
// Make newform 1089.3.b.f in Magma, downloaded from the LMFDB on 18 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1089_b();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1089_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_1089_3_b_f();".
// 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_1089_3_b_f();".
// 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 := [121, 0, -80, 0, 71, 0, -14, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0, 0, 0], [70, 0, -93, 0, 19, 0, -2, 0], [-371, 0, -40, 0, 38, 0, -4, 0], [0, -1623, 0, 714, 0, -160, 0, 13], [782, 0, -1016, 0, 206, 0, -14, 0], [0, 8248, 0, -3535, 0, 493, 0, -25], [0, -51, 0, 2142, 0, -480, 0, 39], [0, 118, 0, -2547, 0, 591, 0, -43]];
    Rf_basisdens := [1, 73, 73, 803, 73, 803, 803, 803];
    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_1089_b();"
function MakeCharacter_1089_b()
    N := 1089;
    order := 2;
    char_gens := [848, 244];
    v := [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_1089_b_Hecke();"
function MakeCharacter_1089_b_Hecke(Kf)
    N := 1089;
    order := 2;
    char_gens := [848, 244];
    char_values := [[-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, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -1, -1], [0, 0, 0, 1, 0, 1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1, 0, 0], [0, -4, 0, 0, -1, 0, 0, 0], [0, 0, 0, -5, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 5], [0, -10, 0, 0, 0, 0, 0, 0], [-25, 0, -1, 0, 0, 0, 0, 0], [-15, 0, -5, 0, 0, 0, 0, 0], [0, 10, 0, 0, -5, 0, 0, 0], [0, 0, 0, 5, 0, -6, 0, 0], [0, 0, 0, 0, 0, 0, -10, 5], [0, 0, 0, 0, 0, 0, 5, -15], [0, 0, 0, 0, 0, 0, -2, 0], [0, 0, 0, -20, 0, 5, 0, 0], [20, 0, -10, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 8, 25], [0, 0, 0, 14, 0, 4, 0, 0], [0, 0, 0, -15, 0, 5, 0, 0], [0, -16, 0, 0, -5, 0, 0, 0], [0, 0, 0, 0, 0, 0, -15, 10], [-55, 0, 15, 0, 0, 0, 0, 0], [0, -10, 0, 0, 5, 0, 0, 0], [-15, 0, 5, 0, 0, 0, 0, 0], [0, -26, 0, 0, 10, 0, 0, 0], [0, 0, 0, -30, 0, 5, 0, 0], [0, 0, 0, 0, 0, 0, -35, 0], [0, 0, 0, -15, 0, -11, 0, 0], [0, 10, 0, 0, -5, 0, 0, 0], [0, 0, 0, 0, 0, 0, 5, 10], [0, 0, 0, 15, 0, -20, 0, 0], [0, -10, 0, 0, 15, 0, 0, 0], [0, 0, 0, 45, 0, 15, 0, 0], [30, 0, -20, 0, 0, 0, 0, 0], [-150, 0, -10, 0, 0, 0, 0, 0], [0, -14, 0, 0, -15, 0, 0, 0], [0, -54, 0, 0, -6, 0, 0, 0], [0, 0, 0, 0, 0, 0, -2, 38], [-175, 0, -21, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 40, 15], [0, 0, 0, 10, 0, -4, 0, 0], [0, -74, 0, 0, 9, 0, 0, 0], [20, 0, 50, 0, 0, 0, 0, 0], [0, 0, 0, -65, 0, -20, 0, 0], [200, 0, 30, 0, 0, 0, 0, 0], [0, 46, 0, 0, -6, 0, 0, 0], [190, 0, 0, 0, 0, 0, 0, 0], [0, -74, 0, 0, -5, 0, 0, 0], [0, 10, 0, 0, 25, 0, 0, 0], [0, 0, 0, 40, 0, 30, 0, 0], [0, 0, 0, 0, 0, 0, -18, 0], [0, 0, 0, 0, 0, 0, 45, -40], [0, -74, 0, 0, -11, 0, 0, 0], [0, 0, 0, 0, 0, 0, 37, -25], [0, 0, 0, 45, 0, 15, 0, 0], [0, 0, 0, 36, 0, 31, 0, 0], [0, -100, 0, 0, -15, 0, 0, 0], [0, 0, 0, -81, 0, 14, 0, 0], [0, -46, 0, 0, 30, 0, 0, 0], [0, 0, 0, 1, 0, -34, 0, 0], [0, 0, 0, 0, 0, 0, -52, -75], [-330, 0, 10, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 45, 75], [-300, 0, 44, 0, 0, 0, 0, 0], [0, 0, 0, -20, 0, 4, 0, 0], [0, 4, 0, 0, -9, 0, 0, 0], [0, 0, 0, 60, 0, 5, 0, 0], [0, 0, 0, 0, 0, 0, -15, -110], [0, -120, 0, 0, -30, 0, 0, 0], [170, 0, 60, 0, 0, 0, 0, 0], [0, 0, 0, 36, 0, -49, 0, 0], [-30, 0, 50, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -20, 45], [0, 0, 0, 0, 0, 0, 127, 25], [-345, 0, 5, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 13, -72], [0, 0, 0, -30, 0, 40, 0, 0], [0, 0, 0, 0, 0, 0, 128, -32], [-275, 0, -9, 0, 0, 0, 0, 0], [0, -10, 0, 0, -15, 0, 0, 0], [-260, 0, -80, 0, 0, 0, 0, 0], [0, 0, 0, -15, 0, 15, 0, 0], [0, 0, 0, 0, 0, 0, -20, 90], [0, 0, 0, 0, 0, 0, 55, 30], [0, 0, 0, -24, 0, -44, 0, 0], [0, 150, 0, 0, -60, 0, 0, 0], [-5, 0, -5, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -10, 40], [0, 50, 0, 0, -15, 0, 0, 0], [-60, 0, -50, 0, 0, 0, 0, 0], [0, 150, 0, 0, 0, 0, 0, 0], [-548, 0, 0, 0, 0, 0, 0, 0], [0, 66, 0, 0, 9, 0, 0, 0], [0, 0, 0, 0, 0, 0, -87, -75], [0, 0, 0, 0, 0, 0, 127, 12], [0, 0, 0, 89, 0, -56, 0, 0], [0, 0, 0, -30, 0, -65, 0, 0], [0, 0, 0, 171, 0, 6, 0, 0], [0, -114, 0, 0, 30, 0, 0, 0], [0, 104, 0, 0, -10, 0, 0, 0], [0, -100, 0, 0, 15, 0, 0, 0], [0, 0, 0, 75, 0, 10, 0, 0], [-395, 0, 15, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -100, -80], [0, -124, 0, 0, 15, 0, 0, 0], [0, 0, 0, 0, 0, 0, -138, -3], [0, 0, 0, -90, 0, -110, 0, 0], [0, 0, 0, 85, 0, -29, 0, 0], [0, 0, 0, 80, 0, 31, 0, 0], [0, 0, 0, 0, 0, 0, -5, 190], [200, 0, -6, 0, 0, 0, 0, 0], [765, 0, 25, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -33, -48], [460, 0, -110, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 90, -45], [0, 0, 0, 0, 0, 0, -25, -65], [0, -220, 0, 0, 0, 0, 0, 0], [780, 0, -50, 0, 0, 0, 0, 0], [0, 0, 0, 214, 0, -66, 0, 0], [0, 106, 0, 0, 45, 0, 0, 0], [0, 0, 0, 0, 0, 0, -130, -100], [-482, 0, 100, 0, 0, 0, 0, 0], [0, 110, 0, 0, 50, 0, 0, 0], [-100, 0, 14, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 20, 195], [-410, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 106, 0, 1, 0, 0], [0, 0, 0, 95, 0, 20, 0, 0], [0, -84, 0, 0, 54, 0, 0, 0], [27, 0, -125, 0, 0, 0, 0, 0], [-655, 0, 95, 0, 0, 0, 0, 0], [0, 280, 0, 0, -75, 0, 0, 0], [0, 0, 0, 100, 0, 120, 0, 0], [0, 0, 0, 0, 0, 0, 75, 245], [0, 0, 0, 255, 0, 6, 0, 0], [0, 0, 0, 0, 0, 0, 95, 165], [0, -30, 0, 0, -75, 0, 0, 0], [0, 0, 0, -125, 0, 0, 0, 0], [0, -110, 0, 0, -5, 0, 0, 0], [1110, 0, -40, 0, 0, 0, 0, 0], [0, -184, 0, 0, 69, 0, 0, 0], [500, 0, 6, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -108, 25], [0, 0, 0, 154, 0, 119, 0, 0], [0, -54, 0, 0, 45, 0, 0, 0], [-300, 0, -196, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -150, -215], [0, 0, 0, 140, 0, 161, 0, 0], [0, 0, 0, 0, 0, 0, -137, 0], [1050, 0, 100, 0, 0, 0, 0, 0], [0, -156, 0, 0, 36, 0, 0, 0], [370, 0, -140, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 268, 183], [0, 0, 0, -95, 0, -25, 0, 0], [0, 0, 0, 0, 0, 0, 37, -150], [0, 0, 0, -30, 0, -56, 0, 0], [0, -290, 0, 0, 85, 0, 0, 0], [0, 0, 0, 0, 0, 0, -20, -80], [0, 46, 0, 0, -91, 0, 0, 0], [0, 0, 0, 75, 0, -9, 0, 0], [0, 0, 0, 0, 0, 0, 48, -162], [0, 0, 0, 0, 0, 0, -45, -30], [0, 0, 0, 0, 0, 0, -280, 105], [-425, 0, 171, 0, 0, 0, 0, 0], [0, 0, 0, -210, 0, 21, 0, 0], [0, 0, 0, 310, 0, -100, 0, 0], [0, 0, 0, 0, 0, 0, 15, 315], [0, 310, 0, 0, -35, 0, 0, 0], [425, 0, -129, 0, 0, 0, 0, 0], [0, -380, 0, 0, -90, 0, 0, 0], [0, 0, 0, 140, 0, -126, 0, 0], [50, 0, -204, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 37, 50], [0, 0, 0, -125, 0, 130, 0, 0], [0, 0, 0, 0, 0, 0, 93, 75], [0, 0, 0, 225, 0, -49, 0, 0], [0, 0, 0, -290, 0, -5, 0, 0], [365, 0, 125, 0, 0, 0, 0, 0], [0, 350, 0, 0, -55, 0, 0, 0], [-285, 0, 25, 0, 0, 0, 0, 0], [0, 26, 0, 0, -15, 0, 0, 0], [0, 0, 0, 0, 0, 0, -170, -15], [0, 0, 0, 0, 0, 0, -205, -105], [0, 0, 0, 94, 0, 59, 0, 0], [-1330, 0, -60, 0, 0, 0, 0, 0], [0, 0, 0, 20, 0, 90, 0, 0], [0, -130, 0, 0, -15, 0, 0, 0], [45, 0, -155, 0, 0, 0, 0, 0], [0, -356, 0, 0, 61, 0, 0, 0], [8, 0, -100, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 303, 75], [0, -436, 0, 0, -70, 0, 0, 0], [0, 0, 0, 0, 0, 0, 355, -140], [0, 0, 0, -10, 0, 30, 0, 0], [-565, 0, 165, 0, 0, 0, 0, 0], [0, 474, 0, 0, 51, 0, 0, 0], [0, 184, 0, 0, 90, 0, 0, 0], [0, -50, 0, 0, 95, 0, 0, 0], [0, 0, 0, -185, 0, -95, 0, 0], [-765, 0, -55, 0, 0, 0, 0, 0], [0, 0, 0, 340, 0, -160, 0, 0], [0, 0, 0, 0, 0, 0, -440, 60], [0, 0, 0, 0, 0, 0, -185, -215], [-1017, 0, -25, 0, 0, 0, 0, 0], [0, 114, 0, 0, 51, 0, 0, 0], [0, -540, 0, 0, -45, 0, 0, 0], [0, 0, 154, 0, 0, 0, 0, 0], [0, 0, 0, -46, 0, 164, 0, 0], [0, 0, 0, 0, 0, 0, 375, -175], [585, 0, 265, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -210, 210], [0, -640, 0, 0, -100, 0, 0, 0], [465, 0, 25, 0, 0, 0, 0, 0], [0, 0, 0, 539, 0, -91, 0, 0], [0, -60, 0, 0, -75, 0, 0, 0], [0, 0, 0, 0, 0, 0, 140, -245], [0, 0, 0, 0, 0, 0, -155, 255], [0, 0, 0, -160, 0, -55, 0, 0], [0, 0, 0, -145, 0, -95, 0, 0], [0, 0, 0, 0, 0, 0, -373, -50], [2070, 0, 20, 0, 0, 0, 0, 0], [0, -284, 0, 0, 94, 0, 0, 0], [0, 0, 0, -110, 0, 205, 0, 0], [0, 0, 0, 0, 0, 0, 105, 130], [0, 0, 0, 0, 0, 0, 88, 125], [0, 0, 0, 321, 0, -59, 0, 0], [0, 170, 0, 0, 80, 0, 0, 0], [950, 0, -60, 0, 0, 0, 0, 0], [0, 0, 0, 15, 0, 110, 0, 0], [0, 0, 0, -255, 0, -125, 0, 0], [0, -80, 0, 0, 25, 0, 0, 0], [140, 0, -280, 0, 0, 0, 0, 0], [0, -484, 0, 0, -136, 0, 0, 0], [-150, 0, 34, 0, 0, 0, 0, 0], [0, 386, 0, 0, -70, 0, 0, 0], [0, 0, 0, 0, 0, 0, 12, 0], [0, 0, 0, 0, 0, 0, -290, 235], [0, 0, 0, 0, 0, 0, -350, -200], [0, 0, 0, -75, 0, 30, 0, 0], [35, 0, -385, 0, 0, 0, 0, 0], [-350, 0, -84, 0, 0, 0, 0, 0], [0, 464, 0, 0, 125, 0, 0, 0], [0, -300, 0, 0, 120, 0, 0, 0], [610, 0, -180, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -318, -300], [0, 0, 0, -195, 0, 190, 0, 0], [0, -704, 0, 0, 0, 0, 0, 0], [0, 0, 0, 304, 0, 119, 0, 0], [0, -430, 0, 0, 45, 0, 0, 0], [0, 0, 0, 0, 0, 0, 220, -135], [-1095, 0, -125, 0, 0, 0, 0, 0], [0, 394, 0, 0, 125, 0, 0, 0], [0, -280, 0, 0, -175, 0, 0, 0], [-600, 0, -46, 0, 0, 0, 0, 0], [0, 0, 0, -111, 0, 4, 0, 0], [0, 0, 0, 0, 0, 0, -135, -255], [0, 0, 0, 310, 0, 46, 0, 0], [0, 0, 0, -309, 0, -169, 0, 0], [0, -354, 0, 0, -75, 0, 0, 0], [0, 0, 0, 150, 0, 255, 0, 0], [0, 0, 0, -124, 0, -69, 0, 0], [0, 0, 0, 0, 0, 0, 115, -190], [1900, 0, -150, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -33, -223], [0, 0, 0, 0, 0, 0, -217, 450], [0, 0, 0, -405, 0, -226, 0, 0], [0, -494, 0, 0, -110, 0, 0, 0], [-455, 0, 175, 0, 0, 0, 0, 0], [290, 0, 0, 0, 0, 0, 0, 0], [785, 0, 65, 0, 0, 0, 0, 0], [0, 0, 0, -315, 0, -315, 0, 0], [0, 0, 0, -486, 0, -86, 0, 0], [1390, 0, 120, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -540, -170], [0, 0, 0, -30, 0, 185, 0, 0], [0, 0, 0, -300, 0, 50, 0, 0], [0, 160, 0, 0, -50, 0, 0, 0], [0, -1026, 0, 0, -75, 0, 0, 0], [100, 0, -234, 0, 0, 0, 0, 0], [0, 836, 0, 0, 50, 0, 0, 0], [0, 0, 0, -270, 0, -285, 0, 0], [0, 0, 0, 45, 0, 54, 0, 0], [0, 0, 0, 0, 0, 0, 302, 387], [55, 0, -395, 0, 0, 0, 0, 0], [0, 586, 0, 0, -6, 0, 0, 0], [-1700, 0, 174, 0, 0, 0, 0, 0], [0, 100, 0, 0, 5, 0, 0, 0], [0, 0, 0, 0, 0, 0, 3, -625], [0, 0, 0, 0, 0, 0, 140, 180], [0, 636, 0, 0, -111, 0, 0, 0], [0, -180, 0, 0, 90, 0, 0, 0], [0, 0, 0, 194, 0, -131, 0, 0], [0, -190, 0, 0, -55, 0, 0, 0], [1400, 0, 350, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 195, 745], [0, -60, 0, 0, -105, 0, 0, 0], [0, 0, 0, 575, 0, 224, 0, 0], [0, 0, 0, 124, 0, 84, 0, 0], [0, 834, 0, 0, -84, 0, 0, 0], [0, 0, 0, 425, 0, -135, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1089_b_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1089_3_b_f();".
// 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_1089_3_b_f(:prec:=8)
    chi := MakeCharacter_1089_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_1089_3_b_f();".
// 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_1089_3_b_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1089_b();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<2,R![27, 0, 16, 0, 1]>,<7,R![5292, 0, -146, 0, 1]>],Snew);
    return Vf;
end function;