
// Make newform 144.2.x.a in Magma, downloaded from the LMFDB on 26 August 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_144_x();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_144_x_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_144_2_x_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_2_x_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, -1, 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_144_x();"
function MakeCharacter_144_x()
    N := 144;
    order := 12;
    char_gens := [127, 37, 65];
    v := [12, 9, 4];
    // 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_144_x_Hecke();"
function MakeCharacter_144_x_Hecke( : Kf := false)
    N := 144;
    order := 12;
    char_gens := [127, 37, 65];
    char_values := [[1, 0, 0, 0], [0, 0, 0, 1], [-1, 0, 1, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := ISA(Type(Kf), Fld), 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]];
    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 := [[-1, 0, 0, -1], [0, 2, 0, -1], [-1, -1, 0, 1], [4, 1, -2, -1], [-1, 0, 1, 1], [3, -3, -2, 1], [4, 0, 0, 0], [-3, 0, 0, 3], [-2, -3, -2, 0], [1, -1, -2, 2], [-4, -3, 4, 6], [-7, 2, 2, -7], [-3, 6, -3, 0], [1, -6, 5, 5], [0, -3, 4, -3], [3, 2, 2, 3], [0, 0, -3, 3], [-7, 1, 8, -8], [2, -2, -5, -3], [-4, 0, 8, -4], [2, 0, -4, -4], [0, -1, 0, -1], [-1, -10, -9, 9], [8, 0, -16, -2], [0, 0, -1, 0], [-1, 1, 0, 0], [4, 9, 4, 0], [-7, 12, 12, -7], [-1, -2, 2, 1], [-9, -2, 9, 4], [-10, 12, 0, -6], [6, -6, 1, 7], [-10, 8, 5, -8], [10, 10, -5, -5], [7, 7, 6, -13], [0, -7, 0, 7], [1, -1, -2, -1], [1, -8, 8, -1], [-2, -13, -2, 0], [7, 3, -4, 4], [-3, -8, 8, 3], [-5, -2, -2, -5], [0, -3, 8, -3], [1, 2, -1, -4], [9, -4, -4, 9], [-8, 0, 16, 8], [4, -4, -11, -7], [0, 9, 0, 9], [1, 14, 13, -13], [11, 11, -8, -3], [-4, 0, 8, -16], [-6, 3, 6, -6], [0, 2, -9, 2], [3, -12, -12, 3], [7, 10, -7, -20], [-8, -3, 4, 3], [5, -2, 2, -5], [-8, 8, 0, -4], [3, -9, 6, 6], [-18, 4, 9, -4], [-10, -10, 11, -1], [-1, -1, 8, -7], [5, 8, -8, -5], [-6, 21, -6, 0], [22, 10, -11, -10], [-15, -19, -4, 4], [-9, 16, -7, -7], [-9, -6, 9, 12], [2, 2, -11, 9], [-5, -1, 4, -4], [0, 2, -27, 2], [-4, 0, 8, 22], [0, -19, -2, -19], [-3, -3, 8, -5], [9, -18, -18, 9], [16, -5, -16, 10], [7, -3, -4, -4], [9, 22, -22, -9], [-15, -10, 15, 20], [-5, -14, -5, 0], [4, -4, -11, -7], [-3, -9, 12, 12], [14, -20, 0, 10], [14, -12, 0, 6], [16, -7, -8, 7], [5, -10, 5, 5], [18, 20, 0, -10], [30, 4, -15, -4], [1, 3, 2, -2], [-2, -5, 2, 10], [13, 24, -24, -13], [0, 5, 10, 5], [-12, 0, 24, -14], [0, 0, 1, -1], [0, 0, -5, -5], [8, 0, -16, 0], [-13, 13, 6, -7], [-8, 0, 16, 28], [3, 14, 14, 3], [-15, 0, 0, 15], [21, 8, -13, 13], [-13, -30, 30, 13], [-16, 16, 1, -15], [-2, -30, 1, 30], [22, 22, -3, -19], [6, -36, 0, 18], [5, 4, -9, -9], [-2, -4, 0, 2], [0, -35, 0, 0], [-38, 2, 19, -2], [-4, 3, 4, -6], [-1, 18, 18, -1], [5, 18, 5, 0], [-17, 18, -1, -1], [14, 0, -28, 8], [0, 2, 15, 2], [-4, 4, 17, 13], [-6, 0, 12, -36], [-5, 5, -24, -29], [-3, 2, 5, -5], [9, 9, 6, -15], [0, 10, 25, 10], [23, -25, 2, 2], [-1, 12, 12, -1], [7, -4, -11, 11], [9, 22, -22, -9], [-1, 29, -28, -28], [6, -20, 0, 10], [-4, 7, 2, -7], [11, -11, 14, 25], [-27, 2, -2, 27], [0, 29, 0, 0], [-20, -17, 20, 34], [7, -4, -4, 7], [3, -24, 3, 0], [-7, 0, 7, 0], [1, -18, -18, 1], [-20, 20, 7, -13], [-7, 7, 24, 17], [-20, 0, 40, 2], [-13, -4, -4, -13], [9, 17, -26, -26], [10, 25, 10, 0], [-15, -12, -12, -15], [-15, 2, -2, 15], [24, -13, -12, 13], [1, -7, 6, 6], [38, -16, -19, 16], [0, 0, -9, 9], [-30, 4, 0, -2], [27, -27, -34, -7], [14, 20, 0, -10], [-3, 0, 0, 3], [10, -7, 10, 0], [1, -8, 7, 7], [0, 1, -10, 1], [2, 0, -4, 40], [0, 16, -9, 16], [-4, 0, 8, -26], [1, -1, 18, 19], [-11, -26, -15, 15], [2, 0, -4, 36], [-14, -7, -14, 0], [17, -30, -30, 17], [-7, -24, 7, 48], [-28, 23, 14, -23], [-34, -12, 0, 6], [3, -7, 4, 4]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_144_x_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_144_2_x_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_2_x_a(:prec:=4)
    chi := MakeCharacter_144_x();
    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_144_2_x_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_2_x_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_144_x();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![1, 2, 5, 4, 1]>],Snew);
    return Vf;
end function;