
// Make newform 144.4.a.a in Magma, downloaded from the LMFDB on 09 January 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_144_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_144_4_a_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_144_4_a_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
        Kf := Rationals();
    end if;
    return [Kf!elt[1] : elt in input];
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_144_a();"
function MakeCharacter_144_a()
    N := 144;
    order := 1;
    char_gens := [127, 37, 65];
    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_144_a_Hecke( : Kf := Kf)
    return MakeCharacter_144_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], [0], [-16], [12], [64], [58], [-32], [136], [-128], [144], [-20], [-18], [288], [200], [384], [-496], [-128], [-458], [496], [512], [-602], [-1108], [704], [960], [206], [-432], [68], [-384], [-518], [960], [-796], [512], [-1824], [2160], [688], [844], [118], [-3576], [384], [-2448], [-4224], [-510], [-384], [-3454], [3216], [-1708], [-2320], [116], [1344], [4594], [-5056], [-3712], [-978], [1856], [-7808], [1024], [-1328], [5812], [8386], [640], [-4832], [-6384], [3312], [9984], [2586], [2832], [5920], [-4674], [-9024], [-4362], [-8768], [-6144], [-4564], [-8770], [1096], [10368], [3248], [-6106], [-7008], [1590], [-192], [9074], [-5248], [-8222], [-16236], [14528], [-6304], [1958], [-4048], [-16988], [6720], [9728], [8444], [-15360], [-6624], [-6912], [19920], [-3680], [11720], [11754], [18904], [-3088], [-21440], [-22624], [6000], [19922], [-3584], [-1984], [14976], [25738], [8548], [8558], [10368], [13088], [4412], [30176], [-21288], [17024], [-2256], [23808], [-26242], [-24590], [2864], [-7616], [-2168], [18000], [3506], [15616], [15036], [-19126], [17392], [-32384], [27708], [-37246], [4192], [26882], [17232], [-31816], [-8272], [9184], [19832], [-15216], [39772], [-18304], [4906], [15360], [-24802], [-15072], [-1800], [7552], [20838], [-47744], [-28280], [7424], [5912], [-26240], [-35620], [3232], [-11478], [39984], [24192], [39456], [41668], [-51648], [-55776], [-36096], [-42532], [29806]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_144_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_144_4_a_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_144_4_a_a(:prec:=1)
    chi := MakeCharacter_144_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_144_4_a_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_144_4_a_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_144_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<5,R![16, 1]>],Snew);
    return Vf;
end function;