
// Make newform 2842.2.a.y 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_y();".
// 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_y();".
// 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 := [-3, 14, 8, -7, -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], [3, -3, -2, 1, 0], [-2, 8, -2, -3, 1], [-2, -9, 3, 3, -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], [1, -1, 0, 0, 0], [2, 0, 1, 0, 1], [0, 0, 0, 0, 0], [1, -1, 1, 1, 0], [1, 1, -1, 1, -1], [4, 1, 0, 1, 2], [0, -1, 0, 0, -2], [-2, -1, -1, -1, -2], [1, 0, 0, 0, 0], [-2, 1, -1, -3, -3], [-1, 0, -1, -1, -1], [2, 2, 1, -2, 0], [-1, 1, 0, -1, -1], [0, -3, -2, 2, -2], [-1, 3, 0, -4, 2], [2, 2, 1, -1, -1], [3, 0, 2, -2, 2], [-2, 2, 0, 3, 1], [-4, 1, -2, 0, -4], [5, -3, -1, 5, 2], [6, -2, -1, 3, 5], [-2, 1, 4, -6, -1], [-1, 1, -6, 0, -3], [2, -5, 0, -3, -3], [3, 1, -1, 3, 1], [7, -2, 2, 4, 0], [-8, -2, 2, -4, -5], [-6, 1, 0, 4, 2], [1, 4, -7, 4, -2], [1, 1, 1, -4, -1], [2, 0, 1, 0, 1], [-11, 2, -3, -1, -4], [-1, 3, -8, 3, -3], [2, 4, 3, -3, 2], [-14, 4, -3, -4, -1], [-7, 0, 4, -2, -3], [1, 6, 1, -2, 1], [7, -2, -3, 1, 4], [2, 3, -3, 4, 1], [1, 0, 0, 6, 5], [10, -2, 2, 4, -1], [-6, -1, -5, 5, 0], [-7, -1, 6, -1, 5], [4, -2, 8, 1, -1], [-5, 2, 4, 2, 8], [13, -6, 6, -1, 0], [8, 1, 7, 2, 9], [0, 10, -2, 0, -2], [5, -3, -3, 0, -1], [-8, 3, 4, -7, -1], [-2, 9, 2, -5, 4], [-11, 6, -3, -9, 1], [-4, -4, -1, -2, -4], [-9, 4, -7, -4, 1], [4, -7, -3, 5, -1], [-4, 9, 0, -5, 1], [-3, 1, -1, -3, 2], [-3, -1, -4, 7, 3], [4, -5, -3, 0, 5], [-11, 4, 3, -6, 4], [-1, 2, 1, -7, -1], [-3, 4, 4, -2, -1], [-15, -2, -3, -2, -9], [2, 6, 5, -8, 2], [-1, -4, 7, -7, 3], [12, -7, 12, -3, 3], [-7, 8, -2, -6, -9], [4, -1, 1, -7, -2], [1, -3, 6, -5, 4], [10, -11, -2, -2, -1], [2, -3, -6, 4, -3], [1, 1, 0, 3, 1], [2, -3, -6, 4, -10], [-4, 5, -3, -1, -1], [13, 6, -3, 0, 5], [5, 1, 1, -2, -4], [9, -6, 7, 4, 9], [-4, -2, 1, 11, 5], [-2, -10, -5, 1, -3], [-15, 7, 2, 5, 8], [5, -11, -2, 7, 5], [6, -1, 2, -10, -2], [10, -3, 0, 4, 3], [7, 1, -5, -2, 5], [-4, -7, -6, 7, -1], [-18, 6, -2, -3, 2], [1, 5, -12, 7, -4], [11, -10, 3, 11, 3], [-2, 0, 4, -1, 1], [-5, -8, -4, 4, 3], [17, -5, -2, 7, 9], [16, -9, -1, 12, 9], [-1, -11, -7, 2, -7], [2, 8, 9, -11, -2], [-1, -2, 4, 4, -1], [13, -4, -6, 3, -9], [3, 9, -2, 0, 2], [-3, -10, 5, -3, 4], [2, -10, -3, 4, 1], [10, -3, 1, 1, 5], [-24, 3, -8, -1, -5], [12, -10, 5, 4, -1], [3, -2, -6, 11, 5], [10, -7, -8, 3, -7], [2, -2, 5, -3, 9], [9, -10, 6, 6, 9], [-2, 3, 6, -10, -6], [-7, 3, -3, -2, -12], [-11, 7, 4, 2, 13], [-7, -5, 5, 6, 11], [-13, 14, 0, -8, -2], [15, -11, 4, -1, 4], [-24, 14, -7, -4, -2], [6, -6, -7, -1, -8], [2, -3, -6, -7, -4], [3, -1, -8, 6, -6], [5, -1, -5, 11, -2], [-3, -7, -5, 2, -12], [-25, -3, 6, -6, 0], [14, 6, 3, -1, 4], [1, 9, -5, 1, 9], [10, 4, 12, -7, -1], [3, -13, 1, -6, -12], [-9, -1, -14, -1, -5], [14, -7, 17, -4, 3], [5, 0, -16, 11, -4], [17, -11, 1, 12, -3], [-2, 5, 2, -9, 2], [-3, -4, -4, 4, 1], [-11, 9, -15, 6, -6], [-6, 6, -1, 0, 7], [-14, -12, 6, -3, -3], [1, 3, -16, 6, -11], [4, 7, -2, 5, 9], [0, 9, 0, 3, 8], [21, 3, 2, -3, -4], [-4, 7, 7, 6, 7], [24, -6, 5, 8, 13], [13, 2, 2, 5, 3], [-6, -8, 1, -7, 4], [26, -2, -5, 8, -2], [-3, 16, -1, 1, 12], [-10, 5, -3, -6, 3], [-17, 8, 3, -10, -10], [9, -7, 15, -5, -1], [-2, -4, 4, -10, -7], [1, 4, -11, -7, -13], [-29, -5, 3, -7, -7], [13, 10, 14, 0, 24], [8, 17, 12, -3, 20], [-8, 10, 1, -16, -4], [-6, 6, -3, 0, 9], [-1, 3, -1, 1, -15], [10, 4, 8, -12, 0], [-4, 1, 12, 0, 10], [2, 0, -7, -2, -11], [-1, -5, -13, -9, -20], [8, 1, -13, 17, -9], [11, -3, -8, 8, 11], [-18, 5, -8, -3, -14], [-13, 0, -8, 12, -3], [-5, -9, 4, -3, -5], [-12, 14, 0, -15, -9], [7, -7, -3, -3, 4], [22, 6, -2, 5, -6], [-8, 3, 7, -11, 2], [6, -5, -5, 9, 8], [-21, -5, -1, -1, 8], [0, -3, -23, 2, -16], [-30, 8, 1, -9, -4], [11, 5, 3, -12, -7], [15, -15, 2, -2, -7], [-26, -12, 3, -10, -21], [6, 1, 6, -6, -5], [21, -8, -2, 0, -3], [2, -3, 11, -1, 12], [8, -1, -11, 22, -2], [39, -5, 2, 0, -1], [-5, 0, 2, 1, -3], [-27, 11, -4, 2, -2], [3, 9, -3, 19, 14], [9, -2, 10, -3, -4], [28, 7, -1, 11, 13], [27, 9, -5, 5, -4], [-16, 14, 0, -14, 3], [-23, -16, -6, 4, -16], [-6, -2, 4, 6, 8], [7, -7, -15, -6, -11], [5, 6, 0, -12, -8], [-2, 5, -1, 0, -12], [-25, 0, -7, 2, -10], [21, 4, -14, 7, 4], [24, -11, 2, 20, 7], [7, 0, 5, 4, 3], [-6, -3, -13, 5, -12], [19, -9, -14, -3, 6], [-20, 10, -8, -10, -9], [24, -10, 20, 3, 7], [34, 2, 7, 4, 12], [-19, -2, 5, -9, 10], [-32, -5, 7, -6, -5], [7, 6, 5, -9, -1], [-9, -10, -15, -2, -19], [-18, 2, -1, 21, 6], [-19, 0, 3, -16, -10], [-7, -4, 13, -23, -3], [12, 8, 1, 2, 4], [7, -4, 20, 4, 19], [-29, -7, 1, -2, -10], [6, -15, -5, 8, -15], [-26, 5, 1, -15, -2], [-10, 9, -9, -14, -10], [29, -16, 10, 2, -6], [0, 6, -6, -3, 12], [32, -4, -6, 2, -5], [-12, 5, 15, 5, 0], [24, -14, 3, 16, 2], [6, -11, -2, 14, -2], [36, 11, -12, 18, 17], [5, -22, -3, 6, 7], [2, -4, -16, 7, -12], [-2, -11, 0, -9, -15], [4, 1, -8, -7, -5], [19, -11, 1, -6, -9], [8, -15, -3, 7, 2], [12, 23, 0, 0, 21], [17, 14, -2, -4, 15], [-12, 8, -5, -8, 8], [-4, 10, -3, -6, 1], [-12, -13, 4, -8, -12], [-10, -10, 16, -3, 20], [32, -8, 4, 20, 21], [18, -29, 7, -7, -8], [-20, 14, -3, -20, -18], [-1, -3, -20, -3, -29], [17, 11, 5, 7, 13], [-7, 16, 11, -3, -8], [-16, 3, -5, 0, -20], [-17, 1, -18, 3, -9], [32, 13, 18, -7, 16], [-3, 8, -15, 4, -5], [0, 8, 9, -13, 9], [28, 0, 7, 1, -5], [30, 10, 5, -14, -5], [-12, 0, -11, 9, 0], [6, 7, 21, 3, 23], [2, 2, 24, 1, 5], [23, 4, -13, 7, -11], [4, 5, -4, -20, -10], [31, -10, -6, 8, 5], [26, -5, 4, -3, -2], [5, -28, -5, 7, -9], [-20, -19, -5, 2, -23], [18, 11, 6, -6, -3], [-9, 7, 3, -10, -9], [26, 11, 10, -1, 13], [-19, 2, -27, -1, -8], [6, -4, -1, 10, 11], [-11, -12, -5, 8, -18], [17, -11, 15, 4, 17], [-4, 0, 2, -5, -5], [5, 0, 3, -11, -9], [13, -2, 10, 3, -11], [-6, -3, 0, -3, 6], [10, -6, -3, 19, -9], [47, -21, 15, 13, 18], [10, -17, -4, -7, -9], [3, -14, 3, 16, -6], [1, 0, -10, -8, 1], [-7, 8, -15, -5, 3], [-25, 9, -7, -15, -7], [10, -15, -1, 7, -7], [-17, 25, 7, 1, 20], [-3, 1, 17, -4, 5], [-13, 20, 3, 14, 19], [-14, -3, 20, -14, 12], [31, -16, 9, 7, 3], [5, -11, -8, 19, 7], [18, -22, 0, 9, 4], [22, 3, 17, -12, 11], [0, 17, -9, 5, 8], [10, -1, -3, -10, -19], [-22, -11, -7, -10, -28], [-6, 5, -1, -16, -4], [5, -10, -25, 10, 0], [-7, 4, 10, 3, -14], [-9, -2, -6, 3, 3], [10, -5, 13, 15, -9], [14, 0, -1, -8, 11], [1, -3, -1, 19, 15], [-9, 4, -8, 19, 2], [12, 22, 18, 2, 36], [29, 3, -19, 21, 9], [7, 11, -13, 2, 0], [-9, -8, 18, -15, -3], [-4, -6, 23, -11, 20], [18, 16, 18, -26, 5], [-29, 9, 14, -10, -16], [-41, 7, 13, -4, 9], [-48, -8, 1, -2, -15], [24, -6, -3, 19, 0], [-59, 11, -2, -17, -21], [-23, -16, 7, -4, -24], [-19, 5, -6, -19, -6], [-5, 12, 12, -3, 15], [4, -9, 0, 2, -12], [25, -14, 3, 5, 15], [4, 12, 17, -15, -7], [-6, -13, 10, 1, 7], [24, -7, -6, -10, -8], [20, -6, 0, 7, -6], [-20, 1, 11, -6, 6], [24, 5, 3, 7, 23], [0, -17, 3, 9, -2], [-41, 9, -5, -11, -10], [23, 6, 4, 0, 4], [10, 1, 13, 1, -10], [5, -3, -7, 19, 6], [-4, -14, -1, -12, -23], [-29, -5, 9, 9, 1], [3, -6, 2, 12, 9], [11, -28, -3, 20, -10], [-21, 18, 7, -5, 22], [-29, 4, -25, -6, -18], [-3, 0, 11, 25, 20], [19, 10, 6, -16, -7], [25, 2, 19, 1, 17], [-10, 0, 2, 2, 23], [8, 15, 5, -18, -12], [17, -15, 8, 16, 16], [-31, 11, 18, 4, 15], [32, -13, -3, 18, 22], [9, -10, -12, 10, -18], [-13, 1, -13, 4, -13], [-4, -13, -4, -8, -14], [-12, -12, 11, -11, -3], [24, 8, -31, 19, -2], [41, 2, -12, 15, -12], [-17, -1, 28, -19, -3], [-5, -13, -9, 13, -13], [-1, 2, 4, 4, -8], [-36, -19, -9, -3, -24], [42, -15, 13, 12, 17], [-13, -1, -3, 5, -9], [-18, 11, -6, 1, 12], [-19, 2, -10, -5, 1], [-24, 4, -28, 15, -8], [15, 10, -32, 22, -16], [-10, -3, -3, 17, 15], [-14, -3, 3, -7, 13], [27, -9, -11, -1, 7], [45, 19, 18, -15, 13], [13, 18, -13, 4, 6], [25, -8, 26, -3, 2], [-47, -15, -26, 19, -9], [-10, 9, -4, -8, 18], [-19, 20, -3, -13, 10], [13, 9, 5, 9, 8], [21, -5, -16, 27, 2], [-51, -4, -13, 14, -5], [-10, 3, -5, -4, -9], [-32, 1, 6, -27, -10], [35, -14, 4, 22, -9], [-18, 10, -17, 4, -11], [-53, 1, -1, -2, -2], [-10, -28, -1, 8, 17], [-50, -16, -15, 20, -3], [-9, -13, 19, -31, -10], [-19, -14, -17, 1, -16], [16, 6, -15, 21, 12], [-50, 8, 7, -3, 2], [-25, 11, -8, 1, -21], [25, 13, -3, 2, -1], [16, -14, 3, 5, -7], [-34, 25, 5, -5, -7], [15, -4, 12, -8, -4], [-32, 4, 9, 0, 8], [-9, 8, -11, 26, 12], [10, -9, 5, 14, 1], [-8, -21, -19, 34, -14], [44, -4, 3, 8, 23], [10, 12, -24, 20, -8], [54, -6, -1, 5, -2], [-26, 5, 13, -3, 27], [43, -21, 0, 12, -3], [-2, 29, -15, 12, -9], [-40, 0, 15, -18, 5], [15, -32, 4, -3, -7], [-36, 28, -1, -33, -29], [-3, 19, -1, -20, 8], [3, 8, -15, -7, -10], [17, 10, 10, 18, 36], [-51, 6, 7, 6, 16], [8, -21, -8, 10, -14], [-12, 6, -23, 17, 0], [-40, -12, -7, 19, -11], [-15, 16, -9, -3, 18], [-11, -16, -3, -8, -8], [21, -3, 4, 3, -15], [-9, -4, -36, 32, -6], [7, -8, 2, -11, -11], [20, -10, 25, -2, 17], [-28, 8, 17, -1, 22], [-15, -11, -5, -7, -9], [40, -18, 5, 17, -2], [28, 7, 5, -17, -4], [-13, -8, 9, -14, 4], [6, -6, -13, 17, -13], [16, -23, -6, -3, -21], [-70, 5, 2, -31, -35], [-51, 32, -5, -11, 8], [2, 25, 19, -8, 18], [-7, -10, 21, -14, 11], [-21, 11, 12, 2, 3], [-2, -8, 7, 6, -19], [8, 18, 0, 2, 8], [-3, -23, 25, -17, -5], [-16, 11, 1, 6, 9], [-45, 10, -14, 8, -8], [-28, 9, -3, 7, -2], [-10, -4, -17, 19, -3], [-41, 29, -6, -4, -1], [-1, 10, 19, 9, 29], [-34, 16, -36, 13, -11], [-35, -16, -3, 16, -4], [54, -20, 17, 30, 16], [34, -9, -16, 12, 10], [-45, -14, 7, -4, -2], [-24, -8, -10, 2, -8]];
    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_y();".
// 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_y(: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_y();".
// 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_y( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2842_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-11, 6, 15, -5, -3, 1]>,<5,R![-3, -5, 9, 10, -7, 1]>],Snew);
    return Vf;
end function;