
// Make newform 6160.2.a.bx in Magma, downloaded from the LMFDB on 24 April 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_bx();".
// 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_bx();".
// 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 := [-28, 24, 22, -12, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [-6, 0, 1, 0, 0], [0, -8, 0, 1, 0], [12, 0, -10, 0, 1]];
    Rf_basisdens := [1, 1, 1, 1, 2];
    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, -1, 0, 0, 0], [1, 0, 0, 0, 0], [-1, 0, 0, 0, 0], [-1, 0, 0, 0, 0], [2, 0, 0, 0, 1], [0, 0, 0, 0, 1], [-2, 1, 0, 1, 1], [-2, 1, -1, 0, 1], [0, 1, 0, 0, 1], [0, 0, 1, 0, 1], [4, -1, -1, 0, -1], [2, -1, 1, 1, 0], [2, 0, 1, 1, 0], [-2, 0, -2, 1, 1], [4, -1, 1, 0, -1], [2, -2, 1, -1, -1], [0, 0, -1, 0, -1], [0, -2, 1, 0, 0], [8, -2, 2, 0, -2], [4, -4, 0, 0, -1], [2, 1, -1, 1, 1], [4, 0, 0, 1, 2], [6, 0, 0, 1, -2], [0, 1, 0, -1, 3], [0, -2, -1, -1, 1], [-6, 0, -2, -2, 1], [2, 2, 2, 1, 0], [-4, 5, -2, 0, 3], [-2, 0, -2, 2, 2], [-8, 0, 0, -2, -2], [2, 3, 2, 0, -1], [-2, -2, -3, 0, -2], [-2, -1, -2, 0, -3], [0, 3, -2, 0, 1], [-10, 3, -1, 1, 1], [2, 0, -2, 1, 0], [0, 2, -1, -2, 0], [-8, 2, -2, -1, 2], [0, -2, -2, 0, 1], [6, -6, -2, -1, 0], [2, -2, 2, 1, -2], [0, 4, -2, 0, -2], [0, -6, -1, 1, -2], [-2, 4, 3, 0, -2], [6, 0, -3, 0, -1], [4, 4, 0, 0, 2], [2, -8, 2, 0, -3], [-8, 8, 0, 0, 4], [8, 4, -2, 1, 0], [-2, -2, 3, 2, 2], [6, -1, 2, 0, 1], [6, -3, -1, -1, -2], [6, -2, 3, -1, -3], [16, -5, -2, -1, -1], [4, 0, 0, 0, 6], [-8, 2, 0, 0, 4], [0, 6, 0, -2, 0], [2, 2, -1, 0, 6], [-6, 6, 0, 0, 6], [-8, 8, 2, 0, 0], [-6, -2, -2, -3, 1], [8, -4, 0, 1, 0], [-12, 2, -3, -1, -1], [8, -3, 0, -2, 1], [0, 5, -2, -2, -3], [6, 2, 0, -3, 2], [10, 6, -1, 2, 0], [-2, 4, -3, 1, 0], [0, 2, 1, 2, 1], [-8, -3, 0, 2, 1], [10, 5, 1, -3, 1], [2, -2, 4, 1, -1], [-6, 4, 1, 0, -2], [-4, -4, 2, 0, 4], [-2, 0, 0, 2, 3], [14, 2, 2, 2, -4], [14, 0, 0, 1, 0], [-4, -4, 0, 1, -2], [-6, -7, -1, 2, -2], [14, 0, -1, 2, 7], [-2, 8, -1, -3, 0], [10, -1, -1, 1, -5], [0, -9, 2, 0, -7], [4, 2, -2, -2, -4], [16, -6, 3, 2, -6], [8, 0, -1, -2, -2], [-10, 6, 3, 0, -4], [-16, 0, 3, -1, -3], [2, 5, -1, 0, -5], [0, -1, 0, -2, -2], [0, -2, 0, 3, 2], [-18, -1, -1, 0, 3], [4, 4, -5, 3, 4], [6, 6, -4, 3, 0], [4, -6, 0, -3, 0], [-10, 8, 0, 0, 2], [-2, 4, -2, 3, 0], [-4, 0, -4, -1, 4], [-4, -3, 2, 0, 3], [-8, 0, -6, -2, -2], [-6, 4, -7, 0, 4], [0, 4, -2, 0, -2], [-2, 0, 0, 2, 2], [-4, -6, -4, 2, 0], [20, -5, 2, -1, -3], [16, -5, -2, -4, -2], [-4, -2, 4, -1, 1], [16, 2, 4, 0, -2], [-2, -3, 3, 3, 2], [-2, -4, 2, -2, 6], [2, 4, -1, 2, 4], [-18, 8, -3, -4, -4], [-6, -4, -1, -2, 5], [4, 4, 4, 0, 4], [2, 4, -3, 3, 8], [-12, -5, -6, 1, -2], [2, 2, 8, 1, 1], [-12, 1, 0, 0, 7], [4, 2, -9, -1, -4], [2, -2, 2, -1, 0], [-10, -2, 7, 0, 0], [0, 0, 2, -5, -5], [16, -4, -1, 0, -2], [6, 6, 1, 0, 5], [12, -7, 4, -2, -5], [-6, -6, -4, -2, 2], [18, -4, -3, -2, -3], [-14, 8, 4, 3, -1], [-16, 2, -4, 0, -5], [8, 12, 0, 2, 0], [0, 8, -2, -2, 0], [4, 2, -4, 4, 8], [16, 3, -2, 2, 5], [-6, 5, -3, 0, -2], [6, -1, 1, 5, 2], [-6, -2, -6, 3, 4], [-8, 4, 2, -2, -6], [-2, -4, -2, 3, 0], [14, -2, 2, -2, -2], [-26, -3, 2, 1, -1], [-20, -3, -4, 0, -1], [-14, -1, -5, 0, -7], [36, -2, 3, 0, -4], [-8, 0, 0, -1, -10], [8, 2, -3, 3, 5], [-4, 4, 4, -3, -1], [4, 4, 6, 0, 3], [6, -6, -3, 1, -11], [-26, 3, -7, -4, 3], [18, -4, -1, -4, 0], [-18, -6, 2, 1, -2], [-20, 12, -3, 2, 6], [-12, 8, 4, 2, -2], [4, 0, 1, 2, 0], [0, 2, -2, 0, 2], [-6, 11, -6, 2, 5], [10, -4, 2, -1, -8], [-12, 8, -4, 1, 3], [-4, 4, 3, 2, -1], [16, 8, -3, 2, 0], [-8, -6, 7, -1, 0], [-24, 14, 1, 2, 8], [14, 0, 1, -4, -3], [14, 8, -2, 2, -2], [-14, -6, 2, 3, 3], [0, 0, -6, -2, -2], [8, -6, 4, -2, -3], [10, -6, 2, 4, 0], [-6, 8, 10, -1, -2], [16, -2, 4, -4, -4], [18, -6, -2, -1, 4], [-22, -3, -1, 3, -1], [-22, 10, -7, -2, 6], [-28, 2, -1, -1, 3], [2, -4, 0, -2, 6], [-4, 4, 5, 3, 4], [-14, -12, 3, -1, 0], [-2, -12, 4, 0, -4], [12, -10, 5, 1, -3], [-6, -7, -1, -8, 1], [26, -3, 6, 1, -7], [-4, -19, 2, 2, -1], [12, 0, 2, 5, 1], [6, -15, 5, -2, -9], [12, -2, -2, 0, -8], [-2, 8, -3, 4, 4], [-12, 9, 2, -1, 12], [-22, -4, 4, -2, 4], [-12, -10, 2, -3, 0], [28, 9, 0, -2, -1], [18, 6, 4, -3, -4], [8, -4, 10, -2, -10], [6, -14, -4, -1, 4], [14, -8, -1, 3, -4], [4, 5, -6, 1, 5], [18, 2, -6, -2, -6], [8, -3, -1, 0, -7], [-4, 12, 4, 1, -1], [-18, -2, -4, 7, 2], [24, -1, -4, 0, 9], [-40, 6, -4, -3, 0], [-10, 0, -2, -2, 2], [-26, 11, 1, 0, 4], [6, -12, -3, -1, -1], [-6, 4, -4, 6, 8], [10, -10, 3, 3, 3], [-2, -6, -4, 3, 2], [2, -12, 0, 0, -10], [-6, -8, 5, -3, -15], [-4, 10, 5, -1, 0], [-20, -4, 6, -3, 0], [-26, 3, 3, -4, -5], [4, -1, -8, -1, -8], [0, -10, 0, -1, -6], [10, 0, 2, 3, 2], [-16, 0, 3, -2, 2], [-6, 13, -7, -1, 8], [-6, 1, 7, 0, 11], [0, -5, -8, -6, -3], [-8, -5, 0, 2, -7], [-16, -4, 4, -7, -8], [0, 0, 5, 0, -6], [-10, -3, -3, 2, -1], [4, -18, 2, -1, -10], [-4, -7, 0, 2, -1], [4, 13, 0, 0, 3], [-12, 2, -4, 0, -2], [-6, -2, -4, 5, 6], [20, -14, -3, -1, 4], [-32, 9, 7, 2, 5], [-14, 1, -2, 4, 9], [-2, -19, 2, 4, -7], [-18, 8, 6, -2, -8], [0, -13, 0, 3, 6], [2, -12, 2, 2, 4], [26, 2, 0, 0, 0], [14, 14, -1, -2, -4], [4, -8, 6, 0, 0], [-6, 12, -9, 2, 5], [12, 0, 7, 6, 2], [-18, 5, -4, 0, 7], [-34, 0, 0, -3, -1], [22, 4, -7, 1, 6], [4, 16, 6, -1, 1], [-36, -4, 4, 1, 2], [-30, -6, 2, -1, -3], [-10, 16, 3, 2, 7], [-4, -12, -4, 0, -4], [-12, 16, 0, -2, 6], [2, -10, 3, -4, -4], [10, 13, 7, -2, 2], [-22, 1, -9, -6, 1], [18, -2, 6, -1, -4], [-2, 14, 6, -1, -7], [24, 16, 0, -2, 0], [-2, -6, -1, -1, -4], [20, -20, 6, 1, -10], [-18, 12, -6, 6, 12], [0, -4, -2, -9, -3], [20, 2, 1, -2, 4], [28, -4, 4, -2, -4], [-4, 18, 3, -1, 7], [4, 10, 2, 0, 5], [20, 5, 2, -1, 1], [-14, 0, 3, 0, 5], [30, 2, -1, -1, 12], [-6, -2, 8, -2, 0], [-20, 2, 5, 0, 12], [30, -14, -1, 0, 0], [54, -8, 14, 4, -8], [-4, -2, -1, 4, 8], [16, -9, -2, 0, -7], [10, -7, 6, 8, 5], [-28, 18, 4, 2, 1], [-26, 2, 4, -2, -1], [-20, 10, -9, -4, 0], [-8, 9, -8, -4, -7], [26, -4, -2, 4, 2], [-14, 3, 0, 0, 9], [0, -10, 2, -3, -10], [8, 22, -2, 0, 4], [14, 20, -2, 0, 0], [-28, 0, -7, 0, 5], [-8, 2, 6, 3, 12], [44, -12, 0, 0, -4], [-2, 8, -5, 8, 4], [26, 2, 13, 2, -4], [6, 4, -7, -6, -9], [30, 1, -9, 0, 0], [2, 12, -7, 1, 10], [-12, 5, -2, 4, 8], [-10, 10, -7, 2, 6], [34, 1, 14, 0, -1], [-26, -6, 5, 0, -8], [-4, -12, 3, -9, -3], [20, 12, -7, -2, 1], [-10, -8, 4, 5, 6], [58, 3, 2, -4, 1], [-60, 10, -10, -2, 8], [-20, 16, -4, -1, 11], [-18, 0, 1, 2, 10], [10, -11, 3, 1, 7], [0, 12, -7, -6, -4], [-8, 12, 8, 0, -6], [10, 14, 1, 4, -2], [8, 12, -3, 2, 4], [-20, 8, -10, -5, -2], [4, -14, -2, 2, 2], [18, 0, 7, 2, 0], [-10, 6, 12, -1, 0], [6, 8, -2, 8, -2], [18, -20, 4, 2, -4], [12, -12, -3, -6, -6], [48, 0, 5, 2, -2], [-2, -5, 7, -1, 2], [26, -2, 3, -8, 3], [22, -1, 6, 4, -7], [4, 1, 8, -6, -9], [34, 10, 2, 4, -4], [-26, -5, 4, 0, 3], [-30, 24, -2, 0, 14], [-16, -8, 9, 0, 3], [-34, 13, 5, 4, 9], [28, 10, -1, -3, -2], [2, 10, -8, -4, 2], [26, -10, -2, -9, -2], [0, -1, -2, 8, -2], [-12, -6, 1, 2, 4], [-22, 4, -7, -2, 4], [0, -17, -6, 4, 5], [28, 7, 3, 0, 11], [-24, -12, 4, 1, 2], [-36, 10, -4, -4, 8], [4, -14, 5, -1, -20], [12, -7, 12, 5, 4], [6, -2, -1, 3, -4], [8, 4, -2, 1, -7], [-14, -6, 2, -6, -2], [6, 20, 4, -4, -4], [-16, -5, 6, 4, 5], [14, -6, -3, 2, -10], [8, -6, -9, -6, -9], [28, 12, -4, 0, 2], [52, -9, 0, 2, 3], [16, -12, 4, 4, -8], [4, 2, -4, 4, 0], [20, 2, 1, -8, 10], [32, -20, 0, -7, -10], [-34, 4, -2, -2, -2], [-2, 11, -2, 9, 3], [-22, 16, -5, 0, 8], [6, 2, 2, 10, 12], [-28, -6, 2, 3, 0], [28, -9, 8, 6, -1], [36, 2, 4, -1, 3], [20, -14, -9, 1, 3], [42, -3, 4, 0, -3], [12, -2, -3, -11, -2], [18, -15, -7, -2, -7], [-14, -2, -13, -4, 0], [-6, -21, 1, 5, 0], [18, 11, -3, -2, 3], [4, -4, -3, 1, 12], [-28, 3, -10, -5, -10], [-6, -16, 13, -8, -12], [22, -2, 10, -3, -1], [32, 18, 0, 4, -4], [22, -2, 16, -4, -10], [36, -4, 8, 2, 0], [-14, 8, -1, 5, 5], [-16, -6, 0, -6, -8], [12, 11, 4, 0, 11], [-32, 4, -6, -1, 6], [0, 1, 9, 6, 5], [30, -6, -1, -2, -1], [14, -15, -1, 1, -1], [28, 2, -4, 4, 3], [-26, -3, -13, 4, 4], [-40, 18, 8, 0, -5], [-28, 8, -4, -1, -6], [42, -20, 3, 4, 0], [12, -16, 13, -6, -14], [6, 2, 5, -6, -10], [-18, -11, 8, 1, -7], [10, 14, 2, 8, -1], [20, -12, -4, -4, 6], [-6, 4, 4, 4, 12], [-14, -10, 4, 7, 6], [-6, -8, -4, 2, -15], [28, 4, 3, -8, 2], [2, -10, -12, 1, 0], [10, -18, 4, 7, 6], [0, -15, 4, -2, -8], [-4, 12, 12, -2, 6], [-40, 6, 1, 3, -2], [4, -18, -2, -2, 4], [-10, 14, -4, -8, 6], [-42, 2, -8, -3, -2], [-44, 13, -6, -2, 11], [8, -18, -5, 3, 3], [14, -6, -2, -6, -2], [24, -12, -4, 0, -12], [-8, 1, -2, 0, -3], [-16, 0, 3, 1, -5], [8, 6, 0, -4, 18], [-12, -9, 2, 2, -1], [-10, 14, -2, -2, 8], [-16, -1, 8, 0, 4], [38, -4, 1, 0, -1], [0, 0, 14, 3, -1], [22, -18, 5, -6, -12], [16, 6, 3, 6, 4], [-24, -6, -19, 3, 2], [-38, 2, 5, -6, -12], [6, -14, 0, -11, -4], [14, -21, 2, 4, -1], [-6, 8, 0, 0, -9], [0, -3, 6, -6, -11], [-10, -2, 4, 1, -4], [26, 16, -8, 4, 16], [2, 2, 4, 9, -7], [18, -7, -1, -4, 11], [-18, 3, -8, -3, 1], [-4, 11, -4, 0, 7], [-30, 6, 0, 3, 8], [-44, 6, 5, 4, 4], [-6, -2, -6, -2, 2], [-18, 2, -1, -1, -9], [52, 2, 6, 3, -2]];
    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_bx();".
// 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_bx(: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_bx();".
// 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_bx( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6160_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![28, 24, -22, -12, 2, 1]>,<13,R![112, -408, 102, 30, -12, 1]>,<17,R![-216, 56, 74, -26, -2, 1]>,<19,R![5488, 392, -532, -62, 8, 1]>,<23,R![56, -104, -208, -42, 4, 1]>],Snew);
    return Vf;
end function;