
// Make newform 6864.2.a.cf in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_6864_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_6864_2_a_cf();".
// 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_6864_2_a_cf();".
// 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, 5, 16, -8, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0], [4, -5, -7, 1, 1], [-2, -5, 7, 1, -1], [-2, -9, 7, 1, -1], [-10, -7, 9, 1, -1]];
    Rf_basisdens := [1, 2, 2, 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_6864_a();"
function MakeCharacter_6864_a()
    N := 6864;
    order := 1;
    char_gens := [2575, 1717, 4577, 4369, 2641];
    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_6864_a_Hecke(Kf)
    return MakeCharacter_6864_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], [1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [1, 0, 0, 0, 1], [1, 0, 0, 0, 0], [1, 0, 0, 0, 0], [1, -1, -1, 1, 1], [1, 1, -1, 0, 0], [1, 0, 1, 1, -1], [1, 0, 1, -2, 0], [2, 0, -1, 0, -1], [0, 0, -1, 3, 0], [1, -2, 0, 2, 1], [-1, 2, 2, 1, 0], [4, -2, -1, -1, 2], [0, 2, -2, -2, 2], [2, 1, 1, 0, -1], [2, -3, 0, -1, -1], [-2, 1, 0, -2, 2], [6, -2, -1, 1, 2], [-3, 2, 0, 0, -1], [1, 1, 1, 1, -3], [4, 2, -2, -2, 0], [-2, 2, 0, 3, 1], [-4, 0, 1, 1, 0], [1, 1, 3, 1, -3], [2, -1, 1, 0, -1], [6, -1, 1, 2, -3], [5, -1, 1, -2, -2], [-4, -1, 3, -4, 1], [3, 1, -2, 0, 3], [4, -1, -1, -2, 1], [-6, 0, -2, -1, -1], [5, 1, 4, -2, -3], [-5, 1, 1, -2, 0], [5, -1, -1, 2, 2], [-1, 1, -2, 5, 0], [2, 1, -1, 1, -4], [4, 1, 1, -2, -1], [-3, -4, 0, -3, -2], [7, -1, 2, -1, 0], [1, -5, -2, -1, 4], [10, -1, -1, 4, -3], [1, -3, 1, 0, 0], [9, -1, 1, 2, -2], [-2, -3, -3, -2, -3], [-9, 1, -3, -3, -1], [2, 0, 1, 0, 5], [15, -3, -1, 4, 2], [-4, -1, 2, -1, 3], [-1, 1, -4, 4, 3], [7, 4, 2, 2, -5], [-4, -2, 2, 2, -6], [4, -6, -2, 2, 4], [-2, -5, -1, -2, 5], [8, -4, -2, 4, -2], [0, 0, 2, 4, -4], [11, -7, -3, 0, 0], [6, -1, 3, -2, -3], [-7, 0, -4, -2, -3], [3, 0, -3, 4, 2], [-11, 1, 3, 2, -4], [-5, 7, 5, 0, 0], [5, -3, 2, -5, 2], [-1, 4, 3, 3, -1], [4, -3, -1, 5, 0], [0, -3, 1, -5, -2], [-2, -4, 0, -2, -2], [8, 6, 1, -1, 0], [-8, 4, -2, -4, 0], [-2, -2, -1, -6, 3], [5, 0, -2, -2, 1], [2, -6, 4, 4, -2], [4, -3, 3, -6, 3], [-12, -8, 2, -3, -3], [22, 0, 0, -2, 0], [6, 0, 2, 4, -6], [-6, -3, 0, -3, 1], [0, -10, -1, -2, 1], [1, 8, 2, 0, -3], [-9, -1, 0, -3, 4], [6, 11, -3, 4, 5], [-6, -2, -2, 4, 4], [16, -5, -1, 6, -1], [5, -5, -3, 1, 9], [16, -4, -4, 0, 0], [-8, 2, 5, -10, -5], [-2, -5, 1, 8, 1], [7, 1, -5, 0, 8], [-10, -4, 0, 1, 1], [-5, 3, 0, 7, -2], [5, 4, 6, 2, 5], [4, -2, 4, 5, -5], [0, 1, 6, 3, 5], [-16, -7, 4, -2, 0], [16, 4, 0, -6, -2], [6, 0, -4, 3, 5], [4, 9, -3, -2, 1], [-17, -3, -1, -1, 3], [6, 0, -4, 4, 8], [1, -2, -1, 4, 2], [11, 7, -5, -6, -2], [-14, 2, 4, -2, 0], [-5, -1, -1, -1, -3], [-5, 2, -8, -3, 2], [16, 8, -2, 2, -2], [-2, -9, -1, -4, -3], [1, 5, -3, 8, -4], [-6, 1, -7, 4, -1], [6, 2, -2, 4, -2], [3, 1, -2, 0, 3], [-9, -3, 3, -2, 2], [-12, 0, 5, 4, -3], [-12, 9, -3, -3, 4], [-26, 0, 0, -3, 5], [-6, 11, 7, 2, -11], [-16, -6, -1, 0, -3], [-8, 0, 0, -2, -6], [8, -2, -2, 0, 0], [6, 4, -1, 1, -6], [0, 2, 0, 0, 6], [-2, 0, -3, -7, -2], [-3, 11, 4, 2, -5], [-4, 1, 4, -7, -3], [2, -6, 4, -9, 3], [-1, -4, 2, -5, 0], [2, -3, -6, -3, -1], [-6, -1, 3, -10, -1], [2, -2, 4, 0, 2], [3, 7, 3, 0, 8], [-23, 3, 5, -4, -6], [4, -3, -5, 6, -3], [-12, 7, -1, 4, -1], [-13, -3, 4, 3, 0], [-11, 12, 4, -6, -3], [0, -1, -1, -2, -1], [-14, 12, 2, 7, -1], [-15, -9, 5, -4, -6], [22, -2, -4, 2, -2], [11, 3, 5, -3, 1], [-5, 1, -1, 0, 2], [19, -5, -1, 4, 0], [-8, -1, 3, -4, -1], [-21, -3, -3, -2, -2], [2, 0, 8, 4, -8], [-2, 0, 6, 10, -10], [-8, -4, 4, 0, 6], [23, -1, -2, -6, 7], [-26, 2, 4, 8, -6], [-6, 2, -4, 8, 2], [-16, 4, 4, -6, -12], [-4, 3, 7, -2, -5], [16, -6, 2, 4, -6], [2, 12, 3, -7, -10], [-26, 10, 4, -12, -2], [8, 4, 2, -8, -2], [15, -9, -8, 4, 5], [-4, -4, -4, 5, -3], [-4, 2, 7, 1, -10], [11, 1, 3, -12, 0], [2, -8, -3, 7, 10], [13, -3, 0, -2, 3], [7, 0, -4, -6, 9], [21, -1, -4, -1, 4], [2, -2, -10, 7, 3], [-2, 0, 2, 6, -2], [-6, -5, -3, 2, 1], [-20, 3, 5, -6, 5], [13, -8, -2, 8, -3], [-16, -12, -2, -2, -2], [-6, 8, -6, -8, 4], [-22, -1, 4, -1, -1], [4, -2, -9, 11, 6], [-10, 1, -9, -6, 5], [-10, 7, 3, -6, -1], [12, -11, -1, 0, -3], [-29, 3, 5, 4, -12], [-2, 5, 7, -11, -6], [3, -7, 3, -5, -5], [-12, 1, 7, -6, -1], [22, 2, -3, 4, 5], [-4, -8, -6, 2, 12], [7, 5, -6, 5, -4], [-3, 8, -2, -2, -9], [14, 6, 3, 3, 4], [0, 12, 4, -6, -12], [-18, -2, -3, -9, 0], [-18, 1, -5, -5, 10], [10, -11, -5, 6, 1], [22, 5, 1, -4, -5], [-17, -7, 0, 3, -4], [18, 0, 2, 2, 2], [-24, 0, 6, -2, 0], [-14, -1, 7, 9, -2], [-2, -14, 6, 0, -8], [-6, 8, -2, 2, 0], [14, 4, -10, 14, 0], [-28, 2, 2, -10, -6], [-5, -10, 0, 0, 13], [10, 14, 5, 5, 4], [-17, -1, -3, 14, 4], [35, -5, 6, 6, -7], [-4, 1, -7, 2, -3], [-6, -6, -3, -11, -6], [-8, -9, -4, -3, 1], [2, -8, 8, -14, -6], [-12, 8, 0, -7, -1], [40, 7, 3, 2, -3], [11, 3, 3, 4, 8], [-16, -14, 8, 2, -2], [0, 0, 5, -7, 8], [6, -6, -6, -4, 6], [-24, -1, 3, -4, -1], [-22, 6, 4, 2, -12], [19, 17, -3, 6, -6], [-10, -4, 1, -9, 12], [-15, 11, -1, -5, 1], [-11, 1, -6, -12, -5], [-2, 2, 5, -9, 6], [-8, -4, 7, -14, -3], [-10, 17, -1, 0, 3], [-11, -10, -6, -6, 5], [-16, -2, -1, -6, 5], [30, 0, -7, -8, 1], [-10, 7, -6, -5, -5], [-8, -2, 10, 14, -2], [-10, 0, 0, 2, -14], [-2, -3, 5, 4, 7], [11, -9, 1, -1, -3], [-4, 4, 2, -12, 10], [-14, 0, -6, -2, -4], [13, -14, -4, 7, 6], [11, 12, 6, -2, -9], [9, -1, 4, 2, -13], [0, 4, -8, -12, 0], [-16, 0, -4, 4, 0], [-6, 12, 6, -8, -6], [-33, -9, 3, 0, 4], [-25, -7, 6, -7, -2], [19, -1, 4, -5, 2], [-48, 3, -1, -4, -1], [-29, -8, 5, -8, 6], [4, 3, 5, 0, -11], [-42, -4, -2, 0, -6], [-13, -4, 0, -6, 7], [2, -6, 13, 5, -4], [-20, 2, 6, 3, 3], [0, -17, 0, -11, 5], [-31, -5, 5, -4, -6], [-6, -12, 9, 3, -10], [-10, -8, -14, -4, 2], [-17, 3, -1, -2, 6], [-2, 6, -7, -5, 6], [-9, -4, 3, -1, 11], [1, -2, 5, -10, 8], [-15, -3, -3, -2, 12], [11, -1, 0, 11, 0], [7, -1, -7, 4, -4], [12, -4, -6, 12, -8], [-24, 7, -1, -8, -3], [15, -1, -4, 18, 3], [-26, 13, 4, 11, -3], [18, 2, 0, 8, 0], [-40, -1, 1, -6, -11], [-16, 0, -2, 11, 1], [4, 6, -2, 0, -2], [-16, 11, -4, -4, -4], [24, -14, -1, 8, -9], [-1, -5, 3, -6, 12], [-17, 9, 6, 16, 1], [-36, 12, 2, -12, 8], [-10, 4, 1, 8, -3], [14, 12, -8, -6, 2], [15, -1, 2, 2, 5], [-16, -8, 0, -6, 12], [-2, 8, 0, -15, -7], [-36, 3, -2, -7, 7], [-5, 9, -5, -2, -2], [-21, 10, 0, -8, 5], [14, -7, -6, 5, 13], [20, -4, -3, -19, 10], [-8, -20, -6, 1, 9], [4, 8, 1, 7, -12], [-35, -1, 5, -10, 2], [19, -11, -3, 12, 0], [6, 17, 1, -2, -7], [-26, -4, -10, 2, 4], [19, -7, -9, 10, -2], [-6, -10, -6, -7, -1], [31, -1, 4, -6, -7], [20, -10, -8, 10, 8], [21, 7, 12, -9, -12], [-1, -8, 8, -8, -11], [-7, -3, -3, 10, 0], [-24, 15, 2, -5, -5], [29, 0, 4, 1, -2], [4, 5, -9, 10, -3], [4, -4, 0, -6, -4], [16, -11, -4, 13, 13], [41, 3, -3, 0, -2], [-4, 2, 0, 10, -4], [-31, -1, -7, 4, -4], [-31, 9, 6, 10, -9], [11, -7, 8, -11, 4], [-2, 4, -10, 10, 10], [22, -10, -1, -3, 12], [-15, 7, 12, -7, -2], [32, 2, -4, -4, 6], [38, 14, -4, 6, -8], [-30, -3, 7, -10, -15], [30, -6, 0, 0, -6], [-26, -3, 10, -2, 6], [-17, -15, 9, 9, 1], [4, 6, -14, 2, 0], [-20, 4, -6, 8, 2], [-2, 14, -4, 16, -4], [-20, 9, 1, -10, -13], [13, 4, -2, -2, 9], [-28, 4, 4, 2, -4], [-19, 5, 8, -8, 1], [33, 1, -4, -4, 3], [38, 2, 4, -2, -2], [11, 22, 11, -2, -20], [26, -12, 0, -7, -3], [-31, 12, 10, 6, -1], [-7, 12, 9, 3, -13], [-12, 17, 12, 6, 4], [-14, -2, 0, 5, -15], [0, -6, -10, 18, 4], [9, -5, -12, 12, 5], [-17, 5, 3, 22, -6], [24, 8, 8, 0, -6], [-1, -3, 4, 0, 9], [35, -7, -1, -8, 10], [1, 1, -11, -4, -2], [-12, 0, 8, 6, -14], [2, -7, 4, 1, -7], [-29, 2, 2, 2, 9], [24, -10, 0, 16, -14], [27, 7, 12, -6, -9], [30, -3, -6, -1, 17], [32, 3, 11, 0, 9], [-9, 5, -3, -14, 4], [-40, -8, 4, -6, 2], [-12, -5, -1, -11, 4], [-24, 15, -7, -14, 15], [-30, 12, -2, -14, 4], [-8, 1, 13, -5, -14], [5, 2, -14, -12, 3], [8, -8, 2, -6, -2], [-25, -13, 5, -8, -4], [24, 10, 2, -2, -6], [30, -6, -9, 4, -3], [51, -7, -3, 7, 5], [-4, -23, -2, -3, -3], [-5, -13, 7, -9, 5], [-10, 0, -1, -7, 14], [-7, 1, 7, -8, 0], [15, 5, -1, 5, 15], [-8, 4, -3, -9, 4], [8, -8, -6, 10, 14], [27, -3, 0, 10, -11], [-18, -5, -1, -4, 1], [-7, -17, 5, -12, 2], [-24, 6, 12, -18, -10], [23, -8, -7, -9, 13], [7, -9, 11, -20, 0], [-27, -8, 8, -2, -9], [24, 10, -5, -15, -2], [0, 2, -10, 18, 0], [-16, -4, -4, 4, -4], [-19, -18, 2, -12, -13], [-7, -7, -1, -17, 3], [45, -7, -1, 13, -1], [-6, 10, 0, 2, -18], [-24, 6, 11, -25, -2], [8, 4, 2, 16, 6], [-39, 1, -7, 12, -4], [29, -3, 8, -12, 5], [44, -6, -9, 11, 4], [-16, -9, 10, 14, -12], [-20, 12, 3, -4, -1], [-3, -2, 4, 2, -13], [-1, 10, -8, -2, -5], [7, -1, -7, -6, 22], [4, 0, 3, 7, -6], [24, 6, 7, -4, 5], [36, 6, -8, 0, 26], [37, 15, 9, -4, -14], [-19, -7, 2, -3, -16], [16, 14, -7, -5, -2], [0, 15, 1, -7, -14], [20, -10, -11, 11, -4], [24, -2, 8, -6, 14], [12, 10, -10, 2, 8], [-16, 14, -3, -15, 4], [29, 14, -12, 6, -9], [32, 5, -11, 10, 17], [-24, 18, 2, 12, 2], [-13, -5, 11, -10, -6], [28, 4, 4, 4, -2], [-30, -19, -1, -4, -1], [11, 12, -8, -4, -9], [-40, 2, 0, -1, -1], [-23, -13, 5, -12, 8], [23, 7, -6, 12, 11], [-42, 11, 6, 3, -11], [-7, 8, -10, 18, -1], [18, 1, 6, 6, 16], [-2, 13, -14, -1, 11], [-2, -6, 16, 10, -10], [-19, -7, 1, -1, -19], [-28, 12, 4, -2, -10], [49, 2, -4, -1, 2], [48, -16, 0, 2, -4], [-24, 10, -6, 8, 4], [3, 8, 4, 4, 11], [-10, -2, 12, 12, 2], [42, -2, -3, -6, -7], [-32, 10, -9, -15, 12], [10, -4, 4, 0, -8], [-12, 0, 14, 8, -16], [-38, -6, 5, 3, -2], [-22, 21, 5, -4, -1], [0, -2, -4, 10, -4], [20, -13, 0, 2, 14], [32, 10, 8, 6, 2], [15, -9, -1, 2, -14], [24, -1, -3, 5, -12], [32, -6, -9, 5, 8]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_6864_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_6864_2_a_cf();".
// 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_6864_2_a_cf(:prec:=5)
    chi := MakeCharacter_6864_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_6864_2_a_cf();".
// 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_6864_2_a_cf( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_6864_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-4, 22, 12, -14, -1, 1]>,<7,R![64, -140, 88, -10, -5, 1]>,<17,R![-256, 492, -22, -52, 0, 1]>,<19,R![32, -152, 164, -40, -4, 1]>],Snew);
    return Vf;
end function;