
// Make newform 560.2.bs.c in Magma, downloaded from the LMFDB on 07 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_560_bs();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_560_bs_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_560_2_bs_c();".
// 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_560_2_bs_c();".
// 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 := [657, -2196, 4737, -6318, 6526, -4844, 3039, -1394, 572, -160, 43, -6, 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, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [6831, -14304, 22560, -18518, 14489, -6886, 3336, -940, 295, -40, 8, 0], [-14895288, 31581987, -50383965, 43891573, -35042994, 17727625, -8701717, 2641205, -826134, 130623, -24323, 845], [-13900563, 28434342, -46228794, 38666825, -30688311, 14561384, -7090142, 1905349, -605637, 66132, -15028, -845], [-125346, 342885, -570384, 580918, -452118, 275445, -127540, 51972, -14492, 4025, -550, 100], [26543106, -70531245, 116264097, -116941719, 93316628, -53971067, 25801813, -9402147, 2753428, -620641, 94591, -11603], [13912749, -20838750, 33581484, -15069787, 11987657, 2137728, -547552, 2195869, -490241, 312896, -33042, 11603], [-14340123, 44621655, -77963187, 88936828, -73420097, 48677439, -23861387, 10164992, -2983999, 821557, -118611, 20098], [-90261, 377016, -682682, 853082, -693355, 480260, -226828, 99908, -27975, 8050, -1100, 200], [-23982993, 63297279, -111279273, 119092836, -98322611, 62528279, -30444649, 12651336, -3681745, 1007233, -143845, 24686], [-9253467, -1703883, 18410913, -50230604, 47573947, -41428071, 20830897, -10342744, 2983517, -926513, 127701, -24686]];
    Rf_basisdens := [1, 1, 306, 617508, 617508, 6054, 617508, 617508, 617508, 6054, 617508, 617508];
    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_560_bs();"
function MakeCharacter_560_bs()
    N := 560;
    order := 6;
    char_gens := [351, 421, 337, 241];
    v := [3, 6, 6, 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_560_bs_Hecke();"
function MakeCharacter_560_bs_Hecke( : Kf := false)
    N := 560;
    order := 6;
    char_gens := [351, 421, 337, 241];
    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, -1, 0, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := ISA(Type(Kf), Fld), 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]];
    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], [0, 0, 0, -1, 1, 0, 0, 0, 1, 0, 0, 0], [0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 1, 0, -1, 0, -1, -1, 1, 0, 1], [1, -1, 0, -2, 0, -1, -1, -1, 0, 0, 0, 1], [0, -1, 0, 1, -1, 0, 0, 0, 0, 1, 1, -1], [2, -1, 0, 1, 0, -1, 0, -1, 0, 0, -1, 0], [2, 0, -1, 1, -1, 0, 1, -1, -1, -1, -1, 0], [-1, 0, -1, -1, 0, 1, 1, 1, 1, -2, -1, 0], [0, 2, -1, -2, -2, 4, 0, 0, 0, -2, -1, -1], [0, -1, 0, -1, 3, 1, 0, 0, -1, 1, 0, 0], [-1, 0, 2, -2, 2, 0, 0, 1, 2, 1, 1, -1], [1, -3, 1, -2, 2, 0, 0, 0, 2, 1, 0, 0], [2, 0, 1, 4, -4, 0, 0, 0, 2, -4, 0, 0], [1, -2, 0, 2, -1, -1, -1, 1, 0, 0, 1, 0], [-1, 1, 0, -1, 2, -1, -1, -1, 0, -1, 2, -1], [2, 1, 0, -1, 5, -1, 2, -2, -3, 3, 0, 2], [-1, 0, 2, 2, 2, -1, 1, 1, -2, -2, -1, 0], [0, -1, -2, -4, -1, -1, 2, -1, -1, 2, -3, -2], [-2, 1, -1, 4, -4, 0, 0, 0, -2, 3, -2, 2], [2, -3, 4, 7, 2, -3, 0, 3, 2, 2, 3, 0], [2, 0, 1, 1, -1, -3, 0, 0, -1, 2, 0, 0], [2, 1, -3, -1, -1, 2, -1, -1, 0, -1, -1, -1], [1, 0, -2, -2, -4, 3, 3, 3, 2, -2, -3, 0], [-2, -2, 0, -4, 4, 0, -2, 2, 0, 2, 0, 0], [1, -4, 2, 3, 1, -4, -3, -1, 1, 2, 2, 3], [-6, -4, 1, 3, -1, -2, -2, 0, 1, 8, 0, 2], [2, 0, -1, -1, -4, -1, 1, 0, 1, 1, 0, 1], [1, -1, 0, 0, 2, 1, 1, 3, -2, -1, -4, 1], [-4, -1, 4, 1, 1, -2, -1, -1, 0, 1, 3, 3], [-4, -1, 2, 9, -9, 0, -2, 2, 4, 5, 3, -3], [4, -2, 6, 2, 2, -1, -1, 1, 6, -3, 1, 0], [2, 0, 0, 2, 4, 0, 2, -2, -8, 0, 0, 2], [0, 2, 2, -2, -2, 4, 2, 2, 0, -2, -2, -2], [-5, 6, -2, -10, 4, 3, 1, -3, -2, 0, -3, 2], [-2, 5, -6, -5, -3, 5, 2, 2, -3, -3, 0, -2], [-9, 0, 0, 6, 0, 0, 1, 2, 0, 5, 1, -1], [0, 0, 0, 0, -3, 3, -3, 0, 0, 3, 0, -3], [-6, -2, 3, 2, 2, -4, 1, 1, 0, 2, 2, 2], [5, 0, 2, 2, 4, 0, 0, -3, -2, 5, 3, 3], [7, -2, 2, -11, 1, -2, 1, 1, 1, -1, 0, -1], [-8, 4, 1, -6, 6, 0, -2, 2, 2, 8, 3, -3], [0, 0, 0, 0, -6, 0, 2, 4, 0, -2, -4, -2], [-2, 0, 0, -8, 14, 0, -2, 2, 2, 2, 0, -2], [4, -2, 0, -4, -4, -4, 1, 1, 0, 2, -2, -2], [2, 0, 0, -4, 10, 0, 2, -2, -2, -2, 0, 2], [-5, -4, 3, 10, -10, 0, -2, 2, 6, 10, 3, -3], [14, -3, 4, -3, -3, -6, -3, -3, 0, 3, 1, 1], [0, 3, 0, -3, 6, -3, 0, -1, 0, -2, 1, 0], [0, 0, -2, -2, 4, 4, -2, 0, 2, 2, 0, -2], [6, 2, 2, 8, -3, 1, 3, -2, 2, -5, -2, -1], [2, -5, 3, -4, 4, 0, 0, 0, 6, 1, -2, 2], [-3, 1, -4, 10, -2, 1, 3, 1, -2, 2, -2, -3], [-17, 0, -3, 0, 0, 0, -4, -4, 0, 0, 3, 3], [4, 0, -2, -2, -8, 4, 0, -2, 2, 4, 2, 2], [-6, 4, -2, -9, -1, 4, -2, -2, -1, -2, 0, 2], [7, 1, 0, 2, 0, 1, 3, 1, 0, -3, -2, -3], [14, 6, 1, -1, 1, 3, 4, -4, 1, -16, -4, 0], [-2, 1, 0, 9, -14, -1, -2, -1, -4, 4, 3, -2], [5, -3, -6, 0, 0, -6, -4, -4, 0, 3, -1, -1], [0, 1, 0, -5, 12, -1, 0, 1, -2, 4, -1, 0], [6, 0, 0, 0, 0, 0, 1, -1, 0, -10, -2, 2], [-4, 1, -5, -7, -7, 2, 3, 3, 0, -1, -5, -5], [4, -1, 0, 1, 1, 1, 4, 0, -3, -1, -4, 4], [0, 0, -4, -4, -4, 8, 2, 6, 4, -2, -6, -4], [0, 6, 2, -20, 11, 3, 3, -2, 2, -3, -2, -1], [-2, 0, 5, 5, 7, -6, 1, -1, -5, -3, 1, 2], [6, -3, 4, 3, 3, -6, 1, 1, 0, 3, -1, -1], [0, 5, 2, -6, 1, 5, 2, 5, 1, -4, 3, -2], [-4, 2, 3, -2, 2, 0, -4, 4, 6, -2, 3, -3], [10, -1, -4, -3, -2, -1, 0, -1, -2, -4, -1, 0], [-6, 0, 3, 3, -7, 1, -4, -6, -3, -2, 6, 2], [1, -3, 0, 2, -5, 3, 1, -5, 1, 3, 4, 1], [-8, 12, 2, -2, 2, 6, 0, 4, 2, -4, 4, -4], [-5, -2, -3, 14, -14, 0, -2, 2, -6, 8, -1, 1], [-13, 6, 3, 3, 0, 3, 1, 5, 3, 8, 5, -6], [2, 2, 0, 10, -14, -2, 2, -4, -6, -8, 2, 2], [-8, 0, -4, -4, 12, 4, -2, 0, 4, -6, 0, -2], [1, -4, 2, 2, 0, -2, -8, 2, 2, -5, 2, 6], [-5, -8, 10, 3, 5, -8, -5, -3, 5, 8, 2, 5], [1, 1, 12, 8, 8, 2, 4, 4, 0, -1, 1, 1], [8, 3, -2, -6, -6, 6, 4, 4, 0, -3, -1, -1], [10, 8, -12, -2, -6, 8, 2, -4, -6, -12, -6, -2], [6, -1, -4, 3, -3, 0, 1, -1, -8, -9, -1, 1], [8, -8, 8, 0, 4, -4, -4, 2, 8, -4, 2, 2], [6, 0, 3, 3, -1, -8, 6, 2, -3, 0, -2, 4], [9, -3, 5, 12, 12, -6, 0, 0, 0, 3, 2, 2], [-6, -4, 4, 12, -4, -2, -4, -2, 4, 6, -2, 6], [-2, -8, 6, 2, -2, 0, -4, 4, 12, 4, -2, 2], [20, 2, -3, 10, -10, 0, 5, -5, -6, -32, -2, 2], [-2, 6, -9, 3, -6, 3, 5, -2, -9, 1, -2, -3], [-10, -5, 0, -1, -7, 5, -10, 6, 9, 7, 4, -10], [12, -9, 4, 7, 2, -9, -6, -3, 2, 0, 3, 6], [-14, -3, 1, -14, 14, 0, -4, 4, 2, 23, 2, -2], [0, 0, -4, -4, 16, 0, -2, 2, 4, 2, -2, -4], [24, 1, 3, 5, 5, 2, 3, 3, 0, -1, -3, -3], [1, 0, 1, 1, 21, -4, 3, -1, -1, -2, 1, 4], [-15, -5, -4, 4, -2, -5, -1, -1, -2, 12, 0, 1], [-24, 10, 4, -2, 3, 5, -1, 6, 4, 13, 6, -5], [-7, 12, -7, -5, -1, 6, 3, -5, -7, -2, -5, 2], [-14, 5, -3, 7, -7, 0, -1, 1, -6, 21, 1, -1], [1, 0, 0, 2, -2, 0, 1, 6, -2, 9, -7, 1], [2, 1, 0, 8, -13, -1, 2, 3, -3, 10, -5, 2], [6, 2, 6, 10, -2, 1, -3, 3, 6, -11, 3, 0], [6, -5, 14, 19, 7, -5, -2, 4, 7, 1, 6, 2], [8, -2, 0, 8, 0, -2, 4, 6, 0, 0, 2, -4], [-12, -1, 0, 1, 1, -2, -3, -3, 0, 1, 3, 3], [-2, 0, -2, -2, -11, 7, -3, 4, 2, 1, -4, -7], [6, -12, 12, 12, 6, -12, -2, 2, 6, 8, 4, 2], [-8, -2, -4, -14, 14, 0, 2, -2, -8, 22, -2, 2], [7, 2, -1, -1, 0, 1, 1, 1, -1, -8, 1, -2], [3, -5, 0, 11, -28, 5, 3, 3, 6, -3, -6, 3], [8, 6, -2, 0, 0, 12, 2, 2, 0, -6, 0, 0], [-1, -9, 0, -8, 2, 9, -1, 11, 14, -12, -10, -1], [-4, 6, -4, 2, -2, 0, 4, -4, -8, 10, -8, 8], [-2, -5, 0, -7, 5, 5, -2, 4, 9, -10, -2, -2], [26, 1, -8, -7, -7, 2, 1, 1, 0, -1, -3, -3], [1, 1, 0, 8, -13, -1, 1, -3, -3, -15, 2, 1], [3, 2, -8, -18, 5, 1, 3, -1, -8, -2, -1, -2], [-2, 1, -7, -2, 2, 0, 0, 0, -14, 3, 4, -4], [-1, -1, -4, -18, -2, -1, -1, -3, -2, 1, -2, 1], [-10, 6, 4, 0, 0, 12, -2, -2, 0, -6, 2, 2], [11, 0, 8, 8, 9, -5, 5, -1, -8, 6, 1, 6], [2, -2, 6, -1, 3, -2, -6, -4, 3, -2, 2, 6], [14, -4, 4, 0, 2, -2, 0, 2, 4, -10, 2, -2], [8, 1, -2, 8, 8, 2, -4, -4, 0, -1, -3, -3], [11, -4, 3, 13, -5, -2, 1, 1, 3, -6, 1, -2], [12, -4, 6, -2, 4, -2, -2, -6, 6, -10, -6, 8], [-18, 4, -1, -4, -4, 8, -1, -1, 0, -4, 4, 4], [13, 0, 0, 0, 4, -4, -4, -3, 0, 17, 3, -1], [-3, 3, 4, 22, 2, 3, -1, 1, 2, -2, 2, 1], [12, 8, -7, -12, 12, 0, 3, -3, -14, -26, -4, 4], [4, 0, 0, 0, 10, -4, 0, -8, 0, 4, 8, 8], [-10, 5, -4, -11, -11, 10, -9, -9, 0, -5, -1, -1], [-16, 0, -4, -4, -8, 3, -7, -5, 4, -9, 5, -2], [7, -9, 2, -4, 4, 0, 2, -2, 4, -1, -1, 1], [23, 1, -8, 9, -4, 1, 11, 5, -4, -7, -6, -11], [0, 1, 0, 6, -17, -1, 0, 1, 5, -2, -1, 0], [4, 4, -6, 20, -20, 0, 6, -6, -12, 0, -4, 4], [-6, 2, 0, 4, -2, -2, -6, 0, -6, 25, 6, -6], [-9, -6, 5, 0, 0, -12, 4, 4, 0, 6, 5, 5], [-10, -8, 12, -4, 8, -4, -4, 8, 12, 14, 8, -4], [-10, -5, -6, 14, -3, -5, -8, -5, -3, 6, 3, 8], [6, -2, 1, -18, 18, 0, 4, -4, 2, -2, 2, -2], [20, -8, 8, -20, 4, -8, -4, -4, 4, -4, 0, 4], [6, -2, -6, 14, 14, -4, -2, -2, 0, 2, 2, 2], [8, 5, 2, -9, 9, 0, 6, -6, 4, -9, 1, -1], [-2, 0, -20, -14, -10, 0, 6, 0, -10, 4, -6, -6], [-14, 8, -10, 2, -6, 4, 2, -2, -10, 8, -2, 0], [9, 0, 3, 3, 9, -12, -2, -1, -3, 11, 1, -1], [17, -10, -2, 0, -1, -5, 1, -3, -2, -6, -3, 2], [-13, -9, 5, 14, -14, 0, -8, 8, 10, 19, 10, -10], [18, -9, -2, 7, -7, 0, 7, -7, -4, -13, -5, 5], [-4, -6, 15, -9, 12, -3, -3, 8, 15, 7, 8, -5], [-4, 12, -18, 21, -9, 12, 4, -4, -9, -8, -8, -4], [-14, 8, 0, -14, 14, 0, 2, -2, 0, 24, -2, 2], [8, 0, 9, 9, 17, 1, 8, 0, -9, 0, 0, 8], [-1, 0, 5, 5, -1, 5, 4, 4, -5, -5, -4, 0], [0, -2, 4, -30, 30, 0, 2, -2, 8, 6, -2, 2], [4, -10, 8, -6, 4, -10, -8, -2, 4, 4, 6, 8], [14, 0, 5, 5, -8, -3, 5, -2, -5, 9, 2, 7], [0, -4, 18, 16, 16, -8, 6, 6, 0, 4, 0, 0], [0, 3, -1, 5, -5, 0, 1, -1, -2, -1, -5, 5], [30, 6, -6, 18, -12, 3, 7, -9, -6, -29, -9, 2], [12, 8, 0, 0, 8, -8, 12, -10, -8, -14, -2, 12], [-1, 6, 0, 15, -27, -6, -1, 0, -3, 17, 1, -1], [12, -7, -2, 17, -1, -7, 0, 4, -1, 1, 4, 0], [-26, 1, 4, -3, 2, 1, -4, -1, 2, 10, 3, 4]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_560_bs_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_560_2_bs_c();".
// 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_560_2_bs_c(:prec:=12)
    chi := MakeCharacter_560_bs();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(997) - 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_560_2_bs_c();".
// 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_560_2_bs_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_560_bs();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![64, 352, 2104, -508, 1489, -494, 824, -272, 175, -28, 16, -2, 1]>],Snew);
    return Vf;
end function;