
// Make newform 2304.3.b.s in Magma, downloaded from the LMFDB on 23 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_s();".
// 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_s();".
// 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 := [16, 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, 5, 0, 0, 0, 1, 0], [8, -4, 0, 7, 16, 4, 0, -1], [0, 4, 0, 7, 0, -4, 0, -1], [0, 0, 3, 0, 0, 0, -1, 0], [-8, -12, 0, 21, -16, 12, 0, -3], [0, 44, 0, 13, 0, 4, 0, 5], [0, 44, 0, -13, 0, 4, 0, -5]];
    Rf_basisdens := [1, 6, 12, 6, 1, 12, 12, 6];
    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, 0, 1, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0, -1, 0], [0, 0, 0, 0, 1, 0, 0, 0], [0, 3, 0, 0, 0, 0, 2, 0], [0, 0, 0, -1, 1, 0, 0, 0], [12, 0, 0, 0, 0, 0, 0, 1], [0, 0, -5, 0, 0, -1, 0, 0], [0, 0, -1, 0, 0, -2, 0, 0], [0, 9, 0, 0, 0, 0, 1, 0], [0, -1, 0, 0, 0, 0, 2, 0], [0, 0, 0, 7, 1, 0, 0, 0], [20, 0, 0, 0, 0, 0, 0, -1], [0, 0, -7, 0, 0, 5, 0, 0], [0, 0, 1, 0, 0, 6, 0, 0], [0, 0, 0, -8, 2, 0, 0, 0], [0, -7, 0, 0, 0, 0, -2, 0], [24, 0, 0, 0, 0, 0, 0, -6], [0, 0, -8, 0, 0, -8, 0, 0], [38, 0, 0, 0, 0, 0, 0, 6], [0, 29, 0, 0, 0, 0, -3, 0], [0, 0, 0, 16, 1, 0, 0, 0], [0, 0, 0, 14, -6, 0, 0, 0], [14, 0, 0, 0, 0, 0, 0, 4], [0, 0, 11, 0, 0, 8, 0, 0], [0, 57, 0, 0, 0, 0, 1, 0], [0, 0, 0, -24, -2, 0, 0, 0], [0, -5, 0, 0, 0, 0, -14, 0], [0, 0, 0, -2, 18, 0, 0, 0], [0, 67, 0, 0, 0, 0, -5, 0], [0, 0, 0, 0, -16, 0, 0, 0], [0, 0, 0, 21, 11, 0, 0, 0], [120, 0, 0, 0, 0, 0, 0, -6], [0, 0, 21, 0, 0, 0, 0, 0], [0, 35, 0, 0, 0, 0, -21, 0], [0, -3, 0, 0, 0, 0, 14, 0], [172, 0, 0, 0, 0, 0, 0, -7], [0, 0, 27, 0, 0, -1, 0, 0], [0, 0, 17, 0, 0, -6, 0, 0], [0, 0, 0, 8, -2, 0, 0, 0], [0, 73, 0, 0, 0, 0, -2, 0], [0, 0, -17, 0, 0, 3, 0, 0], [-18, 0, 0, 0, 0, 0, 0, -10], [0, 0, -19, 0, 0, 10, 0, 0], [0, 133, 0, 0, 0, 0, 13, 0], [168, 0, 0, 0, 0, 0, 0, -2], [0, 91, 0, 0, 0, 0, 11, 0], [0, 0, 0, 0, 23, 0, 0, 0], [0, -55, 0, 0, 0, 0, 46, 0], [0, 0, 0, 10, -2, 0, 0, 0], [0, 0, 13, 0, 0, -23, 0, 0], [-14, 0, 0, 0, 0, 0, 0, -6], [0, 0, 0, 16, 21, 0, 0, 0], [0, 0, 0, 24, -16, 0, 0, 0], [0, 0, -48, 0, 0, 16, 0, 0], [0, 0, -43, 0, 0, 12, 0, 0], [0, 35, 0, 0, 0, 0, -45, 0], [0, 177, 0, 0, 0, 0, 14, 0], [0, 0, 0, -40, -16, 0, 0, 0], [256, 0, 0, 0, 0, 0, 0, 8], [0, 0, 13, 0, 0, -30, 0, 0], [120, 0, 0, 0, 0, 0, 0, -14], [0, 0, 22, 0, 0, 30, 0, 0], [2, 0, 0, 0, 0, 0, 0, 10], [0, 0, 17, 0, 0, 20, 0, 0], [16, 0, 0, 0, 0, 0, 0, 12], [-6, 0, 0, 0, 0, 0, 0, 2], [0, 0, 0, 64, -21, 0, 0, 0], [0, -63, 0, 0, 0, 0, 14, 0], [0, 0, 0, 50, 22, 0, 0, 0], [0, 0, -36, 0, 0, 12, 0, 0], [0, 9, 0, 0, 0, 0, 49, 0], [0, 211, 0, 0, 0, 0, -30, 0], [20, 0, 0, 0, 0, 0, 0, -17], [0, 0, -29, 0, 0, -25, 0, 0], [0, 0, -45, 0, 0, -38, 0, 0], [0, 125, 0, 0, 0, 0, -18, 0], [0, 0, 0, 45, -29, 0, 0, 0], [-46, 0, 0, 0, 0, 0, 0, -20], [0, 0, 0, -112, -1, 0, 0, 0], [0, -311, 0, 0, 0, 0, -18, 0], [0, 0, 1, 0, 0, 13, 0, 0], [162, 0, 0, 0, 0, 0, 0, 28], [0, -91, 0, 0, 0, 0, 61, 0], [0, 0, 0, 32, 27, 0, 0, 0], [0, 0, 0, 23, 33, 0, 0, 0], [226, 0, 0, 0, 0, 0, 0, 12], [0, 0, -87, 0, 0, 22, 0, 0], [0, -53, 0, 0, 0, 0, -29, 0], [0, 0, 0, 32, -5, 0, 0, 0], [0, 0, 24, 0, 0, -40, 0, 0], [0, 19, 0, 0, 0, 0, 19, 0], [0, 0, 0, 0, -40, 0, 0, 0], [-352, 0, 0, 0, 0, 0, 0, -32], [0, 0, -34, 0, 0, -26, 0, 0], [0, 0, -25, 0, 0, 50, 0, 0], [0, 0, 0, 35, -59, 0, 0, 0], [100, 0, 0, 0, 0, 0, 0, 19], [0, -209, 0, 0, 0, 0, 2, 0], [-812, 0, 0, 0, 0, 0, 0, -17], [0, 0, 43, 0, 0, -40, 0, 0], [0, 0, 0, -48, -45, 0, 0, 0], [0, 0, 0, -21, -35, 0, 0, 0], [-24, 0, 0, 0, 0, 0, 0, 54], [150, 0, 0, 0, 0, 0, 0, -26], [0, 0, 0, 0, 8, 0, 0, 0], [0, 0, 0, -182, -10, 0, 0, 0], [0, 0, 33, 0, 0, 45, 0, 0], [314, 0, 0, 0, 0, 0, 0, 10], [0, -293, 0, 0, 0, 0, -29, 0], [0, -357, 0, 0, 0, 0, 34, 0], [0, 0, 0, -28, -20, 0, 0, 0], [-528, 0, 0, 0, 0, 0, 0, 36], [0, -133, 0, 0, 0, 0, 27, 0], [0, 0, 0, 15, 65, 0, 0, 0], [-188, 0, 0, 0, 0, 0, 0, 11], [0, 0, -55, 0, 0, -75, 0, 0], [0, 0, -99, 0, 0, -30, 0, 0], [0, 0, 0, 16, 12, 0, 0, 0], [0, 467, 0, 0, 0, 0, -14, 0], [-414, 0, 0, 0, 0, 0, 0, 12], [0, 0, 81, 0, 0, 36, 0, 0], [0, 0, 0, -160, 25, 0, 0, 0], [-1108, 0, 0, 0, 0, 0, 0, 9], [0, 0, -51, 0, 0, -108, 0, 0], [0, -399, 0, 0, 0, 0, 62, 0], [0, 0, -62, 0, 0, 58, 0, 0], [0, -465, 0, 0, 0, 0, 7, 0], [0, -209, 0, 0, 0, 0, -78, 0], [-200, 0, 0, 0, 0, 0, 0, -30], [0, 0, -33, 0, 0, -45, 0, 0], [0, -127, 0, 0, 0, 0, 97, 0], [0, 329, 0, 0, 0, 0, -66, 0], [0, 0, 0, 53, 3, 0, 0, 0], [638, 0, 0, 0, 0, 0, 0, 14], [0, 0, 83, 0, 0, 96, 0, 0], [-188, 0, 0, 0, 0, 0, 0, 35], [0, 0, 27, 0, 0, 20, 0, 0], [0, 0, 0, 41, -33, 0, 0, 0], [-532, 0, 0, 0, 0, 0, 0, -47], [0, 0, -125, 0, 0, 60, 0, 0], [0, 91, 0, 0, 0, 0, 75, 0], [0, 0, 0, -152, 30, 0, 0, 0], [0, -93, 0, 0, 0, 0, 146, 0], [0, 0, -144, 0, 0, 48, 0, 0], [0, 403, 0, 0, 0, 0, -94, 0], [0, 0, 0, -21, -91, 0, 0, 0], [-196, 0, 0, 0, 0, 0, 0, 53], [0, 0, 17, 0, 0, 125, 0, 0], [0, 49, 0, 0, 0, 0, -66, 0], [0, 0, 0, -96, 24, 0, 0, 0], [636, 0, 0, 0, 0, 0, 0, 61], [0, 0, -202, 0, 0, 30, 0, 0], [-500, 0, 0, 0, 0, 0, 0, 65], [0, 0, 125, 0, 0, 25, 0, 0], [0, 155, 0, 0, 0, 0, -21, 0], [0, 0, 0, 73, -33, 0, 0, 0], [-722, 0, 0, 0, 0, 0, 0, -40], [0, 0, -19, 0, 0, 50, 0, 0], [0, 0, 0, 56, -22, 0, 0, 0], [0, 0, 0, 247, 17, 0, 0, 0], [0, -59, 0, 0, 0, 0, 53, 0], [0, 0, 0, -144, 41, 0, 0, 0], [0, 0, 0, 133, 83, 0, 0, 0], [0, 0, 134, 0, 0, -50, 0, 0], [0, -135, 0, 0, 0, 0, -151, 0], [0, -425, 0, 0, 0, 0, 34, 0], [-162, 0, 0, 0, 0, 0, 0, -92], [0, 0, -37, 0, 0, 10, 0, 0], [0, 0, 0, 112, -124, 0, 0, 0], [0, -91, 0, 0, 0, 0, -34, 0], [0, 0, -71, 0, 0, 5, 0, 0], [1138, 0, 0, 0, 0, 0, 0, 40], [0, 9, 0, 0, 0, 0, -39, 0], [0, 0, 0, -308, 12, 0, 0, 0], [-944, 0, 0, 0, 0, 0, 0, 4], [0, 0, 77, 0, 0, 32, 0, 0], [0, 573, 0, 0, 0, 0, 21, 0], [0, 321, 0, 0, 0, 0, 174, 0], [0, -379, 0, 0, 0, 0, 5, 0], [0, 0, 0, -112, 73, 0, 0, 0], [0, -451, 0, 0, 0, 0, 30, 0], [0, 0, 0, 32, -8, 0, 0, 0], [0, 0, 27, 0, 0, -65, 0, 0], [0, 0, -129, 0, 0, -90, 0, 0], [0, -497, 0, 0, 0, 0, 2, 0], [192, 0, 0, 0, 0, 0, 0, 8], [-978, 0, 0, 0, 0, 0, 0, -18], [0, 0, 46, 0, 0, 22, 0, 0], [814, 0, 0, 0, 0, 0, 0, -48], [0, 0, 0, -184, -106, 0, 0, 0], [336, 0, 0, 0, 0, 0, 0, -60], [0, 0, 133, 0, 0, -2, 0, 0], [0, 0, 0, -48, -100, 0, 0, 0], [0, 0, 0, -56, 24, 0, 0, 0], [-10, 0, 0, 0, 0, 0, 0, -10], [0, 103, 0, 0, 0, 0, 98, 0], [0, 0, 0, 112, -8, 0, 0, 0], [0, 0, -120, 0, 0, -56, 0, 0], [0, 0, 151, 0, 0, 148, 0, 0], [0, 39, 0, 0, 0, 0, 287, 0], [0, 229, 0, 0, 0, 0, -98, 0], [-294, 0, 0, 0, 0, 0, 0, 82], [0, 0, 0, 336, -29, 0, 0, 0], [0, 0, -285, 0, 0, 74, 0, 0], [0, 411, 0, 0, 0, 0, -21, 0], [0, 0, 0, 112, 99, 0, 0, 0], [0, 0, 0, -98, 162, 0, 0, 0], [-308, 0, 0, 0, 0, 0, 0, -119], [-418, 0, 0, 0, 0, 0, 0, -2], [0, 0, 105, 0, 0, -140, 0, 0], [0, -203, 0, 0, 0, 0, -131, 0], [0, 0, 0, 136, 126, 0, 0, 0], [0, 0, 86, 0, 0, 190, 0, 0], [-674, 0, 0, 0, 0, 0, 0, 52], [0, 129, 0, 0, 0, 0, -231, 0], [0, 0, 0, 119, -15, 0, 0, 0], [0, 0, -303, 0, 0, 29, 0, 0], [0, 0, -113, 0, 0, 4, 0, 0], [0, -251, 0, 0, 0, 0, -2, 0], [0, 81, 0, 0, 0, 0, -223, 0], [0, 0, 0, 193, -97, 0, 0, 0], [0, 1087, 0, 0, 0, 0, -49, 0], [0, 0, 0, -112, 141, 0, 0, 0], [0, 147, 0, 0, 0, 0, -30, 0], [0, 0, 0, -42, -230, 0, 0, 0], [0, 0, -110, 0, 0, -150, 0, 0], [0, -587, 0, 0, 0, 0, -139, 0], [0, 0, 0, -32, 72, 0, 0, 0], [0, -485, 0, 0, 0, 0, -142, 0], [1540, 0, 0, 0, 0, 0, 0, -45], [0, 171, 0, 0, 0, 0, 99, 0], [0, 0, 0, -112, 88, 0, 0, 0], [860, 0, 0, 0, 0, 0, 0, 85], [0, 0, -68, 0, 0, -20, 0, 0], [-786, 0, 0, 0, 0, 0, 0, -92], [0, 0, -223, 0, 0, 156, 0, 0], [0, 0, 0, 224, -11, 0, 0, 0], [0, 0, -85, 0, 0, -17, 0, 0], [0, 0, 0, 48, 115, 0, 0, 0], [656, 0, 0, 0, 0, 0, 0, 20], [0, 863, 0, 0, 0, 0, -169, 0], [0, -35, 0, 0, 0, 0, -194, 0], [0, 0, 0, 0, -80, 0, 0, 0], [0, 0, 48, 0, 0, 80, 0, 0], [0, 521, 0, 0, 0, 0, -39, 0], [0, 0, 0, 120, 146, 0, 0, 0], [1568, 0, 0, 0, 0, 0, 0, 48], [0, 0, -32, 0, 0, 160, 0, 0], [0, 395, 0, 0, 0, 0, 130, 0], [0, 0, 0, -186, -22, 0, 0, 0], [0, 0, -11, 0, 0, -15, 0, 0], [258, 0, 0, 0, 0, 0, 0, -152], [0, 0, 15, 0, 0, -54, 0, 0], [0, 0, 0, 56, 106, 0, 0, 0], [0, -107, 0, 0, 0, 0, 78, 0], [-1144, 0, 0, 0, 0, 0, 0, -138], [0, 0, 21, 0, 0, -106, 0, 0], [-402, 0, 0, 0, 0, 0, 0, 84], [0, 831, 0, 0, 0, 0, 215, 0], [0, 0, 0, -56, -234, 0, 0, 0], [0, -131, 0, 0, 0, 0, -18, 0], [0, -59, 0, 0, 0, 0, 190, 0], [0, 0, 0, 77, 107, 0, 0, 0], [-880, 0, 0, 0, 0, 0, 0, 20], [0, 0, -273, 0, 0, 0, 0, 0], [0, 0, 0, -198, -58, 0, 0, 0], [1148, 0, 0, 0, 0, 0, 0, -19], [0, 0, 5, 0, 0, 80, 0, 0], [0, -55, 0, 0, 0, 0, 366, 0], [1176, 0, 0, 0, 0, 0, 0, 2], [1026, 0, 0, 0, 0, 0, 0, -88], [0, 421, 0, 0, 0, 0, -211, 0], [1470, 0, 0, 0, 0, 0, 0, 56], [0, 777, 0, 0, 0, 0, -231, 0], [0, 0, 0, -144, 71, 0, 0, 0], [0, 341, 0, 0, 0, 0, -18, 0], [1146, 0, 0, 0, 0, 0, 0, -150], [0, 0, 0, 360, -10, 0, 0, 0], [0, 0, 226, 0, 0, 186, 0, 0], [0, -243, 0, 0, 0, 0, -139, 0], [0, 0, -321, 0, 0, 51, 0, 0], [0, 11, 0, 0, 0, 0, -158, 0], [-276, 0, 0, 0, 0, 0, 0, 49], [0, 0, 372, 0, 0, 100, 0, 0], [-46, 0, 0, 0, 0, 0, 0, -6], [0, 0, -251, 0, 0, 204, 0, 0], [0, -401, 0, 0, 0, 0, 159, 0], [0, 0, 0, 334, 90, 0, 0, 0], [0, 0, -155, 0, 0, -30, 0, 0], [0, 0, 0, -56, -66, 0, 0, 0], [0, 0, 0, 514, -66, 0, 0, 0], [0, 0, 0, -168, -78, 0, 0, 0], [0, 73, 0, 0, 0, 0, 382, 0], [0, 0, 73, 0, 0, 62, 0, 0], [0, -1329, 0, 0, 0, 0, 127, 0], [0, 0, 7, 0, 0, 204, 0, 0], [0, 0, 0, 32, 143, 0, 0, 0], [-1644, 0, 0, 0, 0, 0, 0, -33], [434, 0, 0, 0, 0, 0, 0, -110], [0, 0, 69, 0, 0, -170, 0, 0], [0, 833, 0, 0, 0, 0, 193, 0]];
    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_s();".
// 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_s(: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_s();".
// 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_s( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2304_b();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<5,R![400, 0, 72, 0, 1]>,<7,R![2704, 0, 120, 0, 1]>,<11,R![-112, 0, 1]>,<17,R![6400, 0, -288, 0, 1]>,<19,R![-80, -24, 1]>],Snew);
    return Vf;
end function;