
// Make newform 4026.2.a.v in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4026_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_4026_2_a_v();".
// 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_4026_2_a_v();".
// 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 := [-8, 18, 13, -9, -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], [-4, -1, 1, 0, 0], [10, -1, -9, 0, 1], [-6, -11, 7, 2, -1]];
    Rf_basisdens := [1, 1, 1, 2, 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_4026_a();"
function MakeCharacter_4026_a()
    N := 4026;
    order := 1;
    char_gens := [1343, 1465, 3235];
    v := [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_4026_a_Hecke(Kf)
    return MakeCharacter_4026_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 := [[1, 0, 0, 0, 0], [-1, 0, 0, 0, 0], [1, 1, 0, 0, 0], [0, 0, -1, 0, 0], [1, 0, 0, 0, 0], [-1, 1, 0, 0, 1], [0, 0, 0, -1, 0], [4, -1, -1, 0, 0], [0, 1, 0, 2, 1], [1, 0, -1, -1, 0], [1, -2, 0, 0, -1], [2, 1, 2, 0, -1], [1, 2, -1, 0, -1], [-2, 1, -1, 0, 1], [2, -3, 1, 1, -1], [2, 1, 2, 2, 2], [1, 2, 0, 0, -1], [1, 0, 0, 0, 0], [-2, 0, 0, -1, -1], [8, -4, -1, -1, 0], [2, 2, 2, 2, 3], [4, -1, 2, 1, -1], [-6, 0, -2, 0, 3], [3, -1, 2, 2, 2], [7, 0, 1, -2, -2], [5, -3, -1, -2, -1], [-6, 3, 2, 1, 0], [-1, -1, -3, -2, 0], [1, -1, 2, 0, 1], [-4, -3, -2, 2, 0], [0, -2, -2, -1, 0], [9, -1, 1, 3, 1], [6, -2, 1, 1, -1], [5, -1, 1, 2, 0], [0, 0, 1, -1, -1], [8, 0, 0, 2, 0], [1, 0, 1, -1, 0], [1, 4, 3, -1, 0], [-2, -3, -2, -2, -2], [-3, 0, 3, 1, 0], [0, -5, 0, -3, -2], [-5, 0, -4, -2, 0], [-10, 3, -4, -3, -3], [-7, -2, -2, 0, 3], [4, -4, -1, -2, 0], [2, 0, 1, 0, -1], [-5, 2, -4, -3, -2], [-6, 3, 6, 4, 2], [6, 0, 3, 2, 2], [12, 3, 3, 0, 2], [0, 2, 4, 2, 4], [-6, 3, -3, -1, 3], [8, -3, 0, 4, 1], [5, 4, -3, -1, -2], [8, 0, 2, 3, -1], [8, -2, -3, -2, -4], [3, 2, -3, 2, -1], [0, 7, 2, -2, -1], [-2, -1, -4, -3, -1], [6, 0, 3, 2, 2], [-2, 3, 0, 0, -2], [6, -5, 1, -2, 1], [-3, 1, -2, -3, 1], [0, -5, -5, -2, -2], [-4, -1, -1, -4, 0], [5, 5, -1, -1, -3], [-10, 2, 1, 1, -2], [-13, -2, 0, -1, 0], [3, 1, 6, 4, 2], [4, 0, -3, 2, -1], [10, -5, 2, -1, 1], [3, 2, 2, 2, 0], [-18, 0, -3, 1, 5], [2, 0, 1, 7, 2], [-5, -6, 1, -2, 3], [-2, -2, -3, -5, -1], [-2, 0, 1, -2, 2], [3, -8, -5, -2, -4], [-1, 2, 7, 3, 1], [-7, 4, 2, 8, 5], [0, -3, -2, 1, 1], [19, 3, 2, -1, -3], [-6, 0, 1, -7, -1], [-4, -12, -3, -1, -7], [-1, -1, 1, -4, 4], [-8, 2, 5, 1, -2], [4, 8, 3, -1, 3], [-24, 6, -2, -5, 1], [-2, 2, 7, 2, 3], [-2, -5, -1, -3, 0], [-13, 7, 3, 3, -2], [2, -9, 0, -2, -8], [-10, 0, -5, 2, -4], [-6, 1, -6, -2, 0], [-18, 5, 3, 1, 2], [-16, 8, 6, 3, 3], [0, 3, 5, 0, 2], [0, -4, 3, 4, 2], [-3, -5, 1, 1, -1], [-6, -9, 7, 0, 1], [-9, 7, 2, 2, 5], [17, -2, -2, -4, -3], [4, -2, -11, -7, -5], [1, 4, 2, 4, -3], [2, -5, 7, 3, -3], [-16, 9, 5, 0, -2], [3, -4, 2, 0, 1], [8, -5, -1, -4, -7], [-20, 1, 3, -1, 2], [2, -12, 2, -3, 2], [5, -9, -2, 0, -2], [-15, -5, -5, 1, 2], [5, 4, -8, -3, -3], [-5, -1, -3, 2, 7], [-14, -6, 0, -1, -3], [6, -2, -3, -8, -3], [-24, -5, -2, -1, 2], [20, -5, -7, -2, -1], [-12, 2, 3, -5, -3], [-5, -2, -7, 1, 3], [5, 5, 5, -3, 2], [-3, -3, -5, 4, 1], [0, -17, 1, 1, -3], [2, -6, -1, -6, -3], [15, -2, 6, 0, -5], [-13, 5, -3, 1, 6], [-9, 16, 3, 1, -3], [-8, -7, 2, -3, -5], [18, -6, 2, 7, 3], [-24, -6, 2, -3, -3], [-3, -1, -3, 6, 2], [7, 8, 3, 1, 1], [-12, 5, 4, 9, 12], [-10, -2, -1, -4, 0], [8, 3, 0, 3, 3], [5, -5, 5, -1, -6], [5, -9, -5, 4, 4], [7, 3, -3, -4, -5], [-18, 15, 6, 1, -1], [-21, -8, -4, 1, 4], [-18, 8, 0, -2, -1], [-6, 2, 1, 3, 5], [7, -13, -5, 2, -5], [-16, -5, -5, -1, -2], [-10, 2, 0, 1, -5], [-7, 4, -4, 5, 6], [-23, 11, 7, 3, 0], [21, -9, 5, 4, 1], [-5, -1, 10, 2, -3], [-9, 5, -3, -4, 7], [-6, -1, 4, -1, 2], [18, -15, -1, 3, 1], [-8, -5, -2, 2, -3], [20, -7, -3, -3, -6], [-12, -1, 3, -7, -6], [1, 1, 4, 5, -2], [-17, 2, 2, -3, 1], [16, -8, -1, 1, 1], [10, -5, 3, 1, -4], [-5, -2, 1, -4, -7], [-8, 9, -6, -3, -1], [16, -10, -7, -1, -4], [-3, -10, 6, -1, 1], [6, -6, -1, 1, -6], [-10, -13, -3, -1, -7], [22, -5, 4, 3, 3], [-6, 13, -4, -3, 5], [2, -6, -7, -5, -6], [1, 5, 7, 6, 9], [12, 11, 0, -3, 3], [-6, 13, 3, 8, 4], [-14, 6, -5, -3, 1], [3, -5, 3, -8, -2], [-20, 9, -2, -5, 0], [2, -3, 5, 5, -1], [30, 0, -2, -5, -5], [12, 7, 0, 3, 3], [-6, -7, 1, 3, 5], [0, -1, -10, -3, 5], [-10, 7, 4, -8, -1], [-11, -2, 10, -7, -4], [-4, 11, 5, 0, 6], [-30, 6, -5, -10, -2], [6, 5, -4, -1, -5], [7, -12, 4, 2, -9], [2, 16, 7, -2, 4], [-14, 11, -4, -9, -3], [15, -1, 0, 0, -9], [10, -3, 2, 13, 7], [1, -1, 1, 3, 2], [32, -15, -2, 5, -1], [5, -3, 10, 8, -7], [-23, -8, 1, 0, -5], [15, -6, -2, -4, -6], [16, -5, -6, -2, 0], [30, -12, -5, -1, 1], [14, 3, 9, -2, -1], [-1, -3, 1, 8, 2], [16, 3, -4, 0, 8], [0, 8, 4, -1, -7], [-20, 9, 5, -4, 1], [14, 0, 10, 2, -6], [-9, 15, 6, 7, 10], [-1, 4, 0, -4, 5], [20, 2, -3, -4, 1], [3, 0, 4, 5, 11], [-5, -6, -14, -6, -5], [4, 11, 4, 0, 8], [-2, 19, 10, -1, 4], [-8, 12, 8, -2, -1], [-22, -8, -2, 5, 8], [-6, -8, 4, -7, 2], [12, -8, -1, -11, -8], [-11, 15, 7, -7, -2], [19, -1, -7, -4, -4], [22, 1, -4, 10, 4], [21, 7, 1, 4, 1], [0, -1, -7, -5, -3], [-37, -7, -3, -1, -2], [6, -12, -9, -7, -13], [-31, -9, -5, 1, 2], [7, -10, -1, -9, -13], [21, -6, 8, 4, -4], [-2, 1, 11, 9, 0], [4, -8, 4, 7, 1], [26, -4, -5, -1, 4], [13, 3, 6, 2, 5], [12, 2, 4, 6, 0], [6, 20, 3, 3, 13], [-8, -11, -10, 0, -7], [-1, -4, -4, 9, 5], [10, 4, -12, -7, 1], [-2, -9, -9, -4, -4], [14, 13, 9, -5, 0], [-21, 7, 5, 6, 7], [-1, 3, -10, -1, 1], [-8, 3, -1, -3, 5], [15, -18, 0, -1, 4], [9, 5, 4, -11, -8], [47, -6, -1, -2, -5], [-7, 3, 9, -6, -7], [-14, 10, 11, 14, 12], [7, -1, 1, -7, -10], [14, -17, -12, -7, -1], [6, 12, 0, -4, 3], [30, 2, 8, 3, -5], [18, -19, 4, 5, -10], [10, -2, -16, 0, 6], [-4, 12, -14, -6, -2], [4, 16, -1, -5, -9], [0, -23, -8, 1, -7], [-19, 6, 8, -7, -2], [0, 0, 9, 8, -3], [36, -15, -9, -11, -7], [24, -3, -1, -7, -8], [-20, 30, 9, -1, 5], [9, -1, 3, 4, 7], [12, 16, -4, 2, 0], [8, 1, 2, 4, 4], [-1, 11, 9, -6, -3], [-1, 4, -8, -9, 5], [-17, 19, 0, -5, 1], [16, 11, 9, -3, 1], [23, 2, 6, 4, -3], [14, -20, 5, 5, -5], [-1, 1, 7, 7, -8], [7, 1, -7, 5, 4], [-8, -8, -2, 7, 1], [-4, 0, 20, 3, -7], [-14, 8, 2, 4, 1], [-2, 15, 9, -8, -8], [-15, 13, 1, 3, -6], [-17, -5, -3, 0, 1], [-14, -14, -3, -2, 0], [1, -19, -11, 6, -1], [20, 4, 5, 4, 7], [-11, -3, 7, 1, -6], [8, 14, -4, 9, 11], [-8, -7, 11, 3, -3], [21, 0, -2, -4, -7], [-15, -10, 4, 7, 13], [-2, -2, 10, 10, -5], [-6, 10, -2, -1, -9], [-22, -16, 7, -7, -8], [-15, 15, -3, -2, 12], [-11, -14, -11, -7, -10], [18, -9, -5, -11, -15], [20, -3, 6, -3, 8], [30, 4, 11, 1, 10], [14, 2, -7, -1, 5], [-6, -20, -1, -2, -4], [-14, 6, -5, -3, -4], [-17, 14, 1, -3, 0], [-24, -1, 5, 1, -1], [-29, 9, -2, -14, 1], [4, 1, -3, -7, 6], [32, -15, -18, -10, -3], [-8, -1, 13, 1, -5], [26, -5, 3, 1, -3], [12, 7, 10, -10, 1], [-2, -20, -6, 0, -8], [0, -8, 0, 9, 1], [-8, 0, 13, 10, 3], [16, -3, 10, 1, 5], [-12, 20, 7, 0, 0], [-4, 6, 4, 6, -11], [2, 6, -2, -5, -9], [2, -10, 1, -4, -10], [34, 1, 3, 13, 9], [18, 8, 8, -2, -3], [-32, 1, -5, 5, 10], [8, 1, -10, -3, 8], [32, -10, -6, -14, -15], [-5, -10, -5, 7, -10], [12, 21, -5, 5, -1], [-14, 1, -4, -6, 6], [-38, 12, 7, 6, 0], [8, 7, 5, 3, -4], [15, -14, 0, -9, 5], [18, 10, 7, -2, -3], [10, -18, -3, -2, 0], [-10, -12, 8, 3, 5], [-24, 15, -6, -8, 12], [23, -2, -5, -7, -6], [6, 5, 15, 6, -1], [20, 3, -6, 13, 3], [18, -3, -7, -6, -5], [-4, 12, -5, 8, 13], [16, -5, 8, -8, -1], [-8, 2, 20, 1, -5], [-17, 7, -15, -6, 3], [10, 2, 7, 18, 11], [12, 24, -2, 3, 15], [15, -19, -18, 0, 2], [23, -3, 5, 8, 11], [3, 3, -19, -10, -3], [-8, 20, 0, -8, -2], [13, -1, 12, 12, 5], [20, -5, 8, -8, 0], [10, -24, -8, -2, -6], [2, -16, -16, -3, 3], [-21, 13, 8, 4, 12], [-18, 15, 11, -5, -7], [39, 16, -6, 5, 0], [0, 7, 8, 11, -6], [10, -2, 8, -3, 3], [24, 10, -4, 8, 7], [20, 7, -16, -3, 2], [-14, 17, 8, -3, 4], [-25, 11, 3, 5, 1], [0, -9, 2, -4, -3], [6, 8, 17, 2, 2], [22, -3, 7, -8, -10], [10, -10, 1, 3, 4], [-8, -4, -7, 12, -2], [2, -18, -7, 3, -5], [5, 6, 8, 13, 10], [-38, 9, -3, 1, -1], [-30, -1, -15, -4, -9], [-22, 13, -5, -3, 10], [16, -14, -23, -6, -2], [-36, 8, 2, 13, 11], [18, -14, 0, 1, -3], [-31, 2, 11, 8, 1], [1, -9, -8, -3, 8], [-20, 4, -2, 9, 3], [25, -19, 4, 1, 2], [-22, -10, 8, 3, -5], [8, 23, 3, 5, 4], [-17, 11, 5, 9, 8], [18, 9, -8, 5, 5], [8, -3, -20, -5, 3], [-14, 18, -11, -2, -1], [30, 6, -7, 0, 6], [24, -13, 4, -7, -1], [-3, 4, -5, -11, 6], [6, 5, 17, -8, -13], [-21, 4, -14, -11, -14], [-14, 21, -3, 4, 2], [10, -8, 15, 7, 9], [-14, -17, -3, -4, -8], [13, -15, 0, 2, -11], [32, 2, 5, -9, -17], [1, -15, -1, -6, -11], [-8, -11, -3, -2, -8], [40, -1, -3, 0, 1], [26, -16, -10, -3, -4], [-9, 8, -7, 9, 3], [28, -10, 10, 11, -9], [1, 6, -6, 14, 11], [-7, -4, -12, -11, -2], [-22, -2, 6, 7, -6], [-18, -2, -3, -10, -3], [10, -13, -20, -13, -13], [-6, -11, -1, 5, -3], [-9, 0, -6, -14, -11], [2, 6, -9, 3, 13], [34, -19, -11, 0, 2], [-1, 0, 4, 9, -12], [16, -9, 9, 6, -7], [17, -18, -12, -17, -10], [5, 5, -2, 1, -5], [15, 14, 6, 6, 1], [40, -4, 12, 20, 11], [-17, -13, -11, -10, -17], [21, -17, -18, -13, 3], [38, -11, 11, 11, -13], [30, 9, -15, 0, 8], [-13, 22, -8, -7, -2], [-24, 9, -4, -5, 9], [64, -4, -13, -5, -1], [-46, 0, 2, -4, -4], [43, -12, 9, 15, 6], [20, -11, -1, 15, 1], [-20, -9, 7, 9, -5], [18, -12, -10, 2, 8], [10, -5, 1, -2, 10], [26, 10, -3, -1, -12], [9, 4, 6, 3, 16], [-18, 6, 5, 4, 18], [16, -12, -5, -17, -8], [-28, -2, -8, -9, 0], [9, -21, 1, 12, 3], [-8, -11, -11, 6, 2], [-1, 1, -16, -1, 5], [-8, -5, -5, -11, -7], [-17, -12, 5, -13, -12], [21, -8, -1, -2, 9], [36, 14, -6, -2, -3], [5, 10, -5, 6, 3]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4026_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4026_2_a_v();".
// 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_4026_2_a_v(:prec:=5)
    chi := MakeCharacter_4026_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_4026_2_a_v();".
// 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_4026_2_a_v( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4026_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-7, -22, 18, 9, -7, 1]>,<7,R![-32, 50, 3, -21, 0, 1]>,<13,R![-158, 243, -23, -31, 2, 1]>],Snew);
    return Vf;
end function;