
// Make newform 4560.2.d.k in Magma, downloaded from the LMFDB on 20 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4560_d();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_4560_d_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_4560_2_d_k();".
// 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_4560_2_d_k();".
// 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 := [64, -320, 800, -784, 324, 104, 232, -128, 36, 4, 8, -4, 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], [-8758151872, 473528496, -1394812968, 2051812292, -913655768, 705909736, -276648264, 110538924, -16487388, 23703524, -9077802, 2101877], [792458944, -3217347232, 1256839928, -2002990488, -647605528, -770790480, 403872146, -172368288, -24324212, -25107184, 13398997, -3940714], [-2348307072, 9480711840, -7939856812, 5438016344, 2085279108, 2700529808, -1529364412, 520946664, 77899386, 93612964, -49457227, 13468838], [-5758911088, 4138399968, -2051473260, -22613183792, -10495968688, -1224384256, 1427235004, -1515562492, -384771120, -50596168, 58054709, -27908578], [-11077367072, 44433332048, -49363607352, 22961507788, 10709015192, 15077086984, -8645816016, 2588973828, 398111644, 528675316, -276433066, 72213471], [-944943624, -2838067024, 7930481560, -10479489340, -4875049852, -2446008600, 1785177452, -863678932, -182811978, -92253612, 59711704, -19601935], [17856922464, 13331224688, -37124235336, 48690475668, 28520993784, 11008990088, -8435149128, 4059889564, 1041253324, 403947252, -282494178, 95220313], [-17273372000, 69664703472, -63031865576, 39327243748, 15554634344, 20456241944, -11577045064, 3871197868, 580599556, 708970492, -373781562, 101159245], [-68354224, 18945108080, -43407293348, 31981959420, 17019026280, 11475993800, -7896232908, 2978874760, 625404004, 423270284, -253862169, 75048949], [-9750878144, -21947888880, 61113087112, -80129584308, -43724741864, -18054944664, 13878896504, -6736017884, -1628227252, -676865564, 464790034, -155955553], [1161504, -4658896, 6211120, -2406636, -1122984, -1582408, 956992, -272116, -41748, -55332, 29316, -7567]];
    Rf_basisdens := [1, 2555640224, 638910056, 638910056, 1277820112, 2555640224, 638910056, 2555640224, 2555640224, 1277820112, 2555640224, 123664];
    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_4560_d();"
function MakeCharacter_4560_d()
    N := 4560;
    order := 2;
    char_gens := [1711, 1141, 3041, 2737, 1921];
    v := [1, 2, 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_4560_d_Hecke();"
function MakeCharacter_4560_d_Hecke(Kf)
    N := 4560;
    order := 2;
    char_gens := [1711, 1141, 3041, 2737, 1921];
    char_values := [[-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 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 := 2;
    raw_aps := [[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 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, 0, 1, 0, -1, 0, 0, -1, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 0, 0, -1, -1, 0, 0, 0, 0, 0, 0, 0], [0, 0, -1, 1, 1, -1, 0, 0, 1, 1, 0, -1], [0, 0, 0, 1, 0, -1, 0, 0, 1, 0, 0, -1], [0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0], [0, 0, -1, 1, 1, 0, 0, 0, 0, 1, 0, 0], [0, 0, 1, -1, -1, 0, 0, 0, 1, -1, 0, -1], [0, 0, 1, -1, -1, 0, 0, 0, 0, -1, 0, 1], [0, 0, -1, 1, 0, 1, 0, 0, 1, 0, 0, 0], [0, 0, 1, 1, -1, 1, 0, 0, -1, -1, 0, 2], [0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0], [-1, 0, 0, 0, 0, 0, -1, -2, 0, 0, 0, 0], [3, 1, 0, 0, 0, 0, 0, 1, 0, 0, -2, 0], [-1, -1, 0, 0, -1, 0, 0, -1, 0, 1, 0, 0], [2, -2, 0, 0, -1, 0, -1, 0, 0, 1, 1, 0], [0, 0, 2, -2, -1, 0, 0, 0, 3, -1, 0, -1], [0, 0, -3, 2, 1, 0, 0, 0, -1, 1, 0, 0], [0, 0, -1, -1, 1, 2, 0, 0, 2, 1, 0, 0], [4, 1, 0, 0, -1, 0, -1, 1, 0, 1, 0, 0], [1, 2, 0, 0, 2, 0, -1, 0, 0, -2, 0, 0], [2, -1, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0], [0, 0, 1, 1, 0, -3, 0, 0, 0, 0, 0, 0], [0, 0, 1, -1, 1, -1, 0, 0, 3, 1, 0, -2], [-8, 3, 0, 0, 1, 0, -1, 1, 0, -1, 0, 0], [0, 0, -2, 2, 0, 1, 0, 0, 1, 0, 0, -1], [0, 2, 0, 0, 1, 0, -1, 0, 0, -1, 1, 0], [0, 0, 0, 0, 0, -2, 0, 0, 2, 0, 0, -1], [-1, 1, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0], [4, 2, 0, 0, 1, 0, 0, -1, 0, -1, 0, 0], [2, -2, 0, 0, -3, 0, 1, -4, 0, 3, 1, 0], [0, 0, -3, 0, 0, 0, 0, 0, 2, 0, 0, -1], [-1, 0, 0, 0, 0, 0, -2, -1, 0, 0, -1, 0], [0, 0, 1, 1, 2, 1, 0, 0, -1, 2, 0, 1], [0, 2, 0, 0, -1, 0, 1, -2, 0, 1, 1, 0], [0, 0, -1, 5, 3, -1, 0, 0, -4, 3, 0, 1], [0, 0, 0, 2, 0, 1, 0, 0, 1, 0, 0, -3], [0, 0, -1, 3, 1, 2, 0, 0, -4, 1, 0, 0], [-7, 5, 0, 0, 1, 0, -1, 2, 0, -1, -1, 0], [0, 0, -4, 0, 1, -2, 0, 0, 0, 1, 0, -2], [-2, -2, 0, 0, 0, 0, -2, -2, 0, 0, 2, 0], [-3, 3, 0, 0, 3, 0, -2, 1, 0, -3, -2, 0], [-3, 6, 0, 0, 4, 0, 0, 1, 0, -4, -3, 0], [2, 3, 0, 0, -1, 0, 2, 2, 0, 1, -1, 0], [2, -3, 0, 0, -1, 0, 0, 1, 0, 1, -1, 0], [0, 0, 4, -3, -2, 1, 0, 0, 3, -2, 0, 0], [0, 0, 2, 0, -1, 2, 0, 0, 0, -1, 0, 1], [0, 0, 6, -4, -4, 3, 0, 0, 3, -4, 0, 1], [0, 0, -1, 1, 0, 3, 0, 0, 1, 0, 0, 1], [0, 0, 1, -1, -3, 3, 0, 0, -1, -3, 0, 3], [0, 0, -2, 4, 2, 1, 0, 0, -3, 2, 0, -4], [0, 0, -2, -2, -1, 0, 0, 0, 6, -1, 0, -2], [3, -2, 0, 0, -1, 0, 2, -2, 0, 1, -3, 0], [0, 0, -1, 0, 1, -4, 0, 0, 1, 1, 0, 0], [0, 0, -1, 0, -2, -2, 0, 0, 0, -2, 0, 1], [0, 0, -1, 1, 2, -1, 0, 0, 3, 2, 0, -3], [8, -2, 0, 0, 2, 0, -2, 0, 0, -2, 0, 0], [0, 0, 0, 3, 2, -5, 0, 0, 1, 2, 0, 0], [-3, -2, 0, 0, 0, 0, 3, 0, 0, 0, -2, 0], [0, 0, 1, 1, -1, 1, 0, 0, 1, -1, 0, 2], [-2, 6, 0, 0, -1, 0, -1, 1, 0, 1, 1, 0], [0, 0, -1, 2, 2, -4, 0, 0, -4, 2, 0, 0], [0, 0, 0, 0, 2, 0, 0, 0, 1, 2, 0, 2], [10, -4, 0, 0, -3, 0, 0, -2, 0, 3, 2, 0], [8, -1, 0, 0, -3, 0, 2, 1, 0, 3, 1, 0], [0, 0, 0, -2, 0, 1, 0, 0, -3, 0, 0, 3], [0, 0, 6, -3, -3, -1, 0, 0, -2, -3, 0, 0], [0, 0, 1, 4, 2, -8, 0, 0, -2, 2, 0, 0], [8, -6, 0, 0, -1, 0, 3, -1, 0, 1, -1, 0], [-8, 1, 0, 0, 0, 0, -3, -1, 0, 0, 2, 0], [0, -2, 0, 0, -2, 0, 2, -4, 0, 2, 2, 0], [-5, -1, 0, 0, 0, 0, 3, 1, 0, 0, 1, 0], [0, 0, -1, -1, -1, 2, 0, 0, 3, -1, 0, 3], [0, 0, 2, -4, -3, 0, 0, 0, -2, -3, 0, 3], [0, 0, 0, -3, 2, 3, 0, 0, -1, 2, 0, -2], [0, 0, -3, 3, 4, -1, 0, 0, 2, 4, 0, -6], [0, 3, 0, 0, -1, 0, 1, -1, 0, 1, 2, 0], [0, 0, -1, 3, 1, -6, 0, 0, -4, 1, 0, 4], [4, -4, 0, 0, -4, 0, 3, -6, 0, 4, 3, 0], [0, 0, 2, 2, -3, -2, 0, 0, -3, -3, 0, 1], [0, 0, -1, -3, 1, 2, 0, 0, -1, 1, 0, -1], [11, 0, 0, 0, 2, 0, 3, 0, 0, -2, -2, 0], [-5, -1, 0, 0, 0, 0, 0, 3, 0, 0, -2, 0], [0, 0, -2, 3, -1, 1, 0, 0, -2, -1, 0, 2], [0, 0, -6, -2, 2, 4, 0, 0, 1, 2, 0, 0], [0, 0, -2, 1, 2, -3, 0, 0, -1, 2, 0, -2], [6, -6, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0], [0, 0, 0, 5, 0, -5, 0, 0, -3, 0, 0, 2], [0, 0, 0, 4, 1, -2, 0, 0, -6, 1, 0, 4], [0, 0, -1, -5, -2, 3, 0, 0, 5, -2, 0, -2], [0, 0, 2, -3, -2, 5, 0, 0, -3, -2, 0, 7], [0, 0, 3, -4, -5, 4, 0, 0, -5, -5, 0, 0], [-11, 5, 0, 0, 3, 0, 0, 5, 0, -3, -2, 0], [14, -2, 0, 0, -2, 0, 2, 0, 0, 2, -4, 0], [-14, 3, 0, 0, 1, 0, 1, -3, 0, -1, -4, 0], [-16, 7, 0, 0, 4, 0, -3, 1, 0, -4, -2, 0], [3, 4, 0, 0, 0, 0, 0, 1, 0, 0, -3, 0], [0, 0, 1, -7, -3, 0, 0, 0, -1, -3, 0, 1], [0, 0, -4, 2, 2, -2, 0, 0, 2, 2, 0, -3], [-5, 1, 0, 0, -1, 0, 0, 3, 0, 1, -2, 0], [0, 0, 2, 0, -2, -4, 0, 0, 1, -2, 0, -2], [8, -2, 0, 0, -5, 0, 1, -2, 0, 5, 3, 0], [3, -8, 0, 0, -3, 0, 3, -2, 0, 3, 2, 0], [0, 0, -4, -6, -1, 6, 0, 0, 6, -1, 0, -3], [10, -2, 0, 0, -2, 0, 2, -4, 0, 2, -2, 0], [0, -3, 0, 0, -2, 0, 0, -3, 0, 2, 3, 0], [1, 5, 0, 0, 2, 0, -4, 2, 0, -2, 2, 0], [0, 0, -2, 6, 5, 0, 0, 0, -4, 5, 0, 4], [0, 0, -2, 0, 0, -2, 0, 0, -4, 0, 0, -3], [0, 0, 1, 2, 1, -6, 0, 0, -3, 1, 0, 0], [0, 0, 3, -3, 3, -2, 0, 0, 8, 3, 0, -7], [0, 0, 3, 3, 0, -1, 0, 0, 1, 0, 0, 0], [3, 3, 0, 0, 1, 0, 1, -2, 0, -1, -3, 0], [-14, -7, 0, 0, 0, 0, 2, 3, 0, 0, 1, 0], [0, 0, 5, -3, -2, 9, 0, 0, 4, -2, 0, 6], [0, 0, -1, 2, 4, 0, 0, 0, 2, 4, 0, -6], [0, 0, 3, 1, -3, -5, 0, 0, -5, -3, 0, 0], [4, 1, 0, 0, 0, 0, -1, 5, 0, 0, 0, 0], [0, 0, 4, -4, -4, 6, 0, 0, 2, -4, 0, 1], [3, -1, 0, 0, -2, 0, -2, -3, 0, 2, 0, 0], [-6, -1, 0, 0, 6, 0, 0, -2, 0, -6, -1, 0], [0, 0, 2, -5, 0, 1, 0, 0, -5, 0, 0, 4], [0, 0, -2, -4, -2, 3, 0, 0, -2, -2, 0, 6], [-2, -1, 0, 0, -4, 0, 0, 1, 0, 4, 3, 0], [0, 0, 2, -2, 0, 2, 0, 0, -2, 0, 0, 7], [-10, 3, 0, 0, 2, 0, -1, 1, 0, -2, 0, 0], [10, -10, 0, 0, -3, 0, -4, -5, 0, 3, 2, 0], [3, -6, 0, 0, -1, 0, 2, -6, 0, 1, 1, 0], [3, 1, 0, 0, 2, 0, -2, 3, 0, -2, 2, 0], [8, -6, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0], [0, 0, 3, -1, 0, 9, 0, 0, 3, 0, 0, 1], [5, -3, 0, 0, 1, 0, -2, 5, 0, -1, 4, 0], [0, 0, -1, 5, -1, -3, 0, 0, 1, -1, 0, -2], [-10, 1, 0, 0, 1, 0, -1, 1, 0, -1, 6, 0], [14, 2, 0, 0, -3, 0, 0, 2, 0, 3, 2, 0], [8, -1, 0, 0, -3, 0, 3, 1, 0, 3, 2, 0], [0, 0, -2, -7, 1, 3, 0, 0, 4, 1, 0, -2], [-7, 0, 0, 0, -4, 0, 2, -5, 0, 4, 5, 0], [0, 0, -7, 7, 3, -7, 0, 0, -2, 3, 0, -5], [9, -1, 0, 0, 0, 0, 2, -1, 0, 0, 0, 0], [-17, -2, 0, 0, -1, 0, -2, 2, 0, 1, -3, 0], [0, 0, 1, -5, -3, 9, 0, 0, 3, -3, 0, 4], [0, 0, 2, 2, 3, 2, 0, 0, -2, 3, 0, -6], [14, -2, 0, 0, 4, 0, 0, 4, 0, -4, -4, 0], [0, 0, -5, 7, 1, -2, 0, 0, -4, 1, 0, -8], [-10, 0, 0, 0, 4, 0, 2, 0, 0, -4, -2, 0], [0, 0, 1, 4, -2, -2, 0, 0, -6, -2, 0, 7], [-4, -5, 0, 0, -4, 0, -1, -1, 0, 4, -2, 0], [11, -5, 0, 0, -7, 0, 4, 1, 0, 7, 2, 0], [6, 1, 0, 0, 1, 0, 3, 5, 0, -1, 0, 0], [0, 0, 2, -10, -2, -6, 0, 0, 0, -2, 0, -3], [-1, 1, 0, 0, 2, 0, 4, 5, 0, -2, -4, 0], [12, -1, 0, 0, -1, 0, 5, 5, 0, 1, 0, 0], [0, 0, 2, 4, 2, 1, 0, 0, 5, 2, 0, -8], [0, 0, 2, 4, -4, -6, 0, 0, -9, -4, 0, 0], [0, 0, 3, -5, -6, 3, 0, 0, -3, -6, 0, -3], [0, 0, 0, -6, 0, 5, 0, 0, 0, 0, 0, 2], [-24, 0, 0, 0, -1, 0, -1, -4, 0, 1, -3, 0], [0, 0, 3, -1, -5, 7, 0, 0, 1, -5, 0, -2], [-5, 6, 0, 0, 0, 0, 0, 1, 0, 0, 3, 0], [-8, 2, 0, 0, 0, 0, 5, -4, 0, 0, -1, 0], [-10, 7, 0, 0, 4, 0, 2, 9, 0, -4, -3, 0], [0, 0, -4, -4, -1, 0, 0, 0, 0, -1, 0, 3], [0, -7, 0, 0, -4, 0, -3, -5, 0, 4, 0, 0], [-3, -2, 0, 0, 0, 0, 4, -2, 0, 0, -3, 0], [0, 0, -1, 5, 5, -11, 0, 0, 0, 5, 0, 1], [0, 0, -2, 8, 2, -5, 0, 0, -5, 2, 0, 3], [-6, -4, 0, 0, 2, 0, 0, -4, 0, -2, 0, 0], [-10, -4, 0, 0, -5, 0, 3, -2, 0, 5, -1, 0], [1, 6, 0, 0, 5, 0, 3, 0, 0, -5, -4, 0], [0, 0, -8, 4, 8, -2, 0, 0, 2, 8, 0, -5], [2, 3, 0, 0, 3, 0, 1, 5, 0, -3, -6, 0], [-22, -2, 0, 0, 4, 0, 0, 0, 0, -4, 0, 0], [-12, 6, 0, 0, 1, 0, 2, 8, 0, -1, -4, 0], [0, 0, 4, -5, -7, 5, 0, 0, 2, -7, 0, -4], [22, -9, 0, 0, -2, 0, -2, -7, 0, 2, 3, 0], [0, 0, 3, -3, -7, 12, 0, 0, -2, -7, 0, 6], [0, 0, 3, -7, -5, 5, 0, 0, 7, -5, 0, 4], [0, 0, -1, 3, 2, 5, 0, 0, -1, 2, 0, -6], [-7, -7, 0, 0, -2, 0, 4, -7, 0, 2, 6, 0], [0, 0, 5, -3, 3, 6, 0, 0, -2, 3, 0, 2], [7, -8, 0, 0, -6, 0, -1, -8, 0, 6, 0, 0], [0, 0, -4, 4, -1, -6, 0, 0, -8, -1, 0, -3], [0, 0, 0, -4, 2, 11, 0, 0, 3, 2, 0, -1], [0, 0, 3, 6, 4, 2, 0, 0, -4, 4, 0, 2], [0, 0, 0, 0, 3, 0, 0, 0, -1, 3, 0, -3], [6, 2, 0, 0, -4, 0, 4, -4, 0, 4, 4, 0], [0, 0, 6, 4, 2, 5, 0, 0, -5, 2, 0, 2], [0, 0, 10, -4, 0, 4, 0, 0, 1, 0, 0, 0], [0, 0, 7, -5, -6, -5, 0, 0, -3, -6, 0, 5], [4, -5, 0, 0, 0, 0, -4, -6, 0, 0, 1, 0], [8, 3, 0, 0, 0, 0, -2, 7, 0, 0, 3, 0], [9, -3, 0, 0, -6, 0, 0, -12, 0, 6, -2, 0], [0, 0, -9, -3, 1, 3, 0, 0, 5, 1, 0, -1], [0, 0, 0, 0, 0, -5, 0, 0, 3, 0, 0, 2], [-24, -2, 0, 0, 1, 0, -1, -2, 0, -1, 7, 0], [0, 0, 1, -1, -3, -6, 0, 0, -6, -3, 0, 6], [0, 0, 0, -4, 1, 8, 0, 0, 8, 1, 0, 1], [0, 0, 6, -4, 2, 1, 0, 0, 5, 2, 0, -1], [5, 5, 0, 0, 1, 0, 1, -4, 0, -1, -3, 0], [0, 0, -4, -4, -3, 8, 0, 0, -8, -3, 0, 8], [11, 4, 0, 0, -4, 0, 2, -7, 0, 4, -1, 0], [-20, -3, 0, 0, 1, 0, 1, 5, 0, -1, -2, 0], [28, 0, 0, 0, -5, 0, 0, -6, 0, 5, -2, 0], [-9, -11, 0, 0, -5, 0, 0, 1, 0, 5, 6, 0], [-15, 4, 0, 0, -1, 0, -1, 0, 0, 1, 2, 0], [0, 0, 10, -11, -7, -3, 0, 0, 2, -7, 0, 6], [-3, -2, 0, 0, 2, 0, 0, 3, 0, -2, 1, 0], [14, -7, 0, 0, -3, 0, 1, -5, 0, 3, -2, 0], [0, 0, 0, -4, -1, -6, 0, 0, -4, -1, 0, -7], [0, 0, -10, -1, 5, -3, 0, 0, 2, 5, 0, -8], [0, 0, 1, -4, 1, 8, 0, 0, -3, 1, 0, 0], [-25, 4, 0, 0, 2, 0, -4, 5, 0, -2, -3, 0], [-17, 9, 0, 0, 3, 0, 1, 4, 0, -3, -5, 0], [0, 0, -3, -3, 3, 7, 0, 0, 8, 3, 0, 1], [26, -4, 0, 0, -1, 0, 0, -5, 0, 1, 0, 0], [0, 0, -5, 2, 3, -2, 0, 0, 3, 3, 0, -8], [0, 0, 4, 7, 1, 3, 0, 0, 0, 1, 0, 4], [-13, 10, 0, 0, 2, 0, 0, 11, 0, -2, -1, 0], [20, -10, 0, 0, 1, 0, -6, -2, 0, -1, 4, 0], [0, 0, -3, 5, 5, -13, 0, 0, -1, 5, 0, -12], [-14, -1, 0, 0, 1, 0, 3, -5, 0, -1, 2, 0], [-10, 3, 0, 0, 11, 0, -1, 3, 0, -11, -2, 0], [0, 0, -2, -8, -4, 3, 0, 0, 3, -4, 0, 9], [-3, 8, 0, 0, 3, 0, 0, 6, 0, -3, -5, 0], [-6, -2, 0, 0, -7, 0, 1, -5, 0, 7, 3, 0], [-6, -2, 0, 0, 2, 0, -5, 2, 0, -2, 3, 0], [0, 0, -1, -14, -7, 2, 0, 0, 3, -7, 0, -2], [0, 0, -3, 12, 2, -8, 0, 0, -2, 2, 0, -3], [0, 0, -1, 5, -2, 5, 0, 0, -5, -2, 0, 12], [12, -1, 0, 0, -1, 0, 4, -6, 0, 1, -5, 0], [-22, 3, 0, 0, 2, 0, -1, 1, 0, -2, -2, 0], [0, 0, -10, 2, 4, 5, 0, 0, -7, 4, 0, -1], [18, 3, 0, 0, -1, 0, -5, 5, 0, 1, 2, 0], [12, 4, 0, 0, 4, 0, 0, 2, 0, -4, 4, 0], [0, 0, 2, 8, -1, -2, 0, 0, -2, -1, 0, 0], [0, 0, 0, -2, -8, 9, 0, 0, -2, -8, 0, 0], [0, 0, -5, 3, 6, 3, 0, 0, 2, 6, 0, -4], [0, 0, -9, 11, 3, 7, 0, 0, -1, 3, 0, 0], [0, 0, -2, 2, 2, -7, 0, 0, -3, 2, 0, -3], [0, 0, -6, 3, 3, -7, 0, 0, 4, 3, 0, -4], [9, 2, 0, 0, 4, 0, 4, -6, 0, -4, -3, 0], [6, -6, 0, 0, -7, 0, 3, -3, 0, 7, -3, 0], [0, 0, -7, -1, 0, -7, 0, 0, 5, 0, 0, -2], [-13, -1, 0, 0, 0, 0, 7, -1, 0, 0, -3, 0], [2, -2, 0, 0, -2, 0, 2, -2, 0, 2, -4, 0], [0, 0, -3, 11, 6, 5, 0, 0, 1, 6, 0, 0], [0, 0, 6, 0, -4, -8, 0, 0, 4, -4, 0, 8], [19, 1, 0, 0, 1, 0, 5, 6, 0, -1, -1, 0], [0, 0, 6, -10, -10, -3, 0, 0, 3, -10, 0, 3], [-2, 3, 0, 0, 0, 0, 6, 6, 0, 0, -7, 0], [14, -5, 0, 0, -3, 0, -5, 1, 0, 3, -2, 0], [0, 0, -9, -3, 3, -1, 0, 0, 3, 3, 0, -10], [-1, -7, 0, 0, -5, 0, 2, -5, 0, 5, 4, 0], [7, 8, 0, 0, 2, 0, 5, 6, 0, -2, -2, 0], [-12, -4, 0, 0, 2, 0, 2, -8, 0, -2, 2, 0], [0, 4, 0, 0, 3, 0, -6, 0, 0, -3, 2, 0], [0, 0, 5, 9, -1, -10, 0, 0, 0, -1, 0, 4], [17, 3, 0, 0, 0, 0, -8, 2, 0, 0, 0, 0], [8, 6, 0, 0, -2, 0, 1, 5, 0, 2, -1, 0], [0, 0, 8, 2, 2, -7, 0, 0, -7, 2, 0, 2], [5, 2, 0, 0, 5, 0, 5, 0, 0, -5, -2, 0], [-14, -1, 0, 0, -5, 0, -1, -1, 0, 5, 2, 0], [3, -1, 0, 0, 5, 0, -1, 4, 0, -5, -1, 0], [0, 0, 1, 1, 5, 3, 0, 0, 4, 5, 0, -5], [-1, -1, 0, 0, -3, 0, -2, -3, 0, 3, 0, 0], [2, 3, 0, 0, 1, 0, -3, 1, 0, -1, 2, 0], [0, 0, 0, 6, 7, -10, 0, 0, 2, 7, 0, -4], [0, 0, -1, 0, 4, 8, 0, 0, 8, 4, 0, 0], [0, 0, 0, 6, -4, -5, 0, 0, -12, -4, 0, 0], [0, 0, -4, 2, -2, -4, 0, 0, 3, -2, 0, -6], [0, 0, 3, 3, -3, 3, 0, 0, -8, -3, 0, -3], [0, 0, -2, -2, -2, 4, 0, 0, -2, -2, 0, 14], [0, 0, 0, -15, -10, 3, 0, 0, -1, -10, 0, 6], [-8, -4, 0, 0, -2, 0, -4, -8, 0, 2, 4, 0], [0, 0, -2, -4, -4, 6, 0, 0, -4, -4, 0, 1], [0, 0, -3, 7, 7, -7, 0, 0, 7, 7, 0, -9], [0, 0, -7, 17, 8, -1, 0, 0, 0, 8, 0, -10], [0, 0, 5, 2, -2, 0, 0, 0, 4, -2, 0, 9], [0, 0, -4, -3, -4, 11, 0, 0, -3, -4, 0, 6], [8, 16, 0, 0, 2, 0, 0, 0, 0, -2, -2, 0], [0, 0, 5, -9, -9, 9, 0, 0, -1, -9, 0, 4], [0, 0, -2, 4, 0, -2, 0, 0, 0, 0, 0, 5], [0, 0, 9, 7, 3, -2, 0, 0, -11, 3, 0, 1], [0, -8, 0, 0, -4, 0, -2, 0, 0, 4, 0, 0], [0, 0, 10, -8, -7, 4, 0, 0, 1, -7, 0, 9], [0, 0, -11, 5, 4, -1, 0, 0, -1, 4, 0, 3], [3, 4, 0, 0, 2, 0, -4, 0, 0, -2, 9, 0], [0, 0, 1, -2, -4, 4, 0, 0, 2, -4, 0, 8], [1, 9, 0, 0, -8, 0, 2, -1, 0, 8, 4, 0], [20, 12, 0, 0, 8, 0, 1, 2, 0, -8, -7, 0], [-9, -5, 0, 0, -3, 0, 1, -2, 0, 3, 1, 0], [43, -2, 0, 0, 0, 0, 2, -4, 0, 0, -3, 0], [0, 0, -5, 4, -4, 0, 0, 0, -6, -4, 0, 7], [-21, 1, 0, 0, 1, 0, -4, 1, 0, -1, -2, 0], [0, 0, 9, -3, -4, -9, 0, 0, 1, -4, 0, 7], [0, 0, -2, -2, 4, 14, 0, 0, 2, 4, 0, -7], [19, 12, 0, 0, 6, 0, 4, 7, 0, -6, -11, 0], [0, 0, 6, 0, 0, 12, 0, 0, 0, 0, 0, 3], [0, 0, 9, 6, 4, -2, 0, 0, 0, 4, 0, -10], [-4, 4, 0, 0, 6, 0, 0, 6, 0, -6, -8, 0], [0, 0, 7, 3, -6, 5, 0, 0, 4, -6, 0, 4], [0, 0, 0, -3, 4, -1, 0, 0, 13, 4, 0, 0], [13, 0, 0, 0, 1, 0, -4, 0, 0, -1, -5, 0], [0, 0, -7, 5, 4, -13, 0, 0, 5, 4, 0, -6], [2, -4, 0, 0, -2, 0, 4, -4, 0, 2, 4, 0], [0, 0, 5, 5, 3, 2, 0, 0, -11, 3, 0, -5], [20, -1, 0, 0, -1, 0, -3, 1, 0, 1, 2, 0], [0, 0, -11, 9, 9, 7, 0, 0, 5, 9, 0, -5], [0, 0, 4, 4, 1, -8, 0, 0, 6, 1, 0, -5], [0, 0, -2, -8, -6, 1, 0, 0, -3, -6, 0, 3], [-6, -7, 0, 0, -1, 0, -3, -5, 0, 1, -6, 0], [-15, 5, 0, 0, 3, 0, -2, 13, 0, -3, 2, 0], [-1, 9, 0, 0, 8, 0, 2, 7, 0, -8, -2, 0], [28, 2, 0, 0, 2, 0, 3, 7, 0, -2, -7, 0], [10, 3, 0, 0, -3, 0, -1, -1, 0, 3, -4, 0], [0, 0, -2, 14, 6, -7, 0, 0, -1, 6, 0, 0], [7, 4, 0, 0, -2, 0, 3, 4, 0, 2, -8, 0], [-3, 1, 0, 0, 4, 0, 2, 2, 0, -4, -4, 0], [0, 0, 2, 8, -2, -2, 0, 0, -6, -2, 0, 6], [20, -2, 0, 0, -10, 0, -2, -6, 0, 10, 0, 0], [-11, 2, 0, 0, 6, 0, -7, 2, 0, -6, 2, 0], [11, -2, 0, 0, -4, 0, 4, -1, 0, 4, -5, 0], [11, -13, 0, 0, -3, 0, -1, -2, 0, 3, 3, 0], [-12, 6, 0, 0, -5, 0, -4, -4, 0, 5, 2, 0], [0, 0, 3, -1, -5, -1, 0, 0, 9, -5, 0, 4], [0, 0, 2, -2, -4, -4, 0, 0, 12, -4, 0, 1], [0, 0, 4, -12, -4, 16, 0, 0, 11, -4, 0, 2], [0, 0, 4, 6, 1, 2, 0, 0, 2, 1, 0, 4], [0, 0, -2, 8, 1, 0, 0, 0, -9, 1, 0, -5], [0, 0, 5, -7, -3, 11, 0, 0, 12, -3, 0, 3], [0, 0, 3, 3, 5, 11, 0, 0, 5, 5, 0, -4], [46, 3, 0, 0, 0, 0, 2, 2, 0, 0, 1, 0], [0, 0, 6, -8, 0, 3, 0, 0, 10, 0, 0, -8], [0, 0, -5, -5, -3, 12, 0, 0, 0, -3, 0, 2], [18, -3, 0, 0, 1, 0, 0, -5, 0, -1, -5, 0], [0, 0, -4, 1, -4, 5, 0, 0, -11, -4, 0, 5], [-12, 8, 0, 0, 0, 0, 2, 1, 0, 0, 6, 0], [0, 0, -3, -1, 7, 1, 0, 0, 1, 7, 0, 8], [-2, -7, 0, 0, 6, 0, 0, -3, 0, -6, -1, 0], [10, 19, 0, 0, 5, 0, 0, -4, 0, -5, -7, 0], [-7, 4, 0, 0, -8, 0, -3, 0, 0, 8, 4, 0], [-17, 3, 0, 0, 0, 0, -6, -5, 0, 0, -2, 0], [-26, 0, 0, 0, 6, 0, -2, -2, 0, -6, 6, 0], [-28, -4, 0, 0, -2, 0, -5, -5, 0, 2, 11, 0], [0, 0, -1, -1, 1, 8, 0, 0, 10, 1, 0, 0], [-39, 2, 0, 0, 5, 0, 1, 10, 0, -5, 2, 0], [8, 1, 0, 0, 1, 0, 3, 3, 0, -1, -6, 0], [0, 0, -3, -3, -3, -11, 0, 0, 8, -3, 0, 1], [0, 0, 7, -13, 2, 1, 0, 0, 1, 2, 0, 7], [0, 0, -6, 2, 0, 7, 0, 0, 3, 0, 0, -11], [0, 0, 4, -3, -4, 3, 0, 0, -7, -4, 0, -2], [-25, 15, 0, 0, 2, 0, -4, 4, 0, -2, -2, 0], [46, -2, 0, 0, 2, 0, 2, -8, 0, -2, -2, 0], [-31, -1, 0, 0, 6, 0, 1, 5, 0, -6, -3, 0], [-9, -8, 0, 0, -3, 0, 1, -2, 0, 3, 6, 0], [16, -12, 0, 0, -8, 0, -2, -10, 0, 8, 2, 0], [-25, 13, 0, 0, 15, 0, -5, 9, 0, -15, -1, 0], [0, 0, 1, -2, -2, 12, 0, 0, 6, -2, 0, -5], [0, 0, 1, 4, -2, 8, 0, 0, -6, -2, 0, 2], [-4, 3, 0, 0, -4, 0, -1, 7, 0, 4, 0, 0], [-14, -4, 0, 0, -4, 0, -6, -8, 0, 4, 8, 0], [0, 0, 0, -6, -2, -10, 0, 0, -6, -2, 0, 0], [0, 0, -10, -2, -2, 13, 0, 0, -5, -2, 0, 7], [16, -6, 0, 0, -6, 0, -6, 2, 0, 6, 4, 0], [0, 0, 7, -7, -3, 5, 0, 0, 5, -3, 0, 9], [0, 0, -2, -9, 0, -7, 0, 0, -9, 0, 0, -7], [0, 0, -12, 8, 9, 4, 0, 0, 8, 9, 0, 0], [-2, 2, 0, 0, 5, 0, 1, -2, 0, -5, -7, 0], [-21, 3, 0, 0, -5, 0, 1, -5, 0, 5, 7, 0], [0, 0, 4, 4, 2, -3, 0, 0, -7, 2, 0, 9], [-4, 11, 0, 0, 3, 0, -5, -1, 0, -3, -6, 0], [21, -16, 0, 0, -13, 0, 3, -2, 0, 13, 6, 0], [0, 0, 13, -8, -4, 6, 0, 0, 4, -4, 0, 8], [0, 0, -12, 9, 12, 7, 0, 0, -1, 12, 0, -7], [21, -5, 0, 0, 0, 0, 0, -2, 0, 0, -4, 0], [0, 0, 6, 0, -2, -7, 0, 0, -10, -2, 0, -12], [28, 1, 0, 0, -7, 0, 2, -5, 0, 7, -1, 0], [-8, 10, 0, 0, 6, 0, -5, 7, 0, -6, -3, 0], [-15, -8, 0, 0, 0, 0, 4, 1, 0, 0, -1, 0], [0, 0, -8, 2, -5, -2, 0, 0, -12, -5, 0, 14], [-11, -7, 0, 0, 0, 0, 1, 1, 0, 0, 7, 0], [0, 0, 5, -3, 3, -2, 0, 0, 14, 3, 0, -9], [-15, 10, 0, 0, 0, 0, -4, 7, 0, 0, 5, 0], [0, 0, -4, 0, 0, 10, 0, 0, 8, 0, 0, 2], [0, 0, 11, -17, -2, 5, 0, 0, 9, -2, 0, 7], [0, 0, -4, 1, -4, 1, 0, 0, 5, -4, 0, -12], [0, 0, 5, 1, 3, -2, 0, 0, -10, 3, 0, 7], [18, 2, 0, 0, 2, 0, 6, 4, 0, -2, -6, 0], [0, 0, -1, 2, -6, 0, 0, 0, -8, -6, 0, 8], [-18, 8, 0, 0, 2, 0, -2, 3, 0, -2, 4, 0], [0, 0, -3, 10, 7, 8, 0, 0, -5, 7, 0, -10], [-14, -10, 0, 0, -6, 0, 1, -7, 0, 6, -3, 0], [4, 12, 0, 0, 4, 0, 0, 2, 0, -4, -2, 0], [0, 0, -7, 1, 0, -1, 0, 0, -2, 0, 0, 4], [21, -9, 0, 0, -16, 0, 6, -10, 0, 16, 6, 0], [35, -4, 0, 0, -2, 0, -5, -6, 0, 2, 4, 0], [0, 0, -3, 11, 1, -5, 0, 0, -1, 1, 0, -14], [0, 0, -14, 5, 6, -13, 0, 0, -3, 6, 0, 1], [0, 0, -18, 0, 2, 14, 0, 0, 10, 2, 0, 1], [18, -9, 0, 0, 2, 0, -6, 1, 0, -2, -3, 0], [0, 0, -1, -13, -3, 16, 0, 0, 5, -3, 0, -3], [-2, -6, 0, 0, 0, 0, 0, -4, 0, 0, -4, 0], [0, 0, 8, -15, -14, 7, 0, 0, -5, -14, 0, 6], [0, 0, 3, -5, 1, 8, 0, 0, -4, 1, 0, 16], [-9, 5, 0, 0, 7, 0, 1, 6, 0, -7, -1, 0], [10, -6, 0, 0, 4, 0, 6, -4, 0, -4, -6, 0], [0, 0, -6, 8, 5, 0, 0, 0, 4, 5, 0, -4], [-17, 18, 0, 0, 6, 0, 5, 6, 0, -6, -2, 0], [1, 6, 0, 0, -3, 0, 5, -2, 0, 3, -6, 0], [-5, 9, 0, 0, 6, 0, -2, 7, 0, -6, 0, 0], [-30, -7, 0, 0, -3, 0, -1, 1, 0, 3, 4, 0], [50, 1, 0, 0, -1, 0, 4, 5, 0, 1, -1, 0], [-22, 19, 0, 0, 10, 0, 3, 7, 0, -10, -4, 0], [0, 0, 4, -6, 6, 3, 0, 0, 1, 6, 0, 4], [0, 0, 11, -7, -9, 4, 0, 0, -4, -9, 0, 2], [0, 0, -1, 11, 4, -11, 0, 0, -9, 4, 0, 4], [-11, 20, 0, 0, 8, 0, 0, 2, 0, -8, -9, 0], [0, 0, 5, -7, -7, 0, 0, 0, 0, -7, 0, 2], [0, 0, -3, 1, -4, -3, 0, 0, 1, -4, 0, -17], [2, -13, 0, 0, -4, 0, -1, -15, 0, 4, 4, 0], [6, 0, 0, 0, -4, 0, 8, -4, 0, 4, 4, 0], [0, 0, -8, 10, 9, -6, 0, 0, -2, 9, 0, -16], [0, 0, -8, 4, 4, 5, 0, 0, 9, 4, 0, -1]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4560_d_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4560_2_d_k();".
// 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_4560_2_d_k(:prec:=12)
    chi := MakeCharacter_4560_d();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(2999) - 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_4560_2_d_k();".
// 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_4560_2_d_k( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4560_d();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![2304, 0, 11904, 0, 15844, 0, 5920, 0, 876, 0, 52, 0, 1]>,<31,R![16, -576, 276, 88, -48, 0, 1]>],Snew);
    return Vf;
end function;