
// Make newform 2842.2.a.s in Magma, downloaded from the LMFDB on 19 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2842_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_2842_2_a_s();".
// 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_2842_2_a_s();".
// 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, 3, 3, -6, -1, 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, -3, -6, 0, 1], [2, 3, -6, -1, 1], [4, -2, -7, 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_2842_a();"
function MakeCharacter_2842_a()
    N := 2842;
    order := 1;
    char_gens := [1277, 785];
    v := [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_2842_a_Hecke(Kf)
    return MakeCharacter_2842_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], [0, -1, 0, 1, 0], [-1, 0, 1, 1, -1], [0, 0, 0, 0, 0], [0, 2, 0, -1, -1], [1, -1, -1, 2, 0], [-1, 1, 0, -1, 1], [-1, -1, 0, 1, 0], [-3, 0, 0, -3, 1], [1, 0, 0, 0, 0], [2, -1, 1, -2, 0], [-4, 2, -3, -1, 0], [0, -1, 2, 2, 0], [3, -2, -3, 0, 2], [-5, 3, 0, -1, -2], [2, 3, 0, 1, 0], [-3, 2, -1, -1, -2], [3, 4, 0, 2, 2], [5, -1, -3, -1, 2], [-1, -1, -2, -3, 0], [-2, 0, 2, -3, -1], [-1, -2, -1, -5, 0], [-10, 0, -1, -2, 3], [1, 0, -3, -4, 3], [-2, 2, 3, 2, 0], [-1, -3, 3, 4, 0], [-3, -2, 0, -4, 4], [-3, 3, 1, 3, 1], [-1, -3, 2, -3, 2], [1, -1, 4, 4, -4], [-7, 3, 1, -6, 1], [-5, 2, -1, 1, -1], [-1, 1, 4, 2, -7], [-1, -6, -3, -4, 0], [-2, -3, 2, 2, -1], [-3, -6, 3, 5, 1], [0, -1, 1, 3, 1], [2, 2, 5, 5, -5], [-7, 1, 0, -4, 1], [-8, 5, 1, -2, -1], [2, -5, -3, -1, -1], [-1, 1, -3, -9, 1], [-5, -6, 0, 5, 3], [-1, -4, 3, 8, -6], [3, -5, 3, 1, 6], [1, 4, 0, 4, -6], [5, -8, -2, 4, 5], [0, -5, -1, 8, -3], [-8, -2, 2, 0, 2], [9, -1, -3, -6, 3], [8, 0, -1, 10, -2], [-3, -1, -6, -7, -1], [-8, 0, -1, -3, 4], [-4, 3, 0, -2, 4], [-10, -2, -1, -1, -3], [6, 7, -1, 4, 2], [4, 6, -5, -10, 2], [0, 4, -2, 5, 1], [-1, 2, -3, 2, 2], [6, -5, 3, 4, -3], [-9, 5, 2, -6, 6], [-2, -10, 5, 13, -2], [-2, 5, -1, -1, 3], [-6, 6, 7, 3, 1], [-4, 1, 0, -12, 0], [-4, -2, 5, -1, -2], [-2, -8, 3, 2, -2], [-2, 1, -1, -13, 1], [-1, -6, 6, 13, -3], [10, -11, -4, -5, 1], [-6, -4, -5, 8, -1], [-8, 8, -5, -8, 3], [9, 2, 9, 6, 2], [15, -5, -6, -3, 2], [2, 5, -5, -4, 2], [-4, 4, -3, -1, 3], [-14, -6, -4, -7, 14], [4, 4, -9, -11, 5], [-1, 4, -5, -7, 0], [7, -4, -3, -1, -8], [4, 3, 2, -7, -1], [3, -4, -7, -2, -4], [13, 7, -4, -3, 8], [8, -4, -5, 4, 1], [-1, 5, 3, 10, 1], [6, -14, 7, 10, 0], [8, 4, 4, 2, -1], [8, 3, 8, 3, -1], [18, -4, 1, 5, 4], [-13, -9, 5, 3, -10], [-8, -3, -5, -13, 9], [-13, -6, -7, -8, -2], [-10, -7, 3, 2, -9], [1, -1, -1, 4, 1], [6, -5, 4, 6, -7], [-18, -7, 3, -1, -1], [-14, 11, 1, 1, -2], [6, -13, 6, 5, 2], [9, -3, 0, 10, -3], [7, -14, 1, 11, 1], [10, 7, 3, 8, 2], [-14, 2, 7, 4, 0], [-11, 14, -3, 1, -3], [-2, 11, -1, 1, 4], [-8, 2, 15, 6, -14], [7, 2, 5, -1, -8], [0, 3, -3, -3, 7], [11, -11, -6, -1, 4], [24, -12, 2, 15, -4], [29, -4, 1, 10, -7], [-1, -7, 5, 4, -5], [-9, 0, 4, 2, 0], [18, -1, 2, -1, 1], [-8, -11, 2, 10, -8], [0, 3, 8, 14, -1], [-17, 5, 2, -11, 3], [-4, 15, -2, -7, -10], [0, 6, 4, 3, 9], [-2, 8, 0, -11, -2], [-16, -1, -8, -21, 10], [10, 1, -8, -2, 7], [-17, -9, -1, 10, -2], [-13, -5, -11, -7, 4], [4, -10, 0, -11, -2], [5, -2, 7, 6, 2], [4, 5, -9, -2, 5], [19, -4, -4, 12, -3], [5, -5, -5, -10, 7], [1, -15, 6, 9, -1], [20, -5, 9, 15, -5], [-20, 0, -2, 7, 2], [25, 6, -5, -1, 3], [-1, -9, 3, 9, -2], [3, -2, -7, -12, -1], [-10, -8, 9, 8, -8], [13, -5, 6, 1, 13], [4, 7, -2, -3, -7], [-28, -11, 3, 6, 1], [9, 6, -3, 3, -11], [-2, -7, 1, -5, 12], [-12, -3, -6, -8, 11], [-2, 3, 8, 2, -10], [17, -5, 6, 10, -3], [4, 11, 7, 4, -11], [29, -7, -6, -8, -2], [9, 3, -1, 0, 4], [15, -9, -13, -5, 7], [-20, 16, -3, -3, -4], [11, -11, -5, 6, 0], [7, -14, 0, 12, 4], [13, -9, 6, 25, -7], [-10, 7, 6, 2, -10], [-21, -4, -5, 7, 5], [-25, -9, 5, 4, -4], [6, -4, -4, 16, 0], [15, 5, 4, 11, 0], [-13, 0, 1, -9, -7], [14, -16, 10, 21, -7], [-36, 5, 1, -12, 8], [1, -8, -1, -6, 3], [-33, 7, 0, -5, 1], [-16, 5, -11, 3, 3], [-5, 6, 7, -2, -14], [-7, -1, -19, -21, 16], [-12, -8, -6, -5, 3], [10, -4, 0, 10, -1], [19, 8, 0, -3, -13], [1, 0, -10, -9, 11], [-6, 4, 6, -3, -7], [3, -6, -6, -3, 0], [8, 1, 6, 8, -7], [-33, 17, 3, -6, -7], [-9, 16, -1, 10, 1], [-39, 2, -11, 1, -1], [24, -8, -3, 2, 3], [2, 3, -3, -17, 11], [-27, -4, 0, -11, 9], [19, -14, -4, 9, 2], [-25, 2, 17, 4, -19], [10, -3, 7, 3, 14], [12, 5, 4, 1, 2], [10, 2, -2, 7, -1], [-27, -4, 12, 10, -9], [-16, -7, -1, 0, 12], [-18, 14, 6, -13, 5], [17, 3, 5, 7, -5], [19, 14, 2, 0, 6], [-24, 26, 4, -6, -6], [9, -17, -1, 0, 7], [3, 28, -4, 4, -4], [21, -2, 18, 23, -14], [-9, -7, 4, 2, -8], [-5, 10, 6, 2, -19], [28, -12, 1, -4, 5], [0, -6, -17, -27, 7], [-13, 22, 6, -1, -5], [-10, 13, 12, 7, -11], [-7, 7, 9, 7, 3], [-9, -7, -5, 11, -12], [-4, 1, -8, 14, 4], [3, 7, -2, -2, -13], [-4, 3, -3, 8, -1], [2, 10, -11, -3, 0], [20, 0, -1, -1, 5], [-18, 5, 14, 2, -7], [11, 9, -2, -2, 8], [6, -20, -9, -7, 2], [14, -23, -4, 14, -8], [-2, 7, 3, -15, -11], [-22, -8, -2, -9, 6], [30, 5, 3, -14, -3], [-7, 18, -6, 1, -5], [-29, -8, 4, 11, 10], [-3, 22, 0, -8, -2], [-18, 10, 18, 4, -15], [-17, 13, -1, 1, -13], [-21, 0, 10, -19, -3], [8, -17, -8, 8, 12], [-13, -7, -12, 3, 12], [8, -22, -1, 12, -3], [30, 0, -7, 5, 11], [-42, 12, -4, -8, -1], [-32, -8, 7, 0, -2], [-20, 22, 3, -28, 14], [-51, -1, 5, -4, -1], [-5, -2, 0, 11, -1], [-14, -4, -3, -8, 3], [4, 3, -13, -13, 11], [6, -9, 12, 4, 0], [-11, 4, -1, 3, -5], [-1, 7, -12, -11, -2], [-4, 22, -6, 0, 1], [-15, 19, 9, -9, -3], [21, -24, -12, -9, 9], [-22, -11, -6, 6, 8], [17, 8, 7, 0, -22], [11, -9, -7, -4, -8], [27, -23, -4, -2, 7], [13, 10, 2, 9, -2], [13, -6, 3, 8, -8], [-21, 11, 18, -7, -9], [6, 10, -5, 23, -3], [-11, -4, -7, -13, 22], [7, 10, 7, 11, 4], [-27, -2, 1, -5, -3], [-8, 29, -12, -20, 1], [14, -11, 1, 0, 4], [-19, -21, 9, 19, -12], [-6, 18, -13, -1, 2], [11, 9, -4, -5, -12], [-40, -9, -7, -27, 15], [-31, 7, 2, -1, -5], [16, 0, 1, 23, -6], [12, 9, 7, -2, 3], [10, 12, 7, -4, -11], [3, 23, 1, -10, -1], [-16, 14, 7, -4, -6], [-7, 15, -10, -22, 11], [25, 8, 11, 7, -1], [-27, 9, -4, -26, 12], [3, 11, -1, -8, 11], [23, -9, -23, -11, 22], [-10, -16, 3, 7, -8], [-4, 13, 5, -3, -14], [-11, 21, 4, -1, 5], [19, 4, -3, 25, 0], [10, -10, 10, 25, -9], [-8, 16, -7, -24, -2], [9, -19, 12, 16, -2], [-6, 5, -11, -7, 9], [22, 2, 11, 23, -22], [11, -7, 5, -16, 4], [-8, 7, -1, -2, -16], [-4, 18, 0, 3, 7], [-13, 15, 7, -4, -15], [2, 6, 19, 5, -15], [25, -9, -14, -7, 8], [-16, 10, 19, 7, -14], [-21, 10, -15, 0, -8], [12, 6, 6, 12, -11], [-22, -15, 1, -20, 9], [7, 0, -2, 13, 1], [32, 19, 3, 14, -11], [-15, 24, -2, -11, 10], [-19, -6, 2, -1, 20], [23, -1, 8, 6, -6], [3, 10, 16, 6, -15], [24, 7, -5, 31, -4], [14, -15, -3, 8, 0], [-21, 14, 13, -7, -5], [-3, 7, -9, -2, -18], [-35, 8, 2, -8, -13], [-18, 24, 4, 6, -6], [-7, -1, -1, -30, 8], [-6, -10, -12, 7, 14], [-2, -3, -17, -1, 18], [-16, 3, -14, -16, -7], [55, -19, -5, 7, 1], [12, -3, -4, -3, -16], [-1, -11, 3, -18, -5], [25, 18, -9, 3, -3], [-16, 7, 0, 14, 7], [53, -4, -1, 14, -8], [-5, 9, 10, 8, -20], [4, -17, 2, 5, 11], [8, -1, 3, -7, -6], [-23, -3, -10, -1, -4], [32, 2, 11, 25, -12], [-39, 16, -3, -29, 8], [8, 8, 15, -4, 6], [3, -33, 18, 33, -10], [14, 6, -2, -8, -15], [1, -20, 8, 21, 6], [-10, 15, 13, -6, -22], [-1, -24, -4, -11, 7], [26, 4, 12, 9, -7], [3, 8, 2, 8, 18], [-21, 8, 0, -13, 15], [-28, 10, -4, -3, -17], [-5, -30, 7, 17, -9], [-9, 11, -1, -18, -12], [-22, 21, 9, -15, -13], [-11, -25, -6, -26, 8], [15, -5, 0, -18, 9], [-1, -7, 12, 18, -6], [41, 21, -4, -4, -1], [6, -5, 13, 9, 5], [2, 16, -5, -27, -8], [-39, 23, 23, 5, -19], [29, -14, -6, -1, -6], [0, 9, 6, -13, 20], [-19, 24, 3, -20, -9], [-21, 22, -12, -31, 0], [-9, -6, -8, 6, 18], [-1, -13, 1, -10, 13], [-23, 9, 12, 17, 10], [-17, -10, 11, 15, -12], [-40, -7, -2, 12, -5], [-25, 0, 2, -10, 11], [-27, 0, -17, -20, 12], [31, -3, 1, 4, -16], [-23, 14, 4, 10, -4], [33, 6, -6, 21, -9], [30, -3, 15, 8, -1], [37, -13, 7, 16, -4], [-9, 5, -2, -1, -5], [-12, 5, 9, -15, -10], [-8, 2, -16, -24, 3], [-9, -12, 0, 30, 2], [14, 17, -9, 18, -6], [-26, -7, -5, -16, 26], [1, 9, 1, -14, 22], [37, 14, 13, 28, -10], [-3, 21, -10, -2, -6], [-13, -2, -8, -2, 5], [11, -16, 5, 12, 4], [1, -19, -8, -5, -8], [-43, 5, -6, -6, 5], [-12, 32, 0, -13, -7], [-36, 31, 16, -5, 5], [12, 16, -11, -15, -3], [2, -5, 1, 26, 7], [15, 21, -14, -27, 21], [32, -1, -9, 3, -7], [-11, 26, 7, -1, -9], [18, -18, 0, -13, 4], [5, -34, 5, 33, -19], [-45, 12, -19, -13, -1], [-6, 10, 2, 27, -1], [-35, 21, 6, -34, 9], [-44, 15, 10, -6, -17], [-8, 17, 2, -2, 7], [-7, -10, -5, -6, 0], [-1, 1, 1, 0, -13], [9, -24, -9, 5, 0], [14, -9, -11, 10, 16], [-3, 2, -16, -16, 12], [-16, 1, 2, 12, -10], [17, -3, -2, -20, 12], [-16, -9, 7, -4, -5], [-29, 14, 16, 13, 2], [5, 26, -7, -27, 15], [30, 8, 8, -8, -4], [-24, -3, 6, -6, 1], [4, -15, -3, 30, -12], [17, 8, -9, -12, 7], [-20, 13, -7, -34, 5], [-35, 2, 1, 5, -19], [18, 5, -15, 1, 16], [-47, -2, -15, 3, -6], [0, 22, -14, -21, 4], [-1, -1, 12, -12, -15], [21, 18, 14, 24, -4], [9, -9, 18, 34, -2], [21, -9, -8, 9, 2], [-18, -5, 2, -18, 23], [25, 16, 9, 19, -6], [23, 23, -2, 18, -15], [-25, -9, -2, 6, -12], [23, 14, -7, 6, -4], [9, 2, 8, 12, -12], [20, -23, -13, 11, -18], [-7, -10, 5, 17, 7], [-54, 15, -8, -6, -1], [13, 7, 27, 6, -12], [-24, -11, -14, -16, 23], [-41, 14, 4, -39, 19], [17, -31, 22, 30, -8], [-65, -2, -7, -11, 6], [0, -26, 9, 4, -6], [10, 2, 21, -22, 0], [27, 11, 2, -34, -5], [-1, 18, 16, 13, -8], [16, -20, 3, 17, -13], [1, 6, -21, 16, 3], [-27, 18, 21, -3, -17], [-34, 18, -8, -14, -6], [7, -3, -19, 12, 12], [18, -27, -21, -12, 3], [-5, -18, 8, 38, 0], [11, -2, 18, 5, 5], [-5, -4, -19, -19, 0], [-15, -28, 5, -22, 13], [-16, -22, -7, -27, 10], [1, -3, -3, 27, 12], [-3, 9, -8, 12, 2], [20, 29, -12, -42, -4], [9, -7, -14, -13, 20], [-11, -15, -8, -30, 14], [28, -6, 24, 20, 2]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2842_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2842_2_a_s();".
// 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_2842_2_a_s(:prec:=5)
    chi := MakeCharacter_2842_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_2842_2_a_s();".
// 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_2842_2_a_s( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2842_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![5, 2, -13, -5, 3, 1]>,<5,R![21, 3, -31, -4, 5, 1]>],Snew);
    return Vf;
end function;