
// Make newform 6840.2.a.bk in Magma, downloaded from the LMFDB on 17 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6840_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_6840_2_a_bk();".
// 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_6840_2_a_bk();".
// 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, -6, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0], [-4, -1, 1], [3, 3, -1]];
    Rf_basisdens := [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_6840_a();"
function MakeCharacter_6840_a()
    N := 6840;
    order := 1;
    char_gens := [1711, 3421, 5321, 2737, 6481];
    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_6840_a_Hecke(Kf)
    return MakeCharacter_6840_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], [-1, 0, -1], [-1, 0, 1], [-2, 1, 0], [0, -2, 0], [1, 0, 0], [3, 1, 1], [-3, 2, 1], [-2, -2, 0], [-2, -3, 0], [-2, 1, -2], [0, 3, 0], [5, -1, -1], [-7, 1, 1], [-7, -3, 1], [1, 3, 1], [2, 2, 2], [2, -4, -2], [4, -2, 2], [-3, -1, 1], [0, 2, 2], [-8, -3, 0], [-2, -3, 0], [-12, 0, -2], [2, 2, 2], [0, 4, 0], [5, 3, -1], [-11, 3, 1], [-2, -2, -2], [-7, 2, -1], [-5, -1, 1], [6, -4, -2], [-14, 0, 0], [-4, 4, -2], [-5, 1, -3], [8, -1, 4], [1, -3, -1], [-3, -1, -3], [-15, 1, -3], [-13, 1, 1], [5, 0, -1], [2, -1, 0], [-1, 3, 3], [4, 6, 0], [-2, -2, 2], [-16, -4, 0], [5, 1, -1], [13, 3, 1], [5, -3, -5], [-13, -2, 1], [-18, 0, -4], [17, 0, 3], [-1, 1, 3], [3, 3, 1], [3, -6, -5], [-14, 0, -6], [1, 1, -1], [-20, 3, 0], [0, -5, 0], [-11, -1, 1], [2, 4, 2], [5, -2, -1], [2, -6, 4], [1, 1, -3], [5, 3, -1], [-14, -1, 0], [16, 2, -2], [-12, -2, 2], [-3, -3, -1], [-27, 0, -1], [-9, 4, -5], [-6, 5, 0], [15, -3, 1], [-2, -2, -6], [-10, 8, 4], [1, 5, 3], [4, 1, 4], [-8, 0, -6], [-15, 2, -5], [-13, 3, 1], [4, 6, -4], [18, 7, 0], [1, -9, -3], [-6, -10, 4], [0, -9, 4], [-2, -10, 4], [2, -8, -4], [-7, 2, 5], [-14, 6, 0], [11, -8, 1], [-8, 6, 4], [7, 2, 9], [10, 0, -6], [-7, -3, -9], [3, 2, 3], [-8, 11, 8], [12, -2, 8], [18, 4, 4], [-2, 0, -2], [-2, 4, -8], [11, 11, -3], [20, 3, 0], [4, 14, 0], [-24, 0, -6], [4, -12, -6], [1, 9, 3], [6, 10, -2], [6, -2, -4], [-2, -4, -2], [-1, -11, 5], [-22, -4, 2], [36, 2, 0], [-16, -4, 0], [-10, -1, 6], [-4, 1, -4], [-3, 1, 3], [-5, -9, -5], [-9, 1, -9], [3, -3, 9], [-2, -1, 0], [-13, -5, 3], [2, -2, -6], [12, 2, 8], [-24, 2, -2], [-13, -3, 3], [1, -4, -5], [-9, 2, 3], [25, -7, -5], [12, 0, 8], [-2, -2, 2], [-24, 8, 6], [15, -7, -3], [-6, 0, 0], [0, 6, -2], [-9, 11, 3], [4, -10, -8], [-13, 1, 3], [-18, 4, 4], [26, -2, 6], [10, -6, -8], [3, 0, 3], [17, 1, -1], [-17, -11, -3], [-10, -12, 2], [-19, -1, -5], [41, 1, 1], [-2, -6, -2], [-2, -2, -6], [4, 1, -2], [12, 4, -6], [-6, 5, -2], [20, 0, -4], [-10, 0, -2], [24, 4, 4], [14, 2, 10], [16, 4, -2], [4, 8, -2], [-19, 8, -3], [18, -4, 8], [-25, 5, 3], [23, -6, -1], [-1, -3, -1], [-5, -9, 7], [-5, -1, 5], [15, -11, -5], [7, 5, 1], [-28, 6, -2], [6, 8, -8], [-1, 3, 7], [-9, -9, -3], [-25, -2, -3], [12, -6, -6], [11, 13, -1], [10, 10, 0], [0, -10, 4], [12, -8, 2], [-40, -2, 0], [-40, 2, -2], [21, -4, 1], [-3, 11, 5], [4, 5, 2], [-7, -5, 1], [-3, -1, -1], [16, 6, 10], [18, -3, 0], [24, -2, -4], [10, 2, 8], [-5, -8, -7], [36, -5, 2], [6, -6, -4], [54, -2, 2], [-1, -2, 3], [14, -10, -4], [-7, -5, -11], [23, -3, 7], [25, 13, -1], [16, 2, -4], [3, 1, -7], [-25, 2, -5], [-1, 1, 3], [8, 5, -2], [-12, -2, 2], [9, -10, -9], [15, -5, -5], [42, 6, 2], [-23, 5, -1], [18, 18, -4], [-18, 2, 2], [16, -4, -2], [6, 2, 8], [7, 12, 11], [-23, 13, 3], [28, 10, 0], [0, 4, 6], [21, 2, 17], [-16, 7, 0], [17, 17, -1], [27, 15, -5], [-45, -1, -3], [8, -16, -2], [6, -17, -2], [-1, 14, -9], [25, -3, 11], [16, 10, 6], [13, -1, 5], [-4, 6, 4], [36, 0, 8], [-23, -10, -1], [39, -3, 1], [11, -7, 1], [-19, 7, 9], [20, -7, -4], [-12, -7, 0], [-3, 3, -9], [25, -5, 13], [-12, -6, 2], [7, 0, -11], [34, 0, -2], [-30, -6, 2], [32, 20, -4], [-9, -16, -5], [-7, -9, -17], [-13, -9, 3], [-39, -6, 3], [9, -8, 1], [1, -9, -11], [9, 3, -9], [19, -1, 1], [11, 1, 1], [6, 4, 0], [-21, 19, 5], [0, 0, -2], [1, -3, 1], [7, 18, -3], [-1, -7, -13], [-6, -2, 10], [-11, 1, 9], [42, -8, -4], [4, -12, 12], [26, -10, -10], [34, 8, 8], [16, 1, 18], [30, 4, -6], [-37, 9, 1], [39, 2, -5], [4, 12, 10], [-4, 4, -8], [-1, -1, 7], [7, -5, -11], [34, 2, 10], [-6, 10, 4], [40, -6, 4], [2, -5, 0], [-3, 8, -11], [14, -4, -8], [-13, -7, 5], [64, -4, 2], [-15, 16, 7], [-2, -14, -6], [-22, -14, 6], [17, 3, -9], [-45, 7, 1], [10, -1, 10], [21, 18, -9], [-22, 16, 8], [9, -7, 9], [-20, 12, -4], [18, 13, -2], [-28, -8, -14], [-12, -2, 6], [-23, -11, -11], [-29, 3, -13], [-6, 17, 8], [-2, -14, -8], [-43, 3, 1], [3, -1, 7], [-21, 21, 11], [-26, 3, -6], [32, 18, -2], [-39, 1, -7], [2, 2, -2], [33, -3, 15], [-6, 0, 2], [40, 9, -2], [-6, 22, 6], [-25, -9, 1], [37, -2, 7], [19, -19, -7], [-55, -1, -1], [-18, 4, -8], [8, 1, -8], [-18, 12, 6], [-5, -17, -15], [40, 10, -2], [3, 12, 1], [30, 6, -2], [-48, 4, 6], [10, -20, 0], [-15, -5, -9], [48, 0, -2], [17, -8, 5], [-36, -6, 8], [-8, -2, 8], [-10, -8, 4], [-50, -18, 2], [-20, 18, 8], [-21, -9, -3], [21, -7, -7], [6, 4, -12], [-15, 1, 1], [0, -14, 8], [18, 12, 0], [-34, -4, 6], [56, -10, -6], [-33, 19, 9], [-5, 15, -1], [-7, 3, 1], [-23, -14, -11], [40, -3, 2], [-49, -9, 1], [13, -20, 1], [-24, 0, 2], [39, -5, 15], [11, -7, -1], [-15, -5, 9], [16, 4, 4], [0, -16, 8], [-2, 8, -4], [21, -5, -9], [4, -23, -10], [-20, 8, 14], [-46, 4, 2], [-9, -23, -3], [79, 3, 3], [-23, -6, -5], [1, -2, 7], [36, -6, 8], [-30, -6, -2], [-21, 7, 13], [0, -14, 2], [-50, -6, -6], [-3, -3, 1], [-18, 1, -14], [10, 17, -4], [-40, 6, -6], [-16, -6, -12], [-32, 4, -2], [-25, -8, -7], [40, 0, 12], [-23, -11, -1], [21, 16, 1], [24, -8, 20], [-7, 23, -1], [17, 15, 1], [-9, -12, -3], [82, 4, 0], [44, -4, 10], [-8, -11, 6], [-9, -22, 3], [-6, 16, 14], [-1, -6, 11], [-3, -19, 3], [-19, -5, 3], [-19, 1, -13], [-12, -22, 0], [37, -3, -5], [-20, -7, 0], [9, 25, -1], [20, 16, -6], [-21, 7, -5], [-35, -2, -1], [-16, 9, -12], [-2, 8, 6], [-2, -15, 4], [-20, 20, 14], [-2, -11, 2], [-85, 1, -3], [10, -4, -8], [-31, 9, -5], [18, 16, -2], [32, -10, -8], [-57, -3, -5], [-5, -4, -5], [-32, 4, -4], [15, -3, -7], [-48, -5, 4], [30, 4, 6], [-33, -4, 9], [8, -5, 6], [25, -3, -11], [-16, -8, 4], [-8, -18, 12], [20, -6, -14], [-34, 8, 4], [-16, 4, -20], [18, 18, 6], [-43, 9, -1], [-14, 18, 14], [3, -8, 7], [12, -11, 14], [-5, 1, 1], [35, -15, -5], [-12, 13, 10], [-13, -27, -5], [-32, 12, 0], [-22, -4, 4], [0, 2, 4], [-29, -8, 9]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6840_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6840_2_a_bk();".
// 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_6840_2_a_bk(:prec:=3)
    chi := MakeCharacter_6840_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_6840_2_a_bk();".
// 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_6840_2_a_bk( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6840_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![-44, -18, 2, 1]>,<11,R![8, -14, 4, 1]>,<13,R![-4, 2, 6, 1]>],Snew);
    return Vf;
end function;