
// Make newform 6840.2.a.bg in Magma, downloaded from the LMFDB on 10 September 2025.
// 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_bg();".
// 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_bg();".
// 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 := [-1, -4, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0], [-3, 1, 1], [-3, -1, 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 := 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]];
    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], [2, 0, 1], [-2, -1, 0], [1, 0, 0], [0, 1, -2], [-6, -1, 0], [0, -2, 0], [2, -1, -1], [-2, -2, 2], [-4, 0, -2], [0, 2, 2], [-2, -2, 1], [-4, -1, 4], [2, 0, -2], [0, 2, -5], [0, -4, 0], [2, -1, 4], [0, 0, -2], [4, 2, 0], [-2, 2, -2], [-2, -3, 1], [6, 4, 2], [-12, 1, 1], [0, 2, 3], [6, -3, 2], [2, 1, -1], [-4, -3, -1], [4, 4, 0], [-2, -1, -4], [0, 4, 4], [-10, -4, -6], [0, 0, 6], [-2, 2, 2], [-20, 0, -2], [4, 1, 7], [-2, -1, 3], [-4, 2, -2], [6, 0, -4], [-8, -1, -6], [-2, 1, -7], [2, 0, 0], [0, -3, -6], [-4, -3, -4], [-12, 3, 1], [-8, 0, -1], [2, 2, -6], [-2, 8, 0], [0, 3, 2], [-6, -4, 6], [-12, 0, -4], [-6, -3, 13], [0, 8, 0], [-14, 4, 0], [12, 1, -2], [-10, 0, -6], [6, 2, -6], [-4, 2, -4], [-2, 4, -3], [0, 3, 3], [-4, -5, 2], [-6, 9, -6], [-2, -2, -5], [4, -1, 4], [-2, 1, 1], [-20, 2, -4], [-2, -4, 12], [-2, -3, 4], [-20, 1, -2], [-16, -6, -6], [18, 0, 3], [4, -1, 10], [-4, 5, -3], [-6, -2, 4], [6, -8, 2], [22, 0, -4], [2, -4, 4], [-12, -2, -8], [22, 5, 0], [-16, -2, 6], [-2, -9, 9], [0, 0, -10], [4, 8, -4], [6, -4, -4], [-14, 1, -2], [-14, 0, -8], [-16, 0, 8], [4, -4, -4], [-4, -8, 12], [-12, 3, -11], [4, -6, 8], [-16, -2, 4], [-16, -3, -2], [18, -2, 2], [6, -2, 8], [16, 0, 11], [10, 0, 2], [-24, -1, -1], [18, 4, 14], [0, -9, -1], [6, 2, -12], [16, 6, -8], [-14, -1, -14], [-20, -2, 4], [-2, 0, 4], [-16, 0, -2], [34, 2, 6], [-4, 7, -3], [22, 4, 6], [-10, 10, -10], [-32, 0, 4], [12, -8, 4], [-10, 4, 2], [-4, -6, 0], [-16, -3, -2], [18, 4, -6], [-4, 13, 0], [6, 3, -12], [-10, -7, -13], [14, 0, 13], [-8, 3, -13], [-8, 2, 8], [22, 0, -6], [-26, 0, -4], [28, -5, 6], [0, 9, 4], [14, -6, 0], [4, 6, 4], [-4, -1, -11], [16, 4, 14], [14, 8, -12], [10, 5, 12], [-2, -7, 0], [-10, 6, -1], [-16, -4, 9], [-10, -6, 17], [14, 9, 14], [20, 1, -6], [-6, -10, 8], [16, 7, 8], [-16, 2, 9], [22, -7, 6], [-8, 2, -2], [14, -4, 8], [-30, -9, -15], [-12, 0, -12], [4, 1, -11], [-6, -4, -1], [2, -2, -6], [-12, -4, -2], [-12, 13, -3], [-16, -6, 5], [16, 10, -10], [-24, 3, 2], [6, -3, 14], [-14, -1, -8], [10, 5, -10], [-4, -4, 8], [-30, -5, 1], [0, 12, -16], [28, 0, -8], [-6, 1, 5], [-12, -1, 9], [24, 8, 18], [-18, -6, 14], [-6, 0, 2], [34, 2, 10], [4, 3, -18], [38, 3, 10], [-20, 0, -12], [-22, -5, -6], [16, 6, -12], [18, 2, -14], [20, 6, -4], [6, 2, 22], [20, 1, 7], [26, 2, -10], [-8, 1, 10], [-4, -1, 2], [2, -5, 11], [-46, -5, -7], [-16, -7, 22], [10, 4, -12], [-6, 0, -1], [16, -3, 13], [-6, 10, 8], [40, 7, 2], [6, -7, -9], [12, -8, 4], [44, 4, -4], [-6, 5, 4], [28, 2, 2], [26, -9, 3], [26, 3, 6], [-10, -2, -20], [-18, 1, -6], [-40, 5, -8], [34, 2, 6], [24, 2, 6], [10, 8, 13], [-14, 14, -10], [8, 6, 0], [26, 8, 2], [-4, 1, 6], [-24, -8, -17], [-18, 13, -6], [12, 7, 14], [-22, -3, 18], [-10, 0, -6], [8, 8, -8], [32, 0, -3], [-32, -6, 16], [-6, -8, -14], [16, -7, 0], [-10, -16, -8], [28, -1, 9], [18, -12, 4], [-10, -2, -22], [-24, 2, 12], [22, -4, 14], [8, -1, -10], [8, -6, -13], [-10, -8, 0], [2, 1, 13], [16, -12, 4], [-52, -5, 3], [-12, 6, -4], [-2, 6, -4], [20, -5, -14], [16, -14, 6], [22, -14, 10], [28, -6, -8], [-16, 1, -20], [-10, 2, 6], [18, 8, -4], [-40, -10, -20], [8, -8, 10], [-8, 7, -17], [-28, -8, 0], [24, -5, -8], [-2, -11, 12], [-14, 15, -5], [-24, -4, 4], [16, 6, -10], [44, 6, 10], [44, 4, 8], [32, -9, -4], [-34, 6, -8], [6, 11, -14], [-32, -7, 10], [-14, -2, -16], [34, 8, -4], [12, 0, 24], [22, 0, -16], [-16, 5, 5], [-42, -1, -13], [26, 10, 26], [-36, -5, -3], [64, -5, 3], [42, -6, 2], [10, 10, -9], [-2, -4, -16], [-52, 1, -4], [2, 5, -16], [2, -7, 24], [0, -2, 29], [66, -2, -4], [54, -1, 2], [40, 8, -5], [-22, -9, 24], [4, 15, -14], [-34, -5, 1], [-8, 23, 0], [28, 2, 6], [-2, 9, 8], [34, 4, 2], [16, -8, 0], [28, 15, 1], [-20, -11, -10], [16, -11, 2], [22, 7, 18], [-20, -12, 0], [4, 0, 16], [18, 13, -6], [14, 5, 5], [24, -9, 14], [-2, 4, -4], [2, -2, -24], [-28, 2, -18], [42, 5, 11], [-28, 2, 2], [-30, -7, 29], [-58, -2, 10], [-32, 10, -12], [-6, -16, 4], [-28, -13, 10], [36, -6, -2], [-30, -2, -2], [-26, 0, -7], [52, -3, 2], [-32, 4, -13], [4, -18, 0], [-2, 17, -3], [32, -2, -11], [38, 7, -20], [8, 4, 4], [14, -2, 24], [8, 18, 2], [-26, -12, -14], [-10, -6, 20], [-32, -3, -11], [16, 4, 8], [10, 10, 10], [0, -10, 20], [0, -2, 6], [-38, 7, 10], [14, -4, 0], [-12, -11, -6], [-30, 5, -8], [-22, -9, -4], [4, 19, -13], [-2, 19, -16], [18, -8, 2], [28, -15, 4], [0, 4, 15], [52, -5, -3], [-14, 10, 10], [-14, -12, 14], [-42, 4, -15], [-4, -11, 18], [20, -4, 22], [48, -2, 0], [4, -16, -10], [38, 9, 18], [-22, -9, -3], [2, 13, -2], [8, 13, 14], [-14, -4, 7], [-66, 7, -8], [-22, -19, -2], [-32, 10, 0], [22, 14, 25], [44, 14, 10], [38, 0, 8], [-16, 20, -21], [-40, 12, 0], [-22, 12, 10], [20, -12, 12], [14, -11, -13], [-34, -12, 2], [-12, 3, 19], [6, -2, 6], [-14, 11, -11], [16, 15, -22], [-4, -4, -4], [30, 11, -20], [12, 3, -7], [-34, 14, 6], [-30, -18, 2], [52, -9, 3], [-12, 5, 18], [28, 12, 28], [-26, -3, -23], [-54, -6, -6], [12, 3, -13], [-22, -10, 14], [56, -10, 0], [-20, -13, 12], [8, -1, -24], [26, 9, -10], [24, -4, 12], [22, -4, -6], [-36, -9, -2], [-16, 4, -20], [-6, -16, 0], [6, 1, -10], [-74, -3, 5], [18, 4, -24], [-34, -16, -4], [8, -21, 0], [6, 8, 16], [-4, 17, -8], [-20, 1, 15], [4, 11, -2], [6, 10, 18], [-36, 0, -18], [-44, -5, -17], [-38, -16, 8], [-50, -16, -23], [-4, 22, -12], [20, -12, 0], [16, 20, -6], [-50, 3, -1], [32, -8, 10], [14, 16, 6], [44, 8, 0], [10, -4, 8], [22, -4, 20], [-26, -8, 12], [-60, -5, 5], [-14, 9, 17], [50, 0, 20], [-56, 1, 2], [38, 4, -2], [-50, 10, -12], [56, 4, -13], [-20, 6, 4], [-42, -1, -9], [18, -24, 12], [16, 12, 7], [-24, -20, -24], [-14, -17, 12], [2, -4, -16], [16, -10, 18], [4, 5, 27], [6, -9, 12], [-12, -11, 13], [18, 16, 0], [-46, 16, -1], [32, -15, 4], [-36, -5, -2], [6, 27, -7], [22, 4, -31], [-8, -22, 9], [18, -27, 8], [4, 4, -16], [28, -17, 6]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6840_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6840_2_a_bg();".
// 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_bg(: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_bg();".
// 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_bg( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6840_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<7,R![-16, -12, 1, 1]>,<11,R![0, 1]>,<13,R![4, 2, -5, 1]>],Snew);
    return Vf;
end function;