0, 5, 0, 0], [-86, 0, 0, 0, 0, 0, 0, -6], [0, 0, 0, -40, 0, -8, 0, 0], [0, 64, 0, 0, 0, 0, 1, 0], [0, 0, 5, 0, 6, 0, 0, 0], [-6, 0, 0, 0, 0, 0, 0, -4], [0, 0, 0, -24, 0, -12, 0, 0], [0, 0, 37, 0, -13, 0, 0, 0], [0, 18, 0, 0, 0, 0, -22, 0], [0, 0, 0, 33, 0, -19, 0, 0], [-130, 0, 0, 0, 0, 0, 0, -6], [0, 13, 0, 0, 0, 0, -20, 0], [0, 78, 0, 0, 0, 0, -8, 0], [0, 0, 108, 0, -2, 0, 0, 0], [0, 0, 0, -39, 0, 17, 0, 0], [0, 0, 0, -6, 0, -36, 0, 0], [0, -24, 0, 0, 0, 0, -1, 0], [0, 0, 172, 0, 2, 0, 0, 0], [-38, 0, 0, 0, 0, 0, 0, 12], [0, 0, 0, 8, 0, -11, 0, 0], [-190, 0, 0, 0, 0, 0, 0, 4], [0, -14, 0, 0, 0, 0, 14, 0], [-166, 0, 0, 0, 0, 0, 0, -4], [0, 0, 0, -66, 0, 26, 0, 0], [0, 8, 0, 0, 0, 0, 21, 0], [0, 0, -139, 0, -10, 0, 0, 0], [0, 0, -8, 0, -14, 0, 0, 0], [84, 0, 0, 0, 0, 0, 0, -7], [0, 0, 0, 24, 0, -34, 0, 0], [0, 0, 103, 0, -8, 0, 0, 0], [0, 0, -4, 0, -16, 0, 0, 0], [0, 0, 0, -104, 0, -17, 0, 0], [-366, 0, 0, 0, 0, 0, 0, 4], [0, 32, 0, 0, 0, 0, -13, 0], [0, 0, -13, 0, 8, 0, 0, 0], [0, 52, 0, 0, 0, 0, 16, 0], [0, 0, 0, 8, 0, 27, 0, 0], [0, 30, 0, 0, 0, 0, -6, 0], [0, 0, 260, 0, -8, 0, 0, 0], [-310, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 114, 0, 19, 0, 0], [0, 186, 0, 0, 0, 0, 2, 0], [316, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 18, 0, 54, 0, 0], [0, -93, 0, 0, 0, 0, -40, 0], [0, 0, 13, 0, 10, 0, 0, 0], [0, 0, 154, 0, 0, 0, 0, 0], [-294, 0, 0, 0, 0, 0, 0, 8], [0, 69, 0, 0, 0, 0, 28, 0], [0, 144, 0, 0, 0, 0, -14, 0], [0, 0, -248, 0, 10, 0, 0, 0], [0, 0, 0, -55, 0, -71, 0, 0], [0, 103, 0, 0, 0, 0, -32, 0], [0, 0, 255, 0, 27, 0, 0, 0], [48, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, -34, 0, -18, 0, 0], [0, 204, 0, 0, 0, 0, 31, 0], [0, 0, -10, 0, 58, 0, 0, 0], [-58, 0, 0, 0, 0, 0, 0, 20], [6, 0, 0, 0, 0, 0, 0, -12], [0, 0, 0, 30, 0, 18, 0, 0], [0, 0, 65, 0, 15, 0, 0, 0], [-136, 0, 0, 0, 0, 0, 0, -25], [0, 0, 0, -71, 0, -13, 0, 0], [166, 0, 0, 0, 0, 0, 0, -12], [0, 0, 0, 22, 0, -78, 0, 0], [0, 0, 214, 0, 4, 0, 0, 0], [202, 0, 0, 0, 0, 0, 0, -34], [0, 0, 0, 130, 0, -4, 0, 0], [0, -56, 0, 0, 0, 0, -64, 0], [0, 0, 2, 0, 40, 0, 0, 0], [0, 0, 0, 64, 0, 69, 0, 0], [0, 0, 9, 0, 29, 0, 0, 0], [0, 184, 0, 0, 0, 0, -66, 0], [0, -188, 0, 0, 0, 0, -63, 0], [0, 22, 0, 0, 0, 0, -36, 0], [0, 0, 0, -277, 0, 27, 0, 0], [0, 0, 0, 36, 0, 6, 0, 0], [0, 0, 533, 0, -11, 0, 0, 0], [0, -284, 0, 0, 0, 0, -12, 0], [126, 0, 0, 0, 0, 0, 0, -20], [0, 0, 0, -122, 0, -32, 0, 0], [0, 0, -285, 0, -42, 0, 0, 0], [0, 0, -350, 0, -14, 0, 0, 0], [286, 0, 0, 0, 0, 0, 0, 28], [690, 0, 0, 0, 0, 0, 0, -26], [0, -103, 0, 0, 0, 0, 96, 0], [0, 0, -221, 0, 44, 0, 0, 0], [0, -280, 0, 0, 0, 0, -12, 0], [-294, 0, 0, 0, 0, 0, 0, 4], [0, 0, -309, 0, 57, 0, 0, 0], [0, -242, 0, 0, 0, 0, -56, 0], [0, 0, -282, 0, 18, 0, 0, 0], [0, 0, 0, -369, 0, 13, 0, 0], [0, 0, 0, 342, 0, 32, 0, 0], [0, 27, 0, 0, 0, 0, -116, 0], [-94, 0, 0, 0, 0, 0, 0, 4], [0, 0, 0, -257, 0, -37, 0, 0], [0, 0, 0, 222, 0, 42, 0, 0], [0, 22, 0, 0, 0, 0, 40, 0], [0, 0, 0, 176, 0, 73, 0, 0], [0, 92, 0, 0, 0, 0, -59, 0], [0, 0, 194, 0, 56, 0, 0, 0], [-390, 0, 0, 0, 0, 0, 0, -6], [0, -196, 0, 0, 0, 0, -85, 0], [0, 284, 0, 0, 0, 0, 92, 0], [0, 0, -100, 0, -26, 0, 0, 0], [818, 0, 0, 0, 0, 0, 0, 6], [0, -264, 0, 0, 0, 0, 89, 0], [0, 0, 111, 0, -59, 0, 0, 0], [-938, 0, 0, 0, 0, 0, 0, -16], [0, 0, 0, 18, 0, -93, 0, 0], [0, -244, 0, 0, 0, 0, 30, 0], [0, 0, 0, 243, 0, -15, 0, 0], [0, 0, -21, 0, -63, 0, 0, 0], [0, -32, 0, 0, 0, 0, -38, 0], [0, 0, 0, 20, 0, -55, 0, 0], [886, 0, 0, 0, 0, 0, 0, 6], [0, 0, 0, -88, 0, 116, 0, 0], [0, 112, 0, 0, 0, 0, 33, 0], [0, 0, -1094, 0, -8, 0, 0, 0], [0, -429, 0, 0, 0, 0, 20, 0], [0, 0, 411, 0, 27, 0, 0, 0], [0, -606, 0, 0, 0, 0, 4, 0], [0, 0, -796, 0, -26, 0, 0, 0], [0, 323, 0, 0, 0, 0, 48, 0], [0, 0, 223, 0, -64, 0, 0, 0], [0, -238, 0, 0, 0, 0, 52, 0], [0, 0, 284, 0, 8, 0, 0, 0], [0, -118, 0, 0, 0, 0, -20, 0], [0, 0, 244, 0, -50, 0, 0, 0], [-242, 0, 0, 0, 0, 0, 0, -44], [0, 0, -369, 0, 15, 0, 0, 0], [282, 0, 0, 0, 0, 0, 0, 40], [0, 0, 0, -323, 0, -85, 0, 0], [0, 0, 0, 130, 0, 32, 0, 0], [0, 0, -111, 0, 49, 0, 0, 0], [546, 0, 0, 0, 0, 0, 0, 40], [0, 0, 0, -26, 0, 34, 0, 0], [0, 0, -279, 0, 69, 0, 0, 0], [0, 0, -348, 0, 80, 0, 0, 0], [-510, 0, 0, 0, 0, 0, 0, -22], [0, 225, 0, 0, 0, 0, 36, 0], [-960, 0, 0, 0, 0, 0, 0, -17], [0, 0, 0, 443, 0, 47, 0, 0], [0, 0, 0, -282, 0, -68, 0, 0], [0, -391, 0, 0, 0, 0, 104, 0], [0, 0, 472, 0, 26, 0, 0, 0], [-648, 0, 0, 0, 0, 0, 0, 44], [1198, 0, 0, 0, 0, 0, 0, -36], [0, 0, -197, 0, 52, 0, 0, 0], [0, 776, 0, 0, 0, 0, -52, 0], [0, 0, 0, 169, 0, -43, 0, 0], [-202, 0, 0, 0, 0, 0, 0, -2], [0, 297, 0, 0, 0, 0, -44, 0], [-386, 0, 0, 0, 0, 0, 0, -78], [0, 0, 0, 42, 0, -126, 0, 0], [0, 480, 0, 0, 0, 0, -63, 0], [0, 0, 11, 0, -48, 0, 0, 0], [0, 0, -752, 0, 70, 0, 0, 0], [0, 0, 0, -201, 0, 47, 0, 0], [0, -376, 0, 0, 0, 0, -93, 0], [0, -420, 0, 0, 0, 0, -44, 0], [364, 0, 0, 0, 0, 0, 0, 17], [0, 0, -172, 0, -56, 0, 0, 0], [1304, 0, 0, 0, 0, 0, 0, -14], [0, 0, 0, 162, 0, -36, 0, 0], [0, 52, 0, 0, 0, 0, -108, 0], [0, 0, 0, 143, 0, 103, 0, 0], [0, 0, 0, -20, 0, -132, 0, 0], [0, 0, -659, 0, 24, 0, 0, 0], [-194, 0, 0, 0, 0, 0, 0, 32], [0, -604, 0, 0, 0, 0, -89, 0], [0, 0, -819, 0, -32, 0, 0, 0], [0, 0, 872, 0, -76, 0, 0, 0], [0, 0, 0, -379, 0, 157, 0, 0], [-798, 0, 0, 0, 0, 0, 0, -44], [0, 460, 0, 0, 0, 0, 107, 0], [-44, 0, 0, 0, 0, 0, 0, -24], [0, 0, 0, -300, 0, -18, 0, 0], [0, 0, 0, -66, 0, -9, 0, 0], [-1498, 0, 0, 0, 0, 0, 0, -14], [0, 0, 0, -236, 0, -190, 0, 0], [0, 0, 341, 0, -134, 0, 0, 0], [0, 138, 0, 0, 0, 0, -68, 0], [1158, 0, 0, 0, 0, 0, 0, 4], [0, 0, 0, 131, 0, -5, 0, 0], [846, 0, 0, 0, 0, 0, 0, 58], [0, 0, 0, 62, 0, -196, 0, 0], [0, 0, 474, 0, 36, 0, 0, 0], [-1078, 0, 0, 0, 0, 0, 0, 60], [-98, 0, 0, 0, 0, 0, 0, 70], [0, 0, -819, 0, -49, 0, 0, 0], [0, 0, 648, 0, -102, 0, 0, 0], [0, 0, 0, -123, 0, -171, 0, 0], [0, 196, 0, 0, 0, 0, 195, 0], [-570, 0, 0, 0, 0, 0, 0, -6], [0, 0, -245, 0, -161, 0, 0, 0], [-1394, 0, 0, 0, 0, 0, 0, -68], [0, 0, 0, -506, 0, 90, 0, 0], [0, -553, 0, 0, 0, 0, -56, 0], [0, 0, -1263, 0, 2, 0, 0, 0], [0, 0, -110, 0, -148, 0, 0, 0], [158, 0, 0, 0, 0, 0, 0, -54], [0, 0, 0, -334, 0, -128, 0, 0], [0, -132, 0, 0, 0, 0, -59, 0], [0, 590, 0, 0, 0, 0, 96, 0], [-1314, 0, 0, 0, 0, 0, 0, 64], [0, 0, 437, 0, 24, 0, 0, 0], [0, 238, 0, 0, 0, 0, 144, 0], [0, 0, 362, 0, -92, 0, 0, 0], [184, 0, 0, 0, 0, 0, 0, 35], [0, 0, 0, 505, 0, 61, 0, 0], [0, 0, 0, -246, 0, 22, 0, 0], [0, 0, 886, 0, 58, 0, 0, 0], [0, 0, 0, -108, 0, -18, 0, 0], [0, 1218, 0, 0, 0, 0, 50, 0], [1962, 0, 0, 0, 0, 0, 0, 4], [0, 421, 0, 0, 0, 0, -36, 0], [0, 0, -141, 0, 80, 0, 0, 0], [0, 0, 334, 0, 56, 0, 0, 0], [-386, 0, 0, 0, 0, 0, 0, -52], [0, 0, 0, -528, 0, 220, 0, 0], [0, -750, 0, 0, 0, 0, 22, 0], [0, 0, -356, 0, -164, 0, 0, 0], [0, -312, 0, 0, 0, 0, -147, 0], [0, 0, 515, 0, 106, 0, 0, 0], [0, 0, 1102, 0, 98, 0, 0, 0], [960, 0, 0, 0, 0, 0, 0, 10], [0, 0, 0, 316, 0, -141, 0, 0], [0, 0, 0, 104, 0, -20, 0, 0], [0, -176, 0, 0, 0, 0, 171, 0], [0, -40, 0, 0, 0, 0, 248, 0], [0, 0, 0, -138, 0, 103, 0, 0], [-1102, 0, 0, 0, 0, 0, 0, 20], [-914, 0, 0, 0, 0, 0, 0, 98], [0, 0, 0, -334, 0, 12, 0, 0], [0, -1004, 0, 0, 0, 0, 15, 0], [0, 213, 0, 0, 0, 0, 196, 0], [0, 0, 365, 0, -43, 0, 0, 0], [0, 1082, 0, 0, 0, 0, -94, 0], [0, 0, 0, 391, 0, -87, 0, 0], [0, 0, 877, 0, 22, 0, 0, 0], [0, -636, 0, 0, 0, 0, 154, 0], [0, 0, 0, 13, 0, -67, 0, 0], [0, -529, 0, 0, 0, 0, 108, 0], [0, 740, 0, 0, 0, 0, -36, 0], [1344, 0, 0, 0, 0, 0, 0, -20], [1794, 0, 0, 0, 0, 0, 0, 34], [-2248, 0, 0, 0, 0, 0, 0, 44], [822, 0, 0, 0, 0, 0, 0, -118], [0, 0, 0, -32, 0, -94, 0, 0], [0, 1607, 0, 0, 0, 0, -16, 0], [-638, 0, 0, 0, 0, 0, 0, -8], [0, 0, 0, -744, 0, -48, 0, 0], [0, 0, -854, 0, 56, 0, 0, 0], [814, 0, 0, 0, 0, 0, 0, -24], [0, 0, 406, 0, 130, 0, 0, 0], [0, -399, 0, 0, 0, 0, 256, 0], [0, -196, 0, 0, 0, 0, -210, 0], [0, 0, -448, 0, -20, 0, 0, 0], [1652, 0, 0, 0, 0, 0, 0, 43], [0, 0, 0, -405, 0, 265, 0, 0], [-1930, 0, 0, 0, 0, 0, 0, -24], [0, 0, -1555, 0, 94, 0, 0, 0], [0, 0, 0, 79, 0, 117, 0, 0], [0, 0, 0, 414, 0, -120, 0, 0], [0, 0, 863, 0, -118, 0, 0, 0], [0, 0, 0, -330, 0, -300, 0, 0], [0, -703, 0, 0, 0, 0, -140, 0], [0, 0, 0, -375, 0, 531, 0, 0], [-558, 0, 0, 0, 0, 0, 0, -148], [0, 0, 0, -190, 0, -115, 0, 0], [0, 0, 0, -588, 0, -70, 0, 0], [0, 1084, 0, 0, 0, 0, -114, 0], [-1018, 0, 0, 0, 0, 0, 0, -92], [0, 0, 0, -9, 0, 295, 0, 0], [2942, 0, 0, 0, 0, 0, 0, -4]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2304_h_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2304_3_h_j();".
// 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_h_j(:prec:=8)
    chi := MakeCharacter_2304_h();
    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_h_j();".
// 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_h_j( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2304_h();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<5,R![1764, 0, -92, 0, 1]>,<7,R![2, 1]>],Snew);
    return Vf;
end function;