
// Make newform 8036.2.a.k in Magma, downloaded from the LMFDB on 17 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_8036_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_8036_2_a_k();".
// 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_8036_2_a_k();".
// 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, 7, 0, -6, 0, 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], [-2, 0, 1, 0, 0], [0, -4, 0, 1, 0], [3, 0, -5, 0, 1]];
    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_8036_a();"
function MakeCharacter_8036_a()
    N := 8036;
    order := 1;
    char_gens := [4019, 493, 785];
    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_8036_a_Hecke(Kf)
    return MakeCharacter_8036_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, -1, 1, 0], [0, 0, 0, 0, 0], [1, -1, 1, -1, 1], [-1, -1, -1, -1, 0], [0, 0, 0, 1, 1], [0, -3, 0, -1, 2], [0, -2, 1, -2, -2], [2, -3, 0, 0, -1], [-3, -4, 1, -1, 0], [2, 2, -2, 0, 1], [1, 0, 0, 0, 0], [6, -2, 0, -1, 1], [-2, 0, 0, 0, -3], [5, 2, -1, 1, 2], [-1, 2, 1, 5, 0], [3, 3, -3, -1, 1], [3, -4, -5, 3, 2], [3, 7, 1, 1, -5], [3, -5, 1, -1, 1], [2, -4, -2, -4, 2], [1, -3, 1, -3, 1], [5, -2, -3, 1, -1], [6, -4, -1, 4, 4], [0, 3, 1, 2, -1], [-6, 3, -2, -4, -3], [-1, 3, -5, -2, -2], [0, 2, -6, 0, -6], [2, -7, 1, 2, 1], [13, -1, -1, 1, 0], [5, 0, 3, -3, 2], [-1, -6, -1, -3, 0], [-7, -1, 3, -3, -1], [-4, 0, 0, 6, -2], [3, 1, -1, -3, -1], [-5, -1, 5, 2, 0], [-3, 4, -6, 1, -4], [2, 10, -2, 3, -3], [3, -1, -5, 3, -3], [1, -2, 7, 1, -4], [5, -2, 4, -3, -6], [-10, -4, -4, 6, -2], [-1, 5, -5, -1, 1], [6, -1, 0, 1, -2], [-11, 1, 1, -3, 0], [10, 1, 0, 4, 7], [-1, 2, -5, 1, 2], [-5, -3, 1, -7, 1], [3, 6, 6, -5, 2], [-2, 7, -2, 0, -1], [8, -9, -4, 0, 3], [4, 6, 4, -2, -8], [11, 3, -5, 3, -1], [8, 5, 5, -2, 1], [-5, 5, 5, -3, -3], [0, -3, 8, 2, 5], [-3, -7, -1, -7, 3], [4, 9, -1, 4, -3], [3, 1, 3, -7, 3], [-10, 4, 0, 2, 4], [9, 3, -3, 1, 3], [-4, -6, -2, 0, -6], [-7, -8, 7, -6, -3], [10, -4, -6, 5, -1], [7, 5, -5, 5, 5], [2, 0, 4, -6, -4], [1, 0, -2, -5, -4], [-10, 4, 2, -8, -4], [-9, 1, 1, 1, -3], [2, 10, 2, -6, 0], [2, -6, 2, 4, 9], [-10, 3, -6, 0, 1], [14, -13, -8, -1, 0], [9, 0, 4, -7, 8], [-4, -3, -1, -4, -11], [0, 3, 0, 6, -2], [-1, -8, 7, -1, 4], [-22, 2, 2, 0, 3], [1, 3, 3, -5, -9], [4, -13, 4, -2, 11], [10, -4, 0, -4, 0], [-1, 1, 13, -3, 1], [3, -8, 3, -7, 2], [4, -1, -1, -4, -1], [5, -8, 3, 1, -5], [13, 0, -2, 7, 0], [3, 5, 1, 3, 3], [3, 1, -7, -1, -3], [6, 3, 0, -2, -9], [-3, 7, 9, -7, 1], [-8, 2, -1, 8, -2], [1, -3, 1, -4, -8], [-3, -7, -4, 1, -3], [-4, -6, 12, -4, -2], [1, -1, -16, 3, -1], [-14, 1, 8, 4, 4], [7, 4, -5, 7, 1], [-3, 1, 9, -5, -1], [14, -1, 8, -1, 4], [23, 1, -5, 1, -3], [32, -3, 0, 0, 1], [-11, 5, -7, -3, -16], [0, 1, 2, 11, 0], [-2, -12, 4, 0, 14], [-8, 9, 1, 2, 9], [-16, 6, 0, 1, -7], [-12, 12, -2, -2, -1], [4, 4, 12, 2, -3], [7, 16, -7, 0, -9], [-5, -13, 3, -11, 5], [2, 2, 13, -14, -2], [-2, 6, 4, 11, -1], [0, -9, 6, 2, 11], [11, 9, -15, 3, 0], [1, 13, -5, 11, 3], [-10, 9, 0, 8, 0], [-3, -3, 1, -5, -1], [-11, 6, 1, -1, -12], [11, 5, 5, -3, 13], [10, 2, -4, 2, 4], [5, 5, -7, 3, 3], [-13, -2, -5, 5, 12], [3, -5, -3, 9, -1], [-2, -11, 8, 8, 0], [9, 4, -1, -3, 3], [-2, -6, 0, -10, -4], [-5, 13, 5, 9, 1], [-17, 8, -3, 8, -3], [-3, 7, 9, 3, -9], [-12, -9, 4, 2, 4], [-3, -14, 1, 0, 5], [3, 7, -9, -3, -5], [14, 10, -4, 6, -4], [10, 0, -4, 6, -6], [-2, 4, 10, -12, -4], [-14, 6, 2, 7, -9], [-6, 4, 6, -8, -14], [-1, -1, -11, -1, -3], [21, -6, -3, -7, -2], [-9, -17, 9, -15, 5], [-14, -12, 0, -7, 3], [-2, 5, -6, 8, -9], [-15, 15, 5, 9, -5], [1, -3, -5, 5, -11], [1, 2, 6, -1, -14], [-10, 4, 0, 8, 6], [0, 3, -8, 16, 5], [17, 0, -9, 3, 6], [-7, -7, -17, 7, 9], [11, -1, -4, -11, 7], [26, 16, -4, 8, -6], [19, 6, 3, 9, -2], [-8, 20, -2, 6, -15], [-26, 1, 6, -4, -8], [-11, -1, -11, -4, -2], [18, 9, 4, -2, 5], [11, -13, 2, -5, 11], [-7, 3, 5, -7, 0], [5, -2, -15, -3, -10], [7, -13, -8, 11, 7], [8, -11, 0, 6, 6], [8, -6, -2, -10, -8], [-6, 0, -2, -5, -1], [14, 3, -18, 4, -7], [17, -7, -11, -3, -7], [-25, 1, 1, 7, 1], [-13, -9, 10, -5, -1], [-7, 6, 1, 8, 15], [-10, 1, 10, 5, -8], [5, 13, -7, 1, -7], [-18, -12, 6, -14, -10], [7, -24, -3, -1, 14], [-6, -6, 6, -16, 1], [7, -3, -3, -12, -2], [-6, 5, -8, 12, -3], [-26, 1, 24, -2, 7], [-10, 13, -5, -10, -11], [7, -3, -1, -9, -6], [-5, 7, -13, 11, 3], [4, 7, 8, 6, -4], [17, 11, -19, 13, 1], [-18, -6, 6, -8, 8], [-8, 1, -16, 8, -3], [-5, -2, 9, -5, 5], [28, 4, 4, 0, 4], [3, -3, 13, -5, -4], [-2, 3, -8, 0, 13], [15, 7, -5, -3, -13], [-17, 8, 23, -1, 3], [9, -4, 13, -11, 6], [-11, 3, 17, -1, -1], [-6, -4, 12, -3, 17], [-1, -1, -11, -17, 1], [-39, -5, 19, -5, 3], [5, -6, -15, 15, 0], [-17, -13, -3, 9, -9], [11, -10, -1, 3, 0], [-28, -10, 9, -2, -6], [-22, -21, 2, -2, 11], [7, 5, -5, 7, -7], [1, 13, 1, 13, -9], [1, -3, -1, -14, -8], [1, -4, 7, -8, -7], [12, 7, 10, -6, 4], [0, 6, 10, -10, -14], [-7, -5, -11, 5, -5], [3, 9, 5, 11, -3], [-25, -6, 22, -7, 2], [23, -4, -1, -11, -2], [-12, -12, 12, -10, 6], [0, 14, 10, 2, 1], [-11, -23, 3, 1, 4], [11, -5, 5, -3, 13], [3, 16, -2, 1, 12], [4, -3, 0, -16, -3], [19, 9, -7, 11, 15], [-6, 2, 4, -2, -2], [11, -6, 1, -9, -4], [1, 2, -15, -11, 2], [21, -6, -21, 7, 2], [22, -10, 2, 10, 8], [27, 11, -13, 17, -1], [0, 18, -6, 28, -2], [-44, -11, 18, -6, 5], [4, 1, 4, -4, -15], [-16, -6, 6, -8, 8], [4, 11, -22, 11, -8], [-16, -19, 4, 3, -2], [2, 11, -18, -4, 6], [33, -6, 0, 7, 2], [16, -14, 14, 10, -3], [-3, 2, -4, 5, 10], [10, 20, -9, 18, 4], [9, -1, -9, 19, -1], [-10, 4, 20, 4, 11], [18, -8, 16, -4, 11], [21, -10, -19, 1, -4], [6, 2, -6, 16, 5], [-22, -9, 2, -1, -2], [-9, -14, 11, -7, -12], [-22, -3, -5, -2, -3], [-5, -15, -4, -3, -15], [-10, 12, -15, -2, -4], [14, -6, -16, -10, 6], [4, -17, 2, -10, 1], [-34, 9, 8, 4, 7], [-26, 29, 7, 2, 1], [-4, 14, 14, 1, -11], [2, 4, 6, -18, 0], [5, -9, 5, 10, 6], [6, 0, -6, 20, 4], [-25, 4, 7, -16, -7], [-11, 2, -3, 11, -6], [16, -11, -10, 6, -8], [2, 5, -20, 5, 6], [-11, -3, 13, -23, -5], [19, 2, -6, -3, -2], [-13, 10, -12, -9, -14], [0, 8, -4, -7, -11], [-10, 21, -3, 0, 9], [-2, 5, 2, -4, 3], [-2, 13, 22, 1, -12], [27, -11, -19, -4, -8], [-4, -6, -12, -4, -20], [-21, 23, 5, -7, -15], [8, -7, -4, 4, -7], [-3, 5, 11, -13, 1], [-10, 0, -4, -3, -7], [29, -24, -1, 3, 12], [0, -9, -6, 6, 11], [-8, 4, -4, -2, 11], [44, -5, -12, -6, -3], [6, -22, 10, 0, -17], [16, -9, -22, -2, -2], [-23, 36, 7, -1, -4], [0, -13, 24, -4, 3], [-2, -19, -2, -8, -9], [-11, -25, 1, -7, 11], [-1, 3, 24, -7, -1], [-27, -15, 19, -10, 14], [-32, 0, 8, -20, -4], [-1, 6, 9, 11, -18], [27, 2, -17, 19, 8], [-15, -18, 1, 17, -3], [-13, 28, 9, 13, -4], [11, 2, -3, 5, -4], [-38, 4, 4, -4, -4], [24, 9, -17, 12, 5], [-53, 11, -1, -1, -6], [20, 4, -10, 20, 6], [-5, -3, 1, -1, 23], [-10, -2, 28, -10, 8], [-22, 29, 4, 6, -9], [-11, -20, 3, 5, 0], [-22, -25, 10, 1, 14], [3, 11, 15, 7, 2], [14, -2, -4, -12, 0], [-35, -4, 9, -15, -1], [-9, 3, -2, -7, 19], [18, -23, 6, -6, 7], [-11, 25, 7, 1, -15], [-12, 12, -10, -7, -13], [51, -3, -11, -5, -5], [1, 21, -17, 13, -18], [20, 19, 1, 4, -17], [15, -1, -17, 24, 8], [36, -6, 10, -14, -6], [-3, -17, -5, 1, -11], [18, -14, 6, -10, 14], [10, 10, -7, -2, 10], [3, -1, 13, -19, 9], [-28, -5, 6, 3, 10], [5, -20, -21, 5, -3], [21, 18, -7, 16, -11], [49, -17, -1, -7, 9], [10, 31, 8, 2, -19], [-19, 13, -8, 17, 5], [-15, 6, -5, -2, -15], [-19, 1, 29, 1, 3], [30, -4, 2, 6, 6], [-3, -5, 21, 8, 12], [-5, -19, -3, -17, 13], [12, -23, 2, 2, 13], [29, -5, -1, -3, -4], [20, -4, 14, -18, 4], [-2, 3, 6, -20, -3], [-7, 1, -9, -6, -10], [13, 19, -11, 13, -3], [-36, -1, -18, 17, 4], [3, 6, -3, 19, -4], [-45, 9, -1, -9, -15], [19, -11, 19, -1, 3], [32, -12, -16, -3, -13], [-8, 1, -11, -18, -11], [-25, 2, 8, 5, 10], [29, -11, -3, 1, 21], [-3, -8, 5, 11, 20], [36, -8, -2, 22, 11], [10, -3, -4, 14, -3], [-29, 6, 29, 5, 12], [-22, -26, -3, -14, 22], [-22, 13, 20, -16, -3], [3, 8, -17, 23, 8], [-28, -2, -17, 0, 18], [-8, -15, 0, -2, -6], [37, 23, -9, 15, 3], [11, -25, -14, 11, -1], [71, -5, -1, -1, 2], [-3, 15, 7, -23, -5], [6, 10, -22, -3, 11], [-38, 0, 40, -7, 11], [-11, -9, -3, 7, -3], [49, -8, -5, 15, 6], [-9, 15, 7, 15, -17], [-39, -19, 10, -11, 17], [-17, 17, 7, 9, -13], [0, -17, -12, 10, -13], [15, 7, -41, 2, -12], [-25, 9, 11, -6, -16], [13, 18, 19, -19, -7], [8, 29, -16, 5, -2], [26, -40, -14, 16, 16], [-25, -24, 14, -13, 0], [29, 31, -18, 11, -17], [-5, -1, -9, -21, 11], [26, -10, -11, -16, 6], [-26, 7, 20, -1, -10], [-1, -7, -23, 11, -10], [-2, -28, 8, -18, 14], [18, -27, 5, -18, 5], [-44, 1, 11, 10, -3], [-3, 3, -9, 3, 9], [-8, -19, 2, 8, 5], [-25, 39, 5, -3, -13], [8, -21, -10, -22, 3], [8, 24, -20, 20, -4], [-5, 1, 27, -11, 9], [10, -9, 6, 5, 30], [7, -16, 20, -25, 4], [36, -36, -20, 5, 5], [-10, -12, -4, 1, 3], [13, -30, -9, -5, -2], [6, 4, 4, -5, -23], [18, -31, -6, -2, 5], [-34, 40, 14, -12, -8], [-27, -12, 25, -21, -10], [35, 0, -8, -1, 16], [55, 10, -3, 1, -14], [-25, -13, 25, -27, -13], [-22, -29, 8, -8, 25], [-3, 1, -11, -7, 2], [28, -12, 20, 2, 10], [1, -9, 19, -17, 9], [-35, 24, 18, 17, 8], [-10, -9, -32, 18, 15], [7, -19, -9, -13, 18], [-5, -15, 17, -33, -9], [33, 31, -16, 5, -9], [-5, -38, 11, 7, 12], [44, -15, -22, -8, -15], [7, 5, -9, -1, 1], [4, 13, -27, -18, 1], [6, 11, -5, 4, 7], [-14, 1, 12, -12, -15], [-17, -5, 34, -23, -5], [-26, -24, 8, -1, 1], [2, 0, 2, 10, 18], [16, 20, -14, 30, 9], [16, -12, 14, 2, -24], [-31, -11, 9, -31, -13], [28, 23, 11, -20, 3], [5, -23, 1, 15, 1], [3, -22, 3, -27, 8], [15, 5, 13, -3, -5], [31, 6, -23, 3, -18], [-29, 3, 17, -1, 2], [-41, -4, -3, -4, -7], [26, 2, -6, -13, 7], [-13, 24, -11, 37, -8], [14, -25, -6, -36, 6], [23, 31, 0, -5, -5], [-26, -9, 14, -28, -2], [-37, -28, 12, 5, -8], [30, 10, -12, -8, 10], [-39, -25, 11, -7, -15], [8, -8, 26, 12, 14], [39, 1, -3, -3, -5], [18, 8, 8, -4, 15], [-6, -24, -6, -25, 15]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8036_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8036_2_a_k();".
// 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_8036_2_a_k(:prec:=5)
    chi := MakeCharacter_8036_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_8036_2_a_k();".
// 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_8036_2_a_k( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8036_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-1, 9, -8, -6, 2, 1]>,<5,R![8, 28, 12, -9, -3, 1]>,<11,R![24, -84, 68, -7, -6, 1]>],Snew);
    return Vf;
end function;