
// Make newform 21.8.g.a in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_21_g();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_21_g_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_21_8_g_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_21_8_g_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, -1, 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_21_g();"
function MakeCharacter_21_g()
    N := 21;
    order := 6;
    char_gens := [8, 10];
    v := [3, 5];
    // 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_21_g_Hecke();"
function MakeCharacter_21_g_Hecke(Kf)
    N := 21;
    order := 6;
    char_gens := [8, 10];
    char_values := [[-1, 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 := 8;
    raw_aps := [[0, 0], [27, 27], [0, 0], [-1006, 249], [0, 0], [-9073, 18146], [0, 0], [67026, -33513], [0, 0], [0, 0], [-8815, -8815], [0, 335663], [0, 0], [409495, 0], [0, 0], [0, 0], [0, 0], [-307432, 153716], [-4443527, 4443527], [0, 0], [-3770937, -3770937], [0, 4517617], [0, 0], [0, 0], [7599304, -15198608], [0, 0], [19963966, -9981983], [0, 0], [-9921727, 9921727], [0, 0], [31011857, 0], [0, 0], [0, 0], [18768217, -37536434], [0, 0], [77038916, -77038916], [29883684, 29883684], [0, 87366376], [0, 0], [0, 0], [0, 0], [-24701055, 49402110], [0, 0], [199709975, -199709975], [0, 0], [-97420178, -97420178], [220580656, 0], [116307078, -232614156], [0, 0], [-26423746, 13211873], [0, 0], [0, 0], [-248703856, -248703856], [0, 0], [0, 0], [0, 0], [0, 0], [508092460, -254046230], [-609742193, 609742193], [0, 0], [-45576953, -45576953], [0, 0], [-198601957, 397203914], [0, 0], [-1075172118, 537586059], [0, 0], [0, 252321299], [1148094355, 0], [0, 0], [824460860, -1648921720], [0, 0], [0, 0], [-226368333, -226368333], [0, -1349823371], [-1275718967, 0], [0, 0], [0, 0], [-981631706, 490815853], [0, 0], [552683353, 552683353], [0, 0], [-692296939, 0], [0, 0], [-1478439131, 2956878262], [-1631683836, 815841918], [0, 0], [0, 0], [0, -1744992103], [0, 0], [3080116139, 0], [0, 0], [0, 0], [-4480490725, 4480490725], [0, 0], [0, 1919066393], [0, 0], [0, 0], [0, 0], [7428533078, -3714266539], [0, -7365806309], [5985186440, 0], [0, 0], [0, 0], [0, 0], [8666359411, -8666359411], [-4486078277, -4486078277], [0, 0], [0, 0], [0, 0], [-1504440001, 3008880002], [-8576063502, 4288031751], [-11282581790, 11282581790], [0, 0], [6409344065, 6409344065], [-11701693036, 0], [0, 0], [4404962743, -8809925486], [0, 0], [0, 0], [0, 0], [-7146891369, -7146891369], [10494777421, 0], [0, 0], [0, 0], [10396136542, -5198068271], [0, 0], [0, 3033183362], [0, 0], [1507788841, -3015577682], [15162216994, -7581108497], [75608263, -75608263], [0, 0], [0, 8336322581], [-7127874962, 0], [0, 0], [3899740305, -7799480610], [0, 0], [-6617682794, -6617682794], [0, 0], [0, 0], [16826188686, -33652377372], [0, 0], [-31862264564, 31862264564], [0, 0], [-12457778223, -12457778223], [0, 0], [4465143171, -8930286342], [0, 0], [-37038965060, 18519482530], [0, 0], [0, 36275909578], [0, 0], [4168365469, 0], [0, 0], [6807804503, -6807804503], [0, 0], [0, 5421243491], [0, 0], [-12709328213, 25418656426], [0, 0], [0, 0], [0, 0], [-11027918977, 0], [0, 0], [0, 0], [0, 0], [-57705544243, 57705544243], [36054319369, 36054319369]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_21_g_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_21_8_g_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_21_8_g_a(:prec:=2)
    chi := MakeCharacter_21_g();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 8));
    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_21_8_g_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_21_8_g_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_21_g();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,8,sign)));
    Vf := Kernel([<2,R![0, 1]>],Snew);
    return Vf;
end function;