
// Make newform 162.2.e.a in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_162_e();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_162_e_Hecke();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_162_2_e_a();".
// 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_162_2_e_a();".
// 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, 0, 0, -1, 0, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    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_162_e();"
function MakeCharacter_162_e()
    N := 162;
    order := 9;
    char_gens := [83];
    v := [8];
    // 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_162_e_Hecke();"
function MakeCharacter_162_e_Hecke(Kf)
    N := 162;
    order := 9;
    char_gens := [83];
    char_values := [[0, 0, 0, 0, 0, -1]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := true, Kf := Kf); // the value of chi on the gens as elements in the Hecke field
    F := Universe(values);// the Hecke field
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),values);
    return chi;
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, -1, 0], [0, 0, 0, 0, 0, 0], [0, 0, 1, 1, -1, -1], [0, 0, 0, 1, -2, 1], [-1, 1, 2, 3, 0, -3], [-2, 0, -3, 0, -2, 0], [2, -3, -1, -2, 4, 4], [0, -3, 4, 3, 4, -3], [2, 0, 0, -2, -2, -1], [-3, -1, -1, 1, 1, 3], [-3, 3, 0, 0, 0, 4], [5, -2, -1, -5, 3, 3], [3, -5, -1, -5, 3, 0], [-4, 4, -2, 2, 0, -2], [-1, 0, 1, -1, 3, -2], [2, 3, 3, 0, -1, -2], [-1, -1, -1, 4, -3, 1], [5, 0, -5, -4, 2, 1], [0, -3, -1, -3, 0, 0], [-4, 6, -1, 4, -5, -5], [0, -5, -2, 1, -2, -5], [8, 0, 5, -5, 0, -8], [2, 0, -2, 2, 0, -2], [0, -1, -3, 5, -3, -1], [-3, 6, -1, 2, 0, -2], [8, 0, -8, -5, -2, 3], [1, 1, 1, -2, 1, -1], [5, 3, 3, 0, -1, -2], [-1, -5, -5, 0, -2, 7], [5, 5, 0, -10, 5, 0], [0, 8, -4, 1, -4, 8], [-7, -2, 0, 9, 9, 1], [-4, 5, 3, -3, -5, 4], [-3, -7, 0, 10, 10, 2], [5, -2, 1, -2, 5, 0], [5, -5, 1, -4, 0, 4], [-1, -1, 7, 2, -1, -7], [-8, 3, 3, 0, -1, -2], [-8, -8, -12, 7, 1, 12], [-9, -3, -3, 6, 0, -6], [-5, 1, -7, 5, 6, 6], [0, 4, -3, -4, -3, 4], [7, 7, 12, -12, -7, -7], [2, 11, 0, -13, -13, -5], [0, -12, 16, 2, 16, -12], [0, -4, 13, 0, -9, -9], [-2, -2, -9, -6, 8, 9], [4, 0, -4, 1, -10, 5], [7, 9, 2, -5, 0, 5], [3, 15, -7, 15, 3, 0], [1, 5, 0, -1, -5, -5], [9, 0, 0, -9, -9, -6], [-8, 4, -11, 11, -4, 8], [0, 12, -13, -8, -13, 12], [-5, 4, -18, 4, -5, 0], [-13, 0, 13, -2, 4, -15], [-13, 2, 2, 0, -11, 9], [11, -3, -3, 0, 9, -6], [-17, 0, 17, 4, 5, -13], [0, 19, 7, 7, 0, -7], [2, -1, -4, -1, 2, 0], [-13, 3, 0, 10, 10, -10], [-2, 2, -10, 2, 8, 8], [11, -6, 5, -6, 11, 0], [12, 5, 1, -11, 0, 11], [18, 0, -18, -13, -11, 5], [5, 0, -5, -8, 6, -3], [1, -7, 10, -7, 1, 0], [6, 6, 0, -12, -12, 15], [-3, 7, 2, -2, -7, 3], [0, -1, 2, -2, 1, 0], [0, 14, -1, -9, -1, 14], [-6, 4, -3, 3, 0, -3], [-15, -15, 0, 11, 4, 0], [1, -4, -4, 0, 7, -3], [-13, -13, 7, 3, 10, -7], [1, -16, -8, -9, 0, 9], [0, 24, -6, 7, -6, 24], [-3, 8, 0, -5, -5, 22], [13, -15, 0, 2, 2, 11], [5, 14, 0, 14, 5, 0], [4, 2, 6, 2, 0, -2], [7, 3, 3, 0, -14, 11], [-19, -13, -13, 0, 8, 5], [5, 0, -5, -13, 5, -8], [14, -15, 16, 2, 0, -2], [-6, 14, 5, 6, -19, -19], [13, -10, -4, 4, 10, -13], [-2, -19, 1, -1, 19, 2], [4, -10, 0, 6, 6, 12], [0, -1, 4, 31, 4, -1], [10, 0, -10, 1, 6, 11], [6, 0, 0, 0, -14, 14], [-14, -14, 11, 0, 14, -11], [-3, 9, -10, 9, -3, 0], [-9, 5, 5, 9, -10, -10], [9, -19, 0, 10, 10, -5], [0, -6, 1, 17, 1, -6], [1, -14, 7, -1, 7, 7], [4, 11, 11, 0, -2, -9], [-15, 0, 15, 15, 17, 0], [11, 6, -13, -11, 7, 7], [11, 11, 0, -22, -22, -6], [5, -23, 2, -2, 23, -5], [-6, -1, 0, 7, 7, 2], [13, 11, -1, -13, -10, -10], [-18, 0, 18, -7, -8, -25], [-2, -16, -16, 0, -12, 28], [27, 27, 5, -28, 1, -5], [24, 0, -24, -3, -1, 21], [-28, 18, -5, 18, -28, 0], [0, -17, 19, 16, 19, -17], [3, -4, 0, 1, 1, 31], [-7, 2, 16, -16, -2, 7], [-1, 4, -19, 1, 15, 15], [-4, 0, 4, 9, 2, 5], [6, 6, 19, -6, 0, -19], [12, -26, -26, 0, 13, 13], [-16, -16, -1, 10, 6, 1], [-7, 11, -9, -2, 0, 2], [13, -26, -19, -26, 13, 0], [-12, 9, -5, 5, -9, 12], [33, -11, 25, -25, 11, -33], [0, 14, 2, -12, 2, 14], [-1, -5, 10, 11, 0, -11], [-43, -2, -2, 0, 0, 2], [-1, 0, 1, 4, -45, 3], [-12, 22, -17, 12, -5, -5], [0, -22, 3, -3, 22, 0], [-13, -1, 0, 14, 14, -8], [19, 10, 1, -19, -11, -11], [-8, 25, 12, 25, -8, 0], [8, 8, 11, 21, -29, -11], [23, 7, 7, 0, -11, 4], [10, 10, -12, -14, 4, 12], [2, 7, -21, 7, 2, 0], [3, -8, 4, -3, 4, 4], [1, 10, 0, -11, -11, 4], [-13, 17, -21, 17, -13, 0], [-14, -2, -2, 0, 8, -6], [2, 23, 23, 0, -10, -13], [-2, -16, 16, 18, 0, -18], [-5, 0, -6, 0, -5, 0], [-7, -29, 27, 7, 2, 2], [0, 7, -28, 0, -28, 7], [10, 18, 17, -17, -18, -10], [10, -23, 17, 7, 0, -7], [4, 0, -4, -2, 21, 2], [25, 25, 11, -7, -18, -11], [37, -1, -1, 0, 3, -2], [21, -20, -14, -20, 21, 0], [13, 20, -3, -13, -17, -17], [0, -5, -7, -3, -7, -5], [-3, -7, 0, 10, 10, 10], [3, 21, -13, -16, 0, 16], [7, 0, -7, 19, 12, 26], [-28, -13, -13, 0, -13, 26], [18, -6, 0, -18, 0, 18], [0, -14, 30, 2, 30, -14], [-10, 3, 0, 7, 7, 29], [-23, -15, 8, -8, 15, 23], [0, -20, -11, 6, -11, -20], [-22, -22, 1, -13, 35, -1], [3, 3, 3, 0, -6, 3], [-24, -24, -7, 14, 10, 7], [8, -25, -6, -14, 0, 14], [0, -3, -18, -8, -18, -3], [-4, -19, -5, 5, 19, 4]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_162_e_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_162_2_e_a();".
// 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_162_2_e_a(:prec:=6)
    chi := MakeCharacter_162_e();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(997) - 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_162_2_e_a();".
// 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_162_2_e_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_162_e();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![9, -27, 36, -24, 9, -3, 1]>],Snew);
    return Vf;
end function;