
// Make newform 144.2.i.d in Magma, downloaded from the LMFDB on 20 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_144_i();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_144_i_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_i_d();".
// 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_i_d();".
// 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 := [9, -3, -2, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [-3, -2, 2, 1], [3, 2, 0, -1]];
    Rf_basisdens := [1, 1, 6, 2];
    Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf);
    Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]];
    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_i();"
function MakeCharacter_144_i()
    N := 144;
    order := 3;
    char_gens := [127, 37, 65];
    v := [3, 3, 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_144_i_Hecke();"
function MakeCharacter_144_i_Hecke(Kf)
    N := 144;
    order := 3;
    char_gens := [127, 37, 65];
    char_values := [[1, 0, 0, 0], [1, 0, 0, 0], [-1, 0, 1, 0]];
    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, 0, 0, 0], [0, -1, 0, 0], [0, 1, -1, -2], [-1, 2, 2, -1], [0, 0, 1, 0], [-2, -1, 3, 2], [-2, -1, -1, -1], [-4, 1, 1, 1], [2, 1, -3, -2], [-1, 2, 2, -1], [4, -1, -3, 2], [2, 2, 2, 2], [7, -2, -5, 4], [2, -4, 3, 2], [-1, 2, 2, -1], [-6, 2, 2, 2], [7, 0, -7, 0], [1, -2, 0, 1], [1, 2, -3, -4], [4, 0, 0, 0], [-2, -3, -3, -3], [1, -2, -4, 1], [1, -2, 12, 1], [6, 0, 0, 0], [2, -4, 3, 2], [-3, 6, 0, -3], [-10, 1, 9, -2], [-4, 5, 5, 5], [2, -2, -2, -2], [2, -1, -1, 2], [0, -2, -2, -2], [16, -1, -15, 2], [2, -4, -13, 2], [1, -2, 1, 4], [6, -1, -5, 2], [-1, 2, -2, -1], [-4, -3, 7, 6], [12, 0, 0, 0], [-12, 3, 9, -6], [-1, 2, 14, -1], [4, -8, -8, -8], [-14, 4, 4, 4], [-5, 10, -2, -5], [3, 2, -5, -4], [22, -2, -2, -2], [-4, 6, 6, 6], [-12, -1, 13, 2], [7, -14, 2, 7], [0, 0, -15, 0], [-6, 7, -1, -14], [2, 1, 1, 1], [-2, 5, -3, -10], [2, -4, -1, 2], [-20, 3, 3, 3], [5, 4, -9, -8], [3, -6, -14, 3], [-6, -2, -2, -2], [8, 4, 4, 4], [3, -6, -22, 3], [-5, 10, 2, -5], [12, -5, -7, 10], [-22, -1, 23, 2], [-28, -1, -1, -1], [-18, -3, 21, 6], [6, -12, 3, 6], [7, -14, -14, 7], [5, -10, -20, 5], [1, -8, 7, 16], [5, -10, 5, 20], [-3, 6, -24, -3], [-8, 16, 5, -8], [0, -6, -6, -6], [1, -2, 20, 1], [-14, 3, 11, -6], [16, -3, -3, -3], [0, -9, 9, 18], [7, -14, 6, 7], [2, -6, -6, -6], [3, -6, 3, 12], [-1, 2, -1, -4], [4, 7, -11, -14], [-3, 6, 28, -3], [8, -2, -2, -2], [22, 3, 3, 3], [-3, 6, 0, -3], [-10, 20, 7, -10], [-26, 3, 3, 3], [-2, 4, -1, -2], [3, -6, 14, 3], [20, -9, -11, 18], [-4, -1, -1, -1], [7, -14, 6, 7], [0, -2, -2, -2], [5, 2, -7, -4], [-1, 8, -7, -16], [-20, 4, 4, 4], [24, -3, -21, 6], [2, 3, 3, 3], [0, 4, 4, 4], [-18, 6, 6, 6], [-2, 4, -21, -2], [-14, 0, 0, 0], [5, -2, -3, 4], [6, -12, -13, 6], [-17, 4, 13, -8], [-2, 5, 5, 5], [-6, 12, -5, -6], [-26, 0, 0, 0], [28, -5, -23, 10], [8, -16, 13, 8], [-4, 3, 1, -6], [-6, -2, -2, -2], [-19, 8, 11, -16], [-6, 12, 7, -6], [32, 0, 0, 0], [6, -12, 11, 6], [3, 0, -3, 0], [0, -10, -10, -10], [30, 3, -33, -6], [-3, 6, 12, -3], [20, -11, -9, 22], [-1, 2, -2, -1], [-7, 14, 4, -7], [12, 1, 1, 1], [-3, 6, 8, -3], [-22, -4, -4, -4], [-7, 14, 12, -7], [-32, -4, -4, -4], [3, -6, 18, 3], [26, -5, -21, 10], [-32, -3, -3, -3], [38, 1, -39, -2], [-40, -1, 41, 2], [22, -4, -4, -4], [-2, -9, 11, 18], [-26, 3, 23, -6], [2, 8, 8, 8], [12, -13, 1, 26], [-20, -7, 27, 14], [30, -5, -5, -5], [20, -9, -9, -9], [-1, 2, -22, -1], [32, 3, -35, -6], [-4, -8, -8, -8], [2, 4, 4, 4], [-11, 22, 4, -11], [3, -6, -26, 3], [3, -6, -30, 3], [7, 4, -11, -8], [-8, 8, 8, 8], [-16, 5, 11, -10], [-2, 8, 8, 8], [12, -13, -13, -13], [0, -5, 5, 10], [0, 0, 37, 0], [-3, 6, -8, -3], [-16, -6, -6, -6], [15, -30, -2, 15], [30, 0, 0, 0], [4, 13, -17, -26], [2, -4, -49, 2], [-18, 3, 3, 3], [-6, 1, 5, -2], [-8, 4, 4, 4], [-5, -2, 7, 4], [-3, 6, 48, -3], [-24, -8, -8, -8], [9, -18, -28, 9]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_144_i_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_144_2_i_d();".
// 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_i_d(:prec:=4)
    chi := MakeCharacter_144_i();
    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_i_d();".
// 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_i_d( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_144_i();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![64, 8, 9, -1, 1]>],Snew);
    return Vf;
end function;