
// Make newform 220.4.a.b in Magma, downloaded from the LMFDB on 16 March 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_220_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_220_4_a_b();".
// 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_220_4_a_b();".
// 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
        Kf := Rationals();
    end if;
    return [Kf!elt[1] : elt in input];
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_220_a();"
function MakeCharacter_220_a()
    N := 220;
    order := 1;
    char_gens := [111, 177, 101];
    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_220_a_Hecke( : Kf := Kf)
    return MakeCharacter_220_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]];
    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 := 4;
    raw_aps := [[0], [5], [-5], [-19], [-11], [-62], [19], [-131], [138], [-79], [217], [-91], [158], [120], [-546], [-439], [290], [-373], [728], [-709], [850], [-1194], [58], [753], [1228], [1266], [-76], [-1456], [1142], [1764], [144], [-2079], [-636], [-468], [-2797], [-2150], [253], [2649], [-723], [-334], [-1092], [-4230], [-2380], [343], [2336], [-2445], [-5895], [5470], [-3914], [1886], [-749], [-5010], [602], [-462], [-6426], [1151], [860], [-916], [1196], [9078], [-4750], [-2718], [3870], [1433], [1430], [-2265], [8280], [-8731], [-202], [2494], [-26], [-10524], [12276], [6266], [-8654], [-5226], [-6998], [-9150], [-4781], [10228], [-7752], [12112], [-16032], [-13412], [-1768], [-5284], [-10814], [14665], [-481], [-11998], [8045], [10636], [4456], [-2563], [11204], [13584], [13020], [-6514], [-11048], [941], [-9208], [11362], [-630], [-3720], [16811], [814], [2231], [27306], [21681], [3274], [16863], [-15758], [6264], [-19828], [14683], [26335], [1225], [19342], [28207], [-9705], [26510], [18223], [-30868], [1179], [-28242], [-2253], [974], [-1603], [30886], [3212], [17728], [-20171], [-8779], [-3630], [-6582], [-40130], [-31627], [-25408], [33254], [27882], [29319], [16694], [1596], [-3228], [-42716], [-20032], [-1834], [-9621], [8950], [-42870], [-15126], [3282], [-29845], [-18816], [-131], [-45025], [3820], [32163], [-9878], [22629], [-18561], [52461], [7227], [16560], [-51252], [21202], [-61736], [37076]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_220_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_220_4_a_b();".
// 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_220_4_a_b(:prec:=1)
    chi := MakeCharacter_220_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 4));
    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_220_4_a_b();".
// 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_220_4_a_b( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_220_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<3,R![-5, 1]>],Snew);
    return Vf;
end function;