
// Make newform 3381.2.a.y in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_3381_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_3381_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_3381_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 := [-9, 19, 7, -9, -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], [-4, 0, 1, 0, 0], [0, -5, 0, 1, 0], [4, -1, -6, 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_3381_a();"
function MakeCharacter_3381_a()
    N := 3381;
    order := 1;
    char_gens := [2255, 346, 442];
    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_3381_a_Hecke(Kf)
    return MakeCharacter_3381_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, 1, 0, 0, 0], [-1, 0, 0, 0, 0], [0, 0, 0, -1, 0], [0, 0, 0, 0, 0], [0, -1, 1, 0, 0], [0, -1, 1, -1, 0], [-1, 0, -1, 1, 1], [-2, 1, -1, 0, 1], [1, 0, 0, 0, 0], [1, 0, -1, -1, -1], [-3, 0, -1, 1, 0], [-1, 0, -1, -1, -2], [-2, -1, -1, 2, 0], [-4, -1, 1, -1, 0], [-6, 2, 0, 2, 0], [0, -2, -2, 1, 1], [-2, -2, 2, -1, -1], [-5, 4, 1, 0, 0], [-3, 1, 2, 0, -2], [3, 0, -1, 0, -1], [-1, -2, -5, 1, 0], [-1, 2, -5, 5, 2], [-1, -2, 1, -1, -1], [3, 0, -1, 0, -3], [0, -1, -1, 2, -1], [-2, -2, 2, -1, -3], [4, 2, 2, 0, 0], [2, -1, 1, 1, -1], [1, 1, 4, -2, 2], [6, 2, 0, -1, 1], [-3, -2, -3, 2, -2], [-6, 3, -1, 4, 2], [4, -3, 1, 0, 0], [-7, -3, 2, -2, -2], [4, -6, 4, 0, -2], [-2, 0, 2, -4, 0], [8, 4, 0, 0, 0], [-1, 5, -2, 2, 3], [-4, -3, 3, 2, 2], [4, 1, 7, -2, 0], [-4, 2, -2, -1, 2], [-10, -5, -3, 2, -1], [-2, 0, -2, 6, 2], [-10, 4, 4, 2, 2], [9, 2, 1, -2, 1], [-5, 1, 2, 6, -1], [6, 1, -1, -4, 3], [5, -5, 4, -4, 0], [-4, 6, -2, 1, 2], [7, -7, 2, -2, -1], [-2, -2, 2, -1, 2], [-2, 6, 4, 1, 2], [-10, 5, -1, -2, 1], [-8, 0, -4, 0, 2], [4, 0, 4, -4, 4], [-2, -3, -3, 4, -2], [-2, 3, 1, 1, -3], [-3, 2, 1, -5, -2], [1, -8, 1, 0, 0], [-4, 4, -2, 2, -4], [1, -1, 6, -2, -5], [4, 0, 4, 0, 2], [-17, -2, -5, 5, -2], [-14, 5, -5, 1, 3], [4, 6, -2, 8, -2], [8, 4, 6, -1, 0], [16, -6, 2, -2, -4], [-2, -5, 5, -7, 0], [11, 0, 3, -5, -3], [-9, 9, -4, -6, -2], [-10, 5, -7, -4, 0], [-4, 2, -8, 1, 1], [1, 4, -5, 0, -2], [1, -10, -1, -3, 0], [2, -2, 8, 2, -2], [1, 8, -3, 1, -1], [16, -1, 3, 0, 2], [8, -6, -2, 2, 4], [-12, -1, -5, -2, 2], [-7, 8, -5, 1, 2], [6, 8, 2, -1, -2], [0, -1, -5, 5, -2], [-4, 1, 5, -3, 5], [10, 0, 10, 0, 0], [-15, 4, -9, 3, 4], [24, 0, 4, -4, 0], [8, 4, -2, 1, -2], [7, 1, -2, -6, 2], [8, -4, -6, 3, 1], [4, -1, -1, 8, -1], [-1, 0, -13, 1, -3], [7, 6, 13, -5, 1], [-16, 3, 7, 0, 3], [-5, -4, -1, 0, -5], [-5, -9, 4, 4, 1], [2, -8, 2, -3, 0], [18, -2, 4, 0, 2], [12, -2, 6, -4, -2], [0, -6, 2, -6, -2], [8, 2, -4, 2, 8], [1, -11, 4, 4, -1], [6, 0, -8, 10, 4], [-35, 2, -1, 2, 1], [6, -4, 4, 0, 0], [-3, 6, -3, -7, 2], [8, -6, 10, -6, -6], [-2, -1, -7, -1, 1], [26, 2, -4, 2, 0], [-9, 8, -9, 2, 5], [-15, -3, -6, -6, 4], [-10, 1, 9, 1, 6], [-15, 12, -7, 1, 2], [10, -7, -7, -1, 3], [-17, 0, 5, 6, 0], [-13, 8, -3, 7, 4], [6, -1, 15, -5, -1], [1, -10, 1, 2, -2], [-20, -4, -10, 5, -1], [-12, 4, -10, -1, -4], [-16, 5, -9, 4, 6], [2, -5, 9, 0, 3], [2, -7, 1, -10, 1], [15, 6, -7, 4, -5], [7, -6, -3, -11, 1], [0, -1, 9, -9, -6], [2, 0, 6, -11, 1], [-22, -3, -5, 1, 0], [8, -15, 3, -2, -6], [26, 3, 5, -10, 3], [26, 2, -4, 4, 2], [4, -1, -5, 2, 7], [-2, 4, 0, 11, -5], [9, -7, -2, -12, 0], [11, 0, -1, 3, -4], [-8, 8, -4, -6, -2], [-28, 2, -2, 2, 0], [-3, -6, 7, 11, 1], [7, -11, 10, 6, -1], [-10, 12, -2, -6, -2], [23, -2, -5, -6, -3], [1, -6, -3, 7, 2], [18, -2, 2, -4, -6], [-23, 7, 2, 6, 1], [-14, 0, 0, 5, 5], [13, -4, 11, 3, -2], [-3, 2, -9, 0, 5], [10, 4, 6, 6, -2], [-4, -8, -8, 8, -2], [12, -8, -8, 8, 4], [-12, 18, -10, 0, 6], [-28, 6, -4, 12, 2], [-6, 2, 12, -6, -6], [1, 0, -1, -2, 4], [8, -9, 7, -1, 1], [-13, -5, 10, -8, -2], [0, -8, 4, 4, -2], [-13, 0, -3, 1, -4], [2, -2, -10, 3, -3], [2, -11, -1, -2, -5], [-6, -4, 2, 6, 2], [-4, 14, 10, -6, -4], [26, 7, 7, -3, -1], [-10, 4, -6, 0, 2], [-11, 12, 1, 4, 7], [32, -7, 3, -3, -1], [-9, -14, 13, -9, -5], [-9, -14, 11, -2, 4], [-3, -4, -13, -3, -2], [-5, -9, -4, 2, -2], [8, -3, 1, -1, 7], [-10, 3, -7, 3, 3], [12, 4, -8, 2, 0], [2, -1, 9, -7, 1], [-15, -6, -9, 3, 2], [-38, 0, -6, 2, 0], [7, -2, 9, 5, 5], [7, -8, 1, -8, 0], [-18, 8, 4, 11, 2], [3, 3, 8, 2, -5], [15, -5, 16, -10, -1], [-7, -13, -10, -4, 5], [-15, 6, -5, -12, -1], [-15, 5, -10, -4, 5], [-18, -11, 3, 1, -1], [-10, 5, 1, 10, -4], [3, -6, 5, -2, -3], [6, -4, -16, 6, -2], [12, -1, 3, -2, 1], [-11, 5, -6, 2, 7], [-16, 1, 11, -14, -10], [17, -1, 4, 0, 6], [10, 6, 0, -3, 8], [-26, 6, 0, 6, 0], [4, -15, 3, -4, -2], [-14, 3, 5, -5, -5], [1, -4, -15, -4, -1], [6, 1, -9, 9, 4], [-16, 0, 10, 4, 0], [-4, 10, -6, -4, -2], [2, 14, 6, -7, -1], [-24, 0, -6, -4, 6], [8, -6, -2, -8, -10], [-15, -11, -2, 12, 2], [29, 0, -5, 0, 8], [-5, -14, 5, -9, -3], [-10, -10, 16, -2, 2], [-12, -1, -5, 0, -3], [0, 10, -14, 15, 6], [-8, 18, -12, 5, 8], [-7, 8, -5, -5, 8], [-28, -4, 6, 0, 0], [-12, 12, 2, 1, 11], [-5, -16, 5, -5, -4], [-10, -20, -10, 4, 0], [4, 5, -11, -1, 9], [3, 19, -8, 0, 5], [-12, 11, -13, 4, -1], [-10, 13, -1, 13, 5], [-12, 5, -17, 2, -2], [11, 8, 9, -5, 1], [-19, 4, 1, 4, 2], [4, 10, 14, -4, -6], [11, 2, 15, 4, -5], [-4, 16, 0, -6, 2], [10, 7, 7, -10, 4], [17, -5, 12, 8, 7], [33, -14, 3, -17, -7], [0, -12, -8, -4, -2], [-27, -12, 7, -2, 2], [15, -4, 7, 7, -1], [12, 9, -1, -6, -1], [-11, -6, -3, -2, -8], [11, 23, -2, 0, 4], [4, -2, -16, 4, -4], [-19, 18, 1, -4, -2], [9, -16, -7, 0, -1], [2, -6, 8, 18, 4], [32, 5, 5, -18, 0], [-3, -2, 7, -17, -5], [16, -25, 5, 0, -6], [42, 1, 13, -8, 0], [-39, 7, -4, 18, 4], [-12, 6, 6, 10, -2], [-30, 3, -9, 10, 5], [-20, 9, -9, 16, 4], [-20, -3, 9, -11, -5], [26, 12, -14, 12, 6], [16, 2, -18, 6, -4], [15, 14, -9, -5, -4], [6, -11, -7, -6, -4], [3, 4, 3, -9, -6], [28, 10, 2, -18, 2], [-4, 3, -15, 14, -2], [-18, -7, 11, 4, 5], [2, -1, -13, 10, 0], [-18, -4, -10, -6, 6], [-1, 4, -5, -11, 2], [-16, 1, -7, 12, -5], [-2, 15, -3, 1, -7], [-1, -18, -9, -3, 6], [12, -1, 1, 3, 2], [-24, 10, -10, -4, 2], [5, -3, -2, 0, -8], [-24, 13, 5, -12, -11], [30, 1, 3, -1, -3], [10, -11, 3, 2, 7], [20, -4, -2, -6, 0], [-21, 8, -5, 7, -3], [-1, 18, -5, 3, -4], [-2, -2, 6, -3, 6], [-21, -2, 11, -1, -2], [-19, 8, 3, -3, -2], [29, -10, 9, 4, -6], [-24, 8, -8, -4, -4], [-5, -2, -13, -7, -6], [-4, -6, 22, -12, 0], [7, 12, -9, 21, -1], [-2, -25, 11, 10, -1], [29, -2, -1, -1, -4], [-29, 6, 5, -5, -5], [-2, 6, 4, 2, -6], [-6, -7, -15, 1, -6], [-1, 12, -17, 15, 9], [-14, -5, 1, 6, -5], [-9, 17, 0, 18, -2], [17, -20, 5, -9, -3], [-15, 20, -7, 3, 6], [36, -4, 2, -6, 6], [-28, -13, 17, -7, 2], [-6, 6, 4, -18, -6], [20, -1, 13, 7, -7], [11, -10, 5, -8, 5], [9, -8, 7, -6, 11], [-9, -8, 19, -9, 5], [-4, -24, 2, 6, -4], [-11, -2, 3, 7, -3], [-2, 6, -4, 10, 12], [-22, 5, 15, -1, -1], [-10, -29, 11, 6, 2], [9, 14, -19, 13, 8], [11, 14, -1, 4, 0], [-4, 2, -4, 24, 4], [-9, 8, -3, 25, -2], [-18, 6, 16, -17, -10], [30, -6, 4, 0, -12], [26, -9, 15, -24, -5], [-2, 14, 4, 1, 4], [8, -15, 13, -2, -9], [34, -20, 18, -18, -12], [-9, -12, 3, -8, -2], [-34, -1, -5, 6, -4], [-36, -13, -9, 10, 0], [36, -10, 22, -3, -9], [-6, -1, 1, -16, -3], [-18, -7, -7, 0, -6], [2, 14, -4, -8, -4], [-6, -13, 11, -4, 8], [23, -8, 3, 11, 5], [4, -9, -1, 10, 1], [-18, 14, 10, 4, -8], [-24, -28, 0, 12, -6], [-10, 3, 1, -5, 6], [-14, -2, -14, 3, -1], [12, -13, -5, -6, -9], [-2, 5, -17, 0, 2], [17, -7, -12, 4, 5], [-3, 22, 1, -7, 0], [-31, 7, -20, 6, 4], [0, -8, 8, -4, 8], [4, 6, 4, 10, -2], [-6, -18, 2, 6, -10], [29, 8, 19, -11, -1], [16, -27, -1, -1, 4], [-8, -8, 2, 5, -7], [-12, -4, -4, -2, 4], [-6, -6, -8, 4, 0], [28, -6, 12, -2, 4], [18, -2, 0, -2, 2], [27, 15, 10, -16, 2], [17, 23, -2, -16, 3], [20, -7, -1, -20, 5], [-14, -4, 8, -2, -12], [6, 9, -1, -3, 11], [27, 4, -3, -3, 18], [-40, 10, -24, 4, 4], [-1, 10, -11, -14, -1], [21, 23, -8, -12, 2], [15, 7, 12, -2, -5], [46, 4, 2, -13, 8], [-13, 0, 3, 27, 1], [-40, 1, 15, -13, -10], [-20, 21, -9, -6, -3], [11, -12, -3, 15, 11], [7, -9, 4, -8, 13], [-47, 22, 11, 3, 2], [10, 8, 10, 8, -2], [-2, 16, -26, 10, -2], [-11, 18, -13, -3, 1], [-1, -14, 3, 19, 1], [-5, 10, -3, -7, 5], [19, 16, -1, 7, -6], [-10, 11, -1, -6, -10], [-27, 10, 3, -1, 9], [4, -6, 10, 14, 6], [-8, 4, -16, 14, 8], [8, -2, 4, 14, -2], [40, -23, -11, -5, -1], [-38, 22, -8, -4, 0], [-44, -9, -5, 7, -2], [-47, 12, -19, -6, 1], [-27, -8, 15, 5, 4], [-52, -2, -2, -4, -4], [-14, -2, 10, 2, -6], [-36, -14, 6, 6, -2], [-14, 15, -1, 0, -5], [26, -4, 20, -1, -11], [24, 0, 2, -9, -9], [-18, -2, 4, -6, 6], [18, 2, -20, 0, -6], [-5, 14, -13, 9, 6], [5, 18, -25, 17, 9], [6, 9, 5, -15, 1], [48, -8, -4, 0, 2], [-1, -12, -7, 6, 3], [7, -5, -12, -12, -5], [6, -1, -3, 21, 7], [-45, -6, -17, 24, -4], [-20, 10, 2, 18, -4], [-15, -4, 3, 1, 2], [38, -8, 6, -28, -6], [-21, 14, 9, 7, 10], [8, 10, -8, 17, -1], [0, -16, 16, 8, -6], [-20, 4, 4, 6, 12], [10, -19, 11, -17, 1], [1, 23, -10, 4, -4], [-28, -13, -9, -6, 9], [-3, 4, 13, 1, -7], [6, 13, -7, -25, -4], [5, 4, 3, -9, 1], [-27, -6, -3, 5, -2], [23, 2, -5, -4, 7], [-26, 4, 6, -4, 0], [21, 4, 21, -4, 3], [-10, -5, 17, -15, -7], [-4, 3, -3, -19, -4], [-2, -14, 14, -2, 8], [0, 20, 2, -12, 6], [14, -10, 0, -2, 2], [-6, 4, 0, 9, 7], [18, 11, -1, 7, 6], [12, -24, -18, -4, 8], [10, 20, -14, -2, 6], [-26, -7, -7, 1, 2], [-6, 17, -1, 17, 6], [-10, 10, 8, -2, 8], [5, -26, 15, -1, -15], [17, 2, 5, -7, -10], [5, 0, -15, 9, 13], [32, 8, -8, 14, -4], [-12, -7, -3, 2, -6], [46, -11, 7, -18, -7], [-30, -16, 10, -3, 4], [-58, 5, 3, 1, -7], [26, 19, -9, -13, -2], [20, -4, -12, 2, -6], [-2, 11, -11, -11, -11], [10, 2, -18, 18, 10], [53, 19, 12, -16, 4], [-50, 19, 5, -1, 1]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_3381_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_3381_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_3381_2_a_y(:prec:=5)
    chi := MakeCharacter_3381_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_3381_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_3381_2_a_y( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_3381_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-9, 19, 7, -9, -1, 1]>,<5,R![-36, 53, -8, -13, 2, 1]>,<11,R![4, 7, -26, -16, 2, 1]>,<13,R![-40, -145, -104, -17, 4, 1]>],Snew);
    return Vf;
end function;