
// Make newform 2304.3.b.q in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2304_b();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_2304_b_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_b_q();".
// 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_b_q();".
// 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, 0, 0, 0, -1, 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, 0, 0, 0, 0, 2, 0], [0, 0, 8, 0, 0, 0, -4, 0], [-2, 2, 0, -2, 4, -2, 0, 0], [0, 4, 0, 4, 0, -4, 0, 0], [0, 4, 0, 4, 0, 4, 0, -8], [2, 4, 0, -4, -4, 0, 0, 4], [-2, 0, 0, 0, 4, 4, 0, 4]];
    Rf_basisdens := [1, 1, 1, 1, 1, 1, 1, 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_b();"
function MakeCharacter_2304_b()
    N := 2304;
    order := 2;
    char_gens := [1279, 2053, 1793];
    v := [1, 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_2304_b_Hecke();"
function MakeCharacter_2304_b_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, 1, 0, -1, 0, 0, 0, 0], [0, 1, 0, 1, 0, 0, 1, 0], [-4, 0, 0, 0, 0, 1, 0, 0], [0, -3, 0, 0, 0, 0, 0, 2], [-2, 0, -1, 0, -1, 2, 0, 0], [12, 0, -1, 0, 2, 1, 0, 0], [0, -2, 0, 2, 0, 0, 0, -2], [0, -5, 0, -5, 0, 0, -2, -2], [0, 3, 0, 1, 0, 0, -4, -1], [0, -1, 0, 6, 0, 0, -4, -2], [10, 0, -5, 0, -5, -2, 0, 0], [28, 0, -3, 0, -2, -3, 0, 0], [0, 6, 0, 0, 0, 0, 1, 5], [0, -11, 0, -3, 0, 0, 6, -2], [64, 0, -3, 0, -4, 0, 0, 0], [0, 17, 0, -2, 0, 0, -8, 6], [0, 0, -1, 0, 12, 0, 0, 0], [0, 30, 0, 8, 0, 0, -3, 1], [2, 0, 8, 0, -12, -2, 0, 0], [0, -5, 0, -11, 0, 0, 2, 9], [68, 0, 0, 0, -4, 7, 0, 0], [-30, 0, -6, 0, 2, 4, 0, 0], [50, 0, -2, 0, 22, 0, 0, 0], [0, 33, 0, 5, 0, 0, 8, -12], [0, 27, 0, -5, 0, 0, 7, -20], [56, 0, 5, 0, 0, -6, 0, 0], [0, -35, 0, -4, 0, 0, -12, 6], [-42, 0, 2, 0, 10, -8, 0, 0], [0, 41, 0, -17, 0, 0, -10, 11], [64, 0, 3, 0, 24, 0, 0, 0], [-22, 0, 5, 0, -3, 2, 0, 0], [0, 0, 11, 0, -12, 8, 0, 0], [0, -39, 0, 15, 0, 0, -8, -8], [0, 61, 0, 13, 0, 0, -15, 4], [0, 21, 0, -6, 0, 0, 8, -2], [68, 0, 9, 0, 10, -13, 0, 0], [0, 8, 0, 44, 0, 0, -6, 14], [0, -37, 0, 9, 0, 0, -6, 14], [128, 0, 5, 0, -20, 0, 0, 0], [0, -17, 0, -28, 0, 0, -16, -10], [0, 56, 0, -34, 0, 0, 5, 3], [-18, 0, -24, 0, 20, -2, 0, 0], [0, 37, 0, -7, 0, 0, -14, 26], [0, -1, 0, -29, 0, 0, 25, -2], [48, 0, 7, 0, -4, 12, 0, 0], [0, 17, 0, 31, 0, 0, 6, 11], [76, 0, -2, 0, -12, -11, 0, 0], [0, 47, 0, -16, 0, 0, 0, -18], [-38, 0, 26, 0, 10, 8, 0, 0], [0, 36, 0, 50, 0, 0, 13, 23], [70, 0, -6, 0, -18, -26, 0, 0], [156, 0, 4, 0, 32, 9, 0, 0], [62, 0, 20, 0, 12, 8, 0, 0], [0, 136, 0, 14, 0, 0, -31, 7], [0, -49, 0, 27, 0, 0, -20, -24], [0, -47, 0, 19, 0, 0, -12, -35], [0, -105, 0, -12, 0, 0, 0, -10], [-118, 0, 12, 0, -4, -12, 0, 0], [-16, 0, -3, 0, 40, -4, 0, 0], [0, 109, 0, 37, 0, 0, 10, -6], [0, 0, -21, 0, -4, 0, 0, 0], [0, 68, 0, 34, 0, 0, 25, -53], [-142, 0, -8, 0, 28, 2, 0, 0], [0, 111, 0, 23, 0, 0, -4, 0], [152, 0, 11, 0, 48, -6, 0, 0], [-114, 0, -8, 0, -4, -6, 0, 0], [108, 0, -30, 0, 20, -3, 0, 0], [0, -15, 0, 46, 0, 0, 20, -22], [-74, 0, -2, 0, 14, -16, 0, 0], [0, 6, 0, -16, 0, 0, -35, 57], [0, 123, 0, 33, 0, 0, 32, -29], [0, -53, 0, -22, 0, 0, 40, 42], [-140, 0, -9, 0, 38, 7, 0, 0], [0, -56, 0, -2, 0, 0, -43, -13], [0, 149, 0, -25, 0, 0, -14, 6], [0, 77, 0, -26, 0, 0, 4, -22], [46, 0, -35, 0, 45, -10, 0, 0], [14, 0, 22, 0, 46, -16, 0, 0], [220, 0, -62, 0, 4, 17, 0, 0], [0, 103, 0, 52, 0, 0, -20, 26], [0, -30, 0, 76, 0, 0, 45, -27], [66, 0, 12, 0, 4, 44, 0, 0], [0, 91, 0, 3, 0, 0, -13, 16], [-84, 0, 14, 0, -20, -51, 0, 0], [-66, 0, -9, 0, -97, -14, 0, 0], [142, 0, -6, 0, 18, -32, 0, 0], [0, -37, 0, -71, 0, 0, 38, -30], [0, -43, 0, -41, 0, 0, 28, 49], [-388, 0, -18, 0, -44, -7, 0, 0], [0, 2, 0, -34, 0, 0, 12, -10], [0, -31, 0, -51, 0, 0, -21, -10], [-248, 0, 43, 0, -36, 6, 0, 0], [-80, 0, -35, 0, -40, -20, 0, 0], [0, 14, 0, -56, 0, 0, 37, 57], [0, 51, 0, -105, 0, 0, -62, 18], [-158, 0, -49, 0, 23, -22, 0, 0], [-316, 0, 71, 0, -10, 11, 0, 0], [0, 49, 0, 36, 0, 0, -68, 30], [236, 0, 69, 0, -10, 9, 0, 0], [0, -25, 0, 53, 0, 0, -48, -72], [-60, 0, 10, 0, 96, 15, 0, 0], [58, 0, 19, 0, -93, 38, 0, 0], [-312, 0, 9, 0, 12, -66, 0, 0], [-90, 0, 32, 0, -20, -50, 0, 0], [-200, 0, -49, 0, -36, 10, 0, 0], [374, 0, -38, 0, -14, -24, 0, 0], [0, 42, 0, 50, 0, 0, -62, -72], [26, 0, 76, 0, -32, 26, 0, 0], [0, -103, 0, -21, 0, 0, -40, -23], [0, 27, 0, 114, 0, 0, 84, -2], [154, 0, 4, 0, -100, -4, 0, 0], [-240, 0, -47, 0, 24, -20, 0, 0], [0, -191, 0, 13, 0, 0, 83, -18], [-106, 0, -17, 0, 39, 38, 0, 0], [-340, 0, -83, 0, 70, -15, 0, 0], [0, -170, 0, -94, 0, 0, -60, -30], [0, -45, 0, -123, 0, 0, -14, 94], [-216, 0, -33, 0, -4, -18, 0, 0], [0, -181, 0, 150, 0, 0, -72, -62], [-270, 0, 40, 0, 48, 60, 0, 0], [0, 225, 0, 107, 0, 0, 44, 76], [116, 0, 64, 0, 76, -21, 0, 0], [196, 0, -47, 0, 114, 51, 0, 0], [0, 63, 0, 47, 0, 0, 28, -96], [0, 207, 0, -112, 0, 0, -12, -38], [0, -210, 0, 58, 0, 0, 56, -10], [0, -99, 0, 61, 0, 0, 97, 4], [0, 193, 0, -120, 0, 0, -12, 30], [-616, 0, 37, 0, 36, 26, 0, 0], [0, -204, 0, 64, 0, 0, 146, -30], [0, -197, 0, 45, 0, 0, -42, 53], [0, -145, 0, 0, 0, 0, 96, -66], [258, 0, -23, 0, -79, -66, 0, 0], [-274, 0, -28, 0, -64, 22, 0, 0], [0, 545, 0, 9, 0, 0, -40, -20], [116, 0, 23, 0, 22, 39, 0, 0], [0, 31, 0, 29, 0, 0, -36, 92], [42, 0, 145, 0, 65, 10, 0, 0], [-548, 0, -133, 0, 26, -27, 0, 0], [0, -167, 0, 41, 0, 0, 68, -64], [0, -19, 0, 53, 0, 0, 41, -140], [-696, 0, 5, 0, 32, 6, 0, 0], [0, 69, 0, 12, 0, 0, 44, -50], [0, -174, 0, -4, 0, 0, 97, -15], [0, -101, 0, 78, 0, 0, 136, 18], [-550, 0, 135, 0, -81, -50, 0, 0], [-140, 0, 5, 0, 130, -41, 0, 0], [0, -216, 0, 146, 0, 0, -77, -59], [0, -335, 0, -58, 0, 0, -60, -70], [30, 0, -56, 0, -48, -32, 0, 0], [-252, 0, -95, 0, 50, -53, 0, 0], [0, -248, 0, 230, 0, 0, -35, -53], [548, 0, 125, 0, 82, -53, 0, 0], [0, -412, 0, -46, 0, 0, -43, -1], [0, -227, 0, -103, 0, 0, -65, -114], [1230, 0, 21, 0, 13, -10, 0, 0], [-242, 0, -40, 0, 56, -48, 0, 0], [0, -21, 0, -127, 0, 0, -46, -42], [-224, 0, -3, 0, -20, 104, 0, 0], [386, 0, -41, 0, 127, 74, 0, 0], [0, 167, 0, 83, 0, 0, -47, -26], [-380, 0, -80, 0, 180, -9, 0, 0], [302, 0, 141, 0, 85, -42, 0, 0], [0, 272, 0, 166, 0, 0, 73, 39], [0, -501, 0, -63, 0, 0, 36, 7], [0, 175, 0, -54, 0, 0, -48, -62], [-558, 0, 26, 0, -62, 48, 0, 0], [0, -491, 0, 163, 0, 0, 10, -26], [568, 0, 19, 0, 92, -6, 0, 0], [0, 197, 0, -46, 0, 0, -124, 18], [0, -116, 0, -124, 0, 0, 76, -16], [-530, 0, -56, 0, -104, 0, 0, 0], [0, 87, 0, 61, 0, 0, 0, -137], [570, 0, -44, 0, -92, 60, 0, 0], [656, 0, 37, 0, -184, 4, 0, 0], [0, -63, 0, -137, 0, 0, 24, 168], [0, -573, 0, 107, 0, 0, -77, 24], [0, 153, 0, -6, 0, 0, 40, -98], [0, 259, 0, -15, 0, 0, 104, 75], [596, 0, 16, 0, 28, -61, 0, 0], [0, 107, 0, -56, 0, 0, 20, 42], [-758, 0, 140, 0, -52, -12, 0, 0], [0, 520, 0, -190, 0, 0, 43, -3], [0, -315, 0, -149, 0, 0, -82, 66], [0, -311, 0, 0, 0, 0, 192, 2], [384, 0, 113, 0, -80, 56, 0, 0], [666, 0, -10, 0, -30, 42, 0, 0], [0, 120, 0, 84, 0, 0, 114, 86], [-254, 0, -60, 0, 132, -16, 0, 0], [400, 0, -175, 0, -44, -20, 0, 0], [-336, 0, 53, 0, 24, -36, 0, 0], [0, 123, 0, -123, 0, 0, -66, -110], [448, 0, -153, 0, -144, -48, 0, 0], [954, 0, -60, 0, -116, -84, 0, 0], [-442, 0, 144, 0, -148, -34, 0, 0], [0, -167, 0, 32, 0, 0, -100, -42], [-938, 0, -96, 0, -32, 68, 0, 0], [0, 100, 0, -6, 0, 0, 13, 7], [0, -153, 0, -23, 0, 0, -124, -68], [0, 141, 0, 199, 0, 0, 24, -19], [0, -53, 0, -188, 0, 0, -128, -90], [702, 0, -8, 0, -164, -94, 0, 0], [116, 0, 110, 0, 120, 83, 0, 0], [0, 91, 0, 111, 0, 0, 50, -94], [0, -123, 0, 223, 0, 0, -76, -207], [852, 0, 162, 0, -16, 43, 0, 0], [90, 0, -150, 0, 130, -72, 0, 0], [20, 0, 205, 0, -70, -41, 0, 0], [398, 0, -20, 0, -104, 22, 0, 0], [0, 113, 0, -189, 0, 0, 84, -108], [0, 971, 0, -73, 0, 0, 9, -38], [-336, 0, 5, 0, 116, 36, 0, 0], [0, 146, 0, -104, 0, 0, -29, -81], [-146, 0, 172, 0, -44, 84, 0, 0], [0, -177, 0, 97, 0, 0, -106, -111], [-138, 0, -157, 0, -61, 78, 0, 0], [0, 272, 0, 76, 0, 0, 166, 26], [0, 207, 0, -127, 0, 0, 68, 44], [0, 235, 0, 188, 0, 0, -16, 38], [0, -177, 0, -141, 0, 0, -187, 90], [-586, 0, -23, 0, 1, -134, 0, 0], [0, -7, 0, 27, 0, 0, 108, 205], [324, 0, -56, 0, 124, -9, 0, 0], [0, -93, 0, -58, 0, 0, -92, 54], [-110, 0, 98, 0, -78, -20, 0, 0], [0, 218, 0, 250, 0, 0, 186, -144], [0, 851, 0, 43, 0, 0, -33, -76], [936, 0, -9, 0, 68, -98, 0, 0], [0, 133, 0, -244, 0, 0, -72, -110], [-724, 0, 13, 0, -66, 41, 0, 0], [0, 261, 0, -61, 0, 0, -26, -9], [-998, 0, -56, 0, -88, -60, 0, 0], [1588, 0, -59, 0, 170, 47, 0, 0], [0, 498, 0, 250, 0, 0, 166, -68], [-558, 0, 18, 0, -86, -24, 0, 0], [0, -343, 0, 31, 0, 0, -76, -52], [-604, 0, 44, 0, 52, -33, 0, 0], [0, -58, 0, 134, 0, 0, 26, 188], [836, 0, 122, 0, 144, -1, 0, 0], [40, 0, -241, 0, -8, 14, 0, 0], [0, -179, 0, 13, 0, 0, 41, -20], [0, 229, 0, 114, 0, 0, 0, -218], [118, 0, -112, 0, -24, -108, 0, 0], [0, -334, 0, 192, 0, 0, -33, 3], [0, -641, 0, 289, 0, 0, 50, -51], [328, 0, -91, 0, -168, 38, 0, 0], [616, 0, -141, 0, 72, 78, 0, 0], [0, 252, 0, -296, 0, 0, -194, 46], [0, 317, 0, 60, 0, 0, 76, -34], [-2026, 0, 18, 0, -6, 56, 0, 0], [0, 1022, 0, -30, 0, 0, -16, -2], [-594, 0, -12, 0, 244, -32, 0, 0], [0, 363, 0, -117, 0, 0, -54, -78], [584, 0, -135, 0, 40, 86, 0, 0], [0, -317, 0, -236, 0, 0, -52, 90], [1336, 0, 49, 0, -12, -6, 0, 0], [0, 61, 0, -7, 0, 0, -138, -106], [1038, 0, -164, 0, -108, 12, 0, 0], [0, 753, 0, 19, 0, 0, 64, 9], [576, 0, 121, 0, 92, 64, 0, 0], [0, -653, 0, 28, 0, 0, -104, 110], [0, -563, 0, -78, 0, 0, -168, -50], [1766, 0, 129, 0, -79, 114, 0, 0], [1288, 0, 95, 0, -80, 22, 0, 0], [0, 351, 0, -139, 0, 0, 120, 128], [650, 0, -62, 0, -246, 48, 0, 0], [268, 0, 99, 0, -182, -103, 0, 0], [0, 177, 0, 103, 0, 0, -184, 56], [0, -943, 0, 6, 0, 0, 72, 150], [-936, 0, 17, 0, 4, -14, 0, 0], [174, 0, -40, 0, -232, -24, 0, 0], [0, -121, 0, 85, 0, 0, -96, 239], [306, 0, -208, 0, -16, 64, 0, 0], [0, -441, 0, 403, 0, 0, 9, -50], [-476, 0, 218, 0, 0, -89, 0, 0], [0, 245, 0, -242, 0, 0, 180, -86], [-822, 0, 44, 0, -288, -30, 0, 0], [-632, 0, -15, 0, 120, -122, 0, 0], [0, 602, 0, -102, 0, 0, -22, 64], [0, 75, 0, -257, 0, 0, 125, -26], [0, 138, 0, -218, 0, 0, 80, -102], [0, 107, 0, -22, 0, 0, -132, 102], [-708, 0, -89, 0, -334, 13, 0, 0], [0, 268, 0, 64, 0, 0, -54, -62], [518, 0, -394, 0, -6, -18, 0, 0], [0, 673, 0, -157, 0, 0, -4, 92], [0, 1193, 0, 37, 0, 0, 43, 14], [438, 0, 110, 0, -290, -16, 0, 0], [0, 91, 0, 5, 0, 0, -110, 190], [240, 0, 257, 0, 44, -60, 0, 0], [1122, 0, 138, 0, -230, -12, 0, 0], [-1872, 0, -83, 0, 156, 20, 0, 0], [0, -95, 0, -102, 0, 0, -112, 102], [0, -733, 0, 321, 0, 0, 86, -126], [0, 597, 0, -237, 0, 0, -126, 83], [0, -95, 0, -67, 0, 0, -116, 16], [-716, 0, -166, 0, -8, 67, 0, 0], [-156, 0, 155, 0, 78, 91, 0, 0], [-238, 0, -148, 0, -320, 58, 0, 0], [0, 227, 0, 269, 0, 0, -66, 98], [0, 7, 0, -27, 0, 0, 236, -245]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2304_b_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2304_3_b_q();".
// 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_b_q(:prec:=8)
    chi := MakeCharacter_2304_b();
    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_b_q();".
// 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_b_q( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2304_b();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<5,R![16384, 0, 12288, 0, 2048, 0, 96, 0, 1]>,<7,R![135424, 0, 108288, 0, 13664, 0, 240, 0, 1]>,<11,R![-80, 8, 1]>,<17,R![70288, -9824, -904, 8, 1]>,<19,R![-149504, 12288, 320, -48, 1]>],Snew);
    return Vf;
end function;