
// Make newform 6160.2.a.by in Magma, downloaded from the LMFDB on 06 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6160_a();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_6160_2_a_by();".
// 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_6160_2_a_by();".
// 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 := [-2, 4, 4, -7, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0], [3, 4, -7, -2, 1], [6, 0, -15, -3, 2], [-6, 2, 15, 3, -2], [10, -4, -18, -2, 2]];
    Rf_basisdens := [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_6160_a();"
function MakeCharacter_6160_a()
    N := 6160;
    order := 1;
    char_gens := [5391, 1541, 3697, 5281, 5601];
    v := [1, 1, 1, 1, 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;

function MakeCharacter_6160_a_Hecke(Kf)
    return MakeCharacter_6160_a();
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, 1, 0, 0], [-1, 0, 0, 0, 0], [1, 0, 0, 0, 0], [-1, 0, 0, 0, 0], [-1, -1, 1, 1, -1], [-1, 1, -1, 1, 0], [1, 1, 0, 1, 0], [-1, 1, 1, 0, -1], [3, 1, 0, 1, 0], [-1, 3, 0, -2, 0], [1, 1, -1, 0, -1], [2, -2, 0, -1, 1], [0, 0, 1, -1, 2], [-1, -1, 3, -1, 0], [5, 1, -3, -2, 1], [3, 1, 0, 0, -1], [1, 1, -2, -2, -1], [2, 4, -1, -1, -1], [-4, 0, 2, 0, -2], [1, -1, -1, 1, 2], [-7, -1, -1, 0, 1], [8, 0, -2, 0, 2], [4, 2, 2, 0, -2], [3, 1, 0, -3, 0], [7, 3, -2, -2, -1], [-3, -3, 3, 1, 2], [2, 4, 0, -2, 1], [-1, -3, 4, 1, 0], [6, 0, 2, 2, -2], [0, 4, -4, 0, -2], [7, 3, 2, -1, 0], [2, 2, 5, 1, -1], [-7, 1, -2, 3, -2], [3, -3, 0, -1, 2], [-9, -3, 1, 0, 1], [4, 2, -4, -2, 2], [-6, 0, 1, 1, 1], [4, -4, -4, 0, -2], [7, 1, -1, -1, 0], [-2, -4, 0, -2, 3], [4, -2, 2, 2, 0], [-8, -4, 4, 4, -2], [-2, 0, -3, 1, 0], [0, -4, 3, 1, -3], [3, 1, -6, -2, 1], [0, -4, -2, -2, 0], [-11, 1, 1, -1, -4], [8, -4, 0, 4, 2], [8, 0, 0, -4, 3], [0, -4, 1, -3, -1], [-3, 1, -4, 3, 0], [-4, 0, 0, 5, -3], [9, -1, -2, -2, -1], [-1, 1, 4, 3, 2], [12, -4, 0, 4, -4], [-4, -8, 4, 0, -1], [0, 4, -4, 2, 2], [-2, 6, -5, -1, -1], [-2, -4, 0, -2, 0], [4, 0, -4, -4, 2], [9, 5, -5, -1, -1], [6, 2, -2, 0, -4], [-3, 5, 6, 0, 0], [3, 1, -2, 1, 2], [5, -5, -2, -1, 0], [6, 0, 4, -6, 1], [4, 0, 1, -7, 3], [16, 4, -7, -1, 4], [7, 3, 6, 0, -3], [11, -3, -4, -1, 0], [-11, 7, -7, 2, -1], [9, 9, -5, -3, 0], [12, 0, -3, 3, 1], [-10, -6, 6, 6, 0], [11, -1, -5, 1, 4], [-12, -2, 8, -2, -4], [-12, -2, 8, 6, 0], [18, 2, 2, -2, -1], [-8, -8, 0, 7, 1], [3, -3, -2, 6, -1], [-6, -2, -1, -5, 1], [7, 9, -9, -2, 3], [19, 1, -8, -3, 2], [-12, 0, 2, -4, -2], [-12, -2, 11, -1, -5], [18, 0, -3, 5, 2], [2, 2, -5, 3, 1], [7, 7, -6, -4, 5], [7, 1, 1, 0, -1], [6, -2, -3, -4, 0], [14, -6, -4, -4, 0], [1, 7, 1, -2, 1], [14, 8, -1, -1, -1], [-16, -2, -6, 0, -1], [14, 2, -12, -4, 4], [0, 2, -2, -2, -2], [-2, 8, 0, 2, 0], [2, 6, -4, -6, -4], [9, -5, 8, 1, 2], [-12, -4, -2, -6, 2], [-28, 4, -3, 3, -1], [18, 2, -14, -2, 0], [0, -2, -2, -2, 4], [-2, -10, 2, 4, 2], [1, -1, 4, -1, 0], [6, -2, -7, -4, 4], [-19, -1, -7, 3, 2], [-28, 4, 0, -2, 0], [0, -4, 4, -3, 1], [14, -4, -2, -8, 3], [10, -2, 1, -1, 5], [0, 4, 1, -1, 1], [5, 3, 6, -2, 1], [4, 0, 12, -4, -2], [-6, 6, -7, 1, -1], [12, 12, -11, -2, 2], [7, 3, -3, -1, -6], [1, -5, 0, -3, -2], [12, 10, 1, -1, 1], [2, 4, -10, 2, -4], [0, 8, 9, -7, -3], [-13, -3, 7, -1, -2], [22, 0, -1, -3, 7], [-3, 3, -2, 4, -3], [11, -3, -2, -5, -2], [-12, -2, 8, 6, -4], [-7, -9, 12, 4, -3], [13, -3, -11, -3, 4], [5, -9, 3, 3, -4], [-16, 0, 0, 12, -6], [0, -4, -6, 2, 6], [-16, 4, 10, -4, 2], [7, -3, -2, 1, 0], [-24, -8, 8, 3, 1], [-8, 0, -8, -1, -5], [0, 6, 2, -6, 2], [18, -2, 0, 0, 2], [8, -6, 4, 2, 0], [0, 10, -6, 4, -2], [-5, -1, 6, 3, -6], [1, 7, 10, -1, 0], [-1, -3, -9, 8, 1], [26, 4, -11, -3, 1], [-10, 2, 10, -6, -3], [-13, 11, 0, -2, 0], [7, -7, 5, 5, -2], [-11, 11, -1, -3, 2], [1, -1, 2, -2, 3], [-9, 1, 9, -2, -1], [6, 2, 9, 3, 1], [20, 10, 0, -4, -2], [6, -8, -7, 11, -1], [10, -2, 4, 4, -2], [-32, -2, 7, 5, -3], [-4, -8, 6, 8, 2], [-7, 1, -8, -1, -2], [-22, 4, 10, 0, 4], [11, 5, 7, 7, 2], [9, -7, -8, 4, -1], [18, 12, -7, -1, -1], [-22, 0, 5, 5, 2], [6, 0, 7, -9, -1], [13, 3, -8, -8, 9], [10, -4, -2, -10, 2], [5, -3, -17, 1, 0], [-22, 2, 4, -4, 0], [-21, -11, 7, 3, -4], [16, -2, -8, -2, -4], [2, -4, 6, 0, 4], [-34, 2, 6, 0, 0], [18, 4, 10, 2, -4], [-15, -1, 1, 6, 1], [-16, 4, 15, 7, -1], [-3, 5, -6, 0, -4], [28, 6, 2, 2, -2], [10, 0, 1, -3, 1], [-6, 2, -1, -1, 7], [-12, -18, 4, 6, 5], [3, 3, -6, -2, -3], [-7, -1, -5, 2, 1], [25, 5, -16, 1, 6], [1, -1, -2, -1, 0], [-23, -5, 3, -1, -6], [-3, 7, 1, -4, -1], [-8, 8, 0, -8, 1], [24, 4, -11, 3, 5], [-10, -2, 11, 2, -6], [12, -10, -8, 12, 2], [-4, -16, 2, 6, 4], [1, -5, 2, 1, -8], [-6, -8, 2, 0, -5], [-6, 10, -4, -4, 2], [22, 4, -2, -2, -2], [-8, -8, -1, 1, -2], [13, -1, 4, -11, 6], [-6, -4, -6, 8, -2], [11, -5, 7, 6, -5], [-5, 5, -1, -13, 6], [28, -2, 0, 2, 3], [23, 5, -2, 1, -4], [0, 8, -2, 2, -6], [12, 6, 14, 2, 2], [-30, -10, 2, 3, 3], [-5, -11, -2, 0, 5], [-2, 4, -2, -2, 4], [-17, -11, 18, 2, -1], [-6, 8, -2, 0, -5], [10, 0, -18, -2, -4], [5, -1, 4, 2, 1], [8, 6, -7, -7, -2], [-12, 12, -6, -6, -1], [-17, 5, -9, 8, -7], [2, 10, 5, -2, -2], [-2, -2, -18, 2, 2], [-20, -6, -4, -2, -6], [-28, -2, 5, -9, -5], [30, -6, -10, -5, 1], [-9, -7, 5, 8, 3], [15, 1, 8, 7, -6], [-19, 7, -10, -3, -2], [12, 4, 8, -6, -2], [10, 4, -17, -1, 4], [-31, 3, -3, -4, 3], [-20, 4, -4, 12, -10], [-9, 1, 2, 7, -2], [-7, -5, 2, 9, -4], [-14, 2, 8, 2, -4], [-16, -6, 2, 12, -1], [8, 2, 7, 9, -1], [11, 11, 7, -6, -5], [13, 9, 4, -5, -8], [-17, 15, 6, -9, 2], [-14, -8, -6, 14, 0], [-4, 4, -9, -8, -6], [12, 2, -8, -2, 3], [14, -8, 0, 2, 6], [2, 18, -1, -1, -1], [-18, 10, -2, 2, 4], [-11, 7, 10, -2, 7], [24, -10, 5, 3, 5], [-9, -5, -6, 5, -8], [-11, -3, -9, 3, 0], [-2, -2, -1, 3, -5], [13, -1, 11, 3, 2], [-6, -10, 16, 2, -4], [-5, -5, 3, 9, 0], [23, -7, -6, 2, 7], [-4, -8, -4, 8, 4], [20, 4, 4, -4, 8], [26, -2, -13, -5, 1], [0, 0, -12, 3, 1], [19, 1, 5, 8, 5], [30, -4, -6, -6, 2], [1, -3, 5, 1, -9], [48, 0, -4, -8, 6], [-4, 4, -3, -5, -1], [12, -4, -10, 8, -2], [26, 4, -2, -2, -6], [1, 15, -1, 7, 0], [-34, -4, 23, 7, -3], [-20, -8, 4, 20, -4], [-7, 5, -4, 0, 3], [-13, -11, 9, -7, 0], [33, 11, 4, 5, -2], [-27, -1, -12, 4, -1], [-12, -12, 5, 3, 3], [-10, -20, 6, 0, 0], [-10, 12, -13, 3, -1], [4, -12, 1, 1, 3], [2, -8, 8, 4, 12], [-14, 4, -11, -7, 7], [15, 5, -12, 1, -2], [11, 7, -14, -1, -8], [15, 13, -3, -7, -2], [-5, -17, 1, -7, 4], [-28, 6, 3, -5, -1], [41, -13, 4, 5, -2], [-2, 16, -12, -8, -2], [5, 5, 0, -3, -2], [-12, 8, 14, -6, 4], [-36, -16, 0, -2, 2], [0, 6, -6, -16, 3], [-29, 3, 0, -12, 3], [-18, -2, 18, 2, 2], [-6, 2, -8, 8, -8], [0, -8, -9, 9, -1], [-14, 10, -19, 1, -1], [7, -15, 4, 0, -1], [-6, -2, -12, -5, 7], [-14, -10, -5, -5, 1], [-16, 4, 3, 12, -10], [-36, 4, 3, -1, -3], [7, -9, -6, -3, -6], [2, -10, 17, 5, -7], [15, -5, 2, 12, 3], [-9, 11, 10, -4, 4], [10, -4, -4, 10, 0], [29, 5, 8, -7, -2], [-20, -8, 12, 14, 2], [-21, -11, 11, 7, -6], [6, -10, -17, 1, -1], [-23, -9, -7, 10, 5], [8, -14, 7, 9, 3], [14, 2, 14, -2, 4], [-14, -10, 1, -7, -3], [-34, 0, -3, 7, -1], [-18, 10, 8, -8, 9], [-2, -6, -26, 0, -2], [8, -4, -1, -15, 13], [14, 8, 0, -6, 3], [-10, -4, 12, 8, -2], [32, 10, -8, -4, 2], [26, 4, -3, -5, -1], [-40, -10, 23, 13, -7], [-52, -12, 12, 3, -5], [3, 5, -14, 0, 3], [13, 1, -8, -9, 4], [13, 3, 0, -9, 2], [-18, -8, 20, 2, -10], [-21, -5, 6, 15, -6], [26, 0, -10, -18, 8], [-7, 1, 10, 6, 7], [25, -9, 5, 4, -3], [-18, 12, -3, -11, -6], [4, -6, 14, 0, -4], [12, -6, -18, -4, -1], [-22, 2, -11, 4, -8], [6, -4, 3, -5, -3], [-38, 10, 11, -11, -1], [-9, 9, -4, -13, 6], [29, -3, 11, 4, 7], [6, -6, -2, -4, 4], [-4, -20, 14, 0, -4], [-12, 14, -5, 1, 3], [26, 2, -1, -2, 2], [16, 16, 5, -1, -1], [21, 19, -17, -5, 4], [4, 26, -6, -4, -4], [-16, 6, 6, 8, -1], [-13, -7, -2, 21, -4], [-14, -14, 5, 5, 3], [-11, 5, 4, -14, 3], [0, 12, -10, -2, 10], [-1, 1, 8, 11, -14], [0, -16, 20, -4, -2], [24, -4, 6, -10, 7], [14, -8, 7, -5, 3], [22, -6, -2, 0, 16], [-4, -2, 30, -2, -10], [19, -9, -8, 3, -2], [-20, -16, 15, 5, 1], [12, 6, -8, -2, -8], [2, -18, 8, 0, -4], [-13, -3, -12, -5, 2], [-13, 5, -19, 3, 2], [5, 9, -8, -6, -6], [17, 5, -8, -1, 0], [34, -20, 7, 7, 0], [49, 3, 1, -8, 3], [2, -2, -9, -1, 5], [-2, -18, -14, -3, 3], [-25, -19, 17, 2, -11], [14, 4, -15, 1, -7], [24, -12, 3, -8, 0], [4, 12, 1, -13, 3], [7, 11, 13, 5, -9], [18, -6, 8, -2, -4], [-12, 2, -16, 10, 0], [2, -14, -16, 0, 6], [-27, 3, 10, -4, 11], [26, -6, -4, -10, 12], [-9, 1, -24, -1, 10], [2, -2, -14, 12, 0], [31, 27, -23, -8, 5], [3, 9, 10, 12, -3], [-27, 7, 17, -10, -1], [25, -9, 5, -19, 6], [-18, 2, -8, -9, 3], [-19, 7, 9, 5, -4], [26, -2, -16, -4, -3], [-20, -16, -9, -3, 9], [28, 6, 7, -7, -5], [18, 10, 1, 5, -7], [53, 13, -24, -11, 8], [-9, 11, -1, 7, -8], [-28, 0, -6, 2, -6], [6, 0, 12, 0, -6], [6, 8, -2, 8, 1], [-19, -7, 5, 7, -4], [-14, -20, 9, 13, 4], [12, 6, 4, -4, -4], [-8, 2, -8, 2, -15], [4, -16, 13, 2, -6], [24, -4, 6, 2, 14], [-16, -2, 9, 1, -6], [36, 12, 12, -2, -4], [16, 18, -2, -8, -10], [36, 2, -4, -14, 3], [-1, 5, -16, -3, -8], [-11, 1, -6, 2, 11], [-24, -22, -10, 0, 6], [-4, -4, -4, 8, 0], [7, 1, 28, -1, -8], [-11, 13, -2, 0, -7], [-22, -6, 2, 0, -12], [15, -3, 10, 1, -4], [-16, 2, 0, 2, -10], [16, 12, 19, -6, -2], [19, -15, -12, 0, -3], [3, 1, 15, 3, -4], [-16, -12, -13, -9, 17], [4, 2, -17, -13, -3], [24, -18, -3, 7, -11], [40, -8, -13, 3, -7], [18, -4, 0, 4, -8], [41, -11, -2, 5, 0], [-5, 15, -3, 3, -4], [23, -15, 4, -13, 8], [14, 4, 2, 0, -11], [-34, 12, 0, 4, -2], [1, -3, 7, 11, -9], [3, -3, -15, -12, -1], [1, -15, 14, 9, -12], [33, 19, 12, 1, -8], [36, 22, 12, 4, -8], [6, 20, -5, 3, -7], [-30, -4, 2, 0, -2], [31, 5, 10, -2, -1], [10, -10, 6, -6, 8]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6160_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6160_2_a_by();".
// 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_6160_2_a_by(:prec:=5)
    chi := MakeCharacter_6160_a();
    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_6160_2_a_by();".
// 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_6160_2_a_by( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6160_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-8, 24, 12, -10, -2, 1]>,<13,R![-416, 336, 116, -52, -2, 1]>,<17,R![-208, -432, -268, -40, 6, 1]>,<19,R![-1088, 160, 200, -34, -6, 1]>,<23,R![32, -112, 128, -50, 2, 1]>],Snew);
    return Vf;
end function;