
// Make newform 1344.2.q.y in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1344_q();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1344_q_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_1344_2_q_y();".
// 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_1344_2_q_y();".
// 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 := [4, -18, 27, -19, 12, -3, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0], [3, -1, 1, 0, 0, 0], [-18, 43, -28, 22, -5, 2], [22, -56, 37, -31, 7, -3], [50, -121, 82, -64, 15, -6], [24, -62, 44, -33, 8, -3]];
    Rf_basisdens := [1, 1, 2, 1, 2, 1];
    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_1344_q();"
function MakeCharacter_1344_q()
    N := 1344;
    order := 3;
    char_gens := [127, 1093, 449, 577];
    v := [3, 3, 3, 2];
    // 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_1344_q_Hecke();"
function MakeCharacter_1344_q_Hecke(Kf)
    N := 1344;
    order := 3;
    char_gens := [127, 1093, 449, 577];
    char_values := [[1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 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, 0], [-1, 0, -1, 0, 0, 0], [0, 0, 0, 0, 1, 0], [1, 0, 0, -1, 1, 0], [0, 1, 1, 2, -2, 1], [0, -2, 0, 1, 0, -1], [-2, -2, -2, 0, 2, 0], [1, 1, 2, 1, 0, 2], [0, 0, 2, 0, -2, 0], [-4, 1, 0, 0, 0, 0], [1, 0, 0, -2, 1, -1], [-1, -1, -2, -1, 4, -2], [-4, 0, 0, 2, 0, -2], [4, 0, 0, 1, 0, -1], [0, 0, -4, 0, 2, 0], [2, 1, 4, 4, -3, 2], [-4, 1, -3, 2, -2, 1], [-2, -2, 4, -2, 2, -4], [-3, 1, -4, -2, 0, -1], [0, 2, 0, 0, 0, 0], [-11, 1, -10, 2, -2, 1], [-1, -1, 8, -1, 3, -2], [5, -4, 0, 1, 0, -1], [0, 0, -4, 0, 2, 0], [1, 0, 0, -1, 0, 1], [6, 0, 8, 4, -2, 2], [1, 1, -4, 1, -2, 2], [1, 1, 9, 1, 2, 2], [3, -5, 2, -2, 6, -1], [8, 0, 0, 0, 0, 0], [-6, 1, 0, 1, 0, -1], [-1, -1, -1, -1, 2, -2], [4, -2, 4, 0, 2, 0], [2, -2, 0, -1, 0, 1], [0, 0, -4, 0, -2, 0], [-4, 5, -4, 0, -5, 0], [6, -2, 4, -4, 4, -2], [-2, -2, -2, -2, -2, -4], [2, 4, 0, 2, 0, -2], [2, 2, 8, 2, -2, 4], [-8, 4, -8, 0, -4, 0], [10, 0, 0, -1, 0, 1], [4, 4, 4, 4, -4, 8], [-3, -6, -5, -4, 8, -2], [-12, 2, 0, -2, 0, 2], [-12, -4, -12, 0, 4, 0], [2, 0, 0, -2, 0, 2], [20, 1, 0, 0, 0, 0], [4, -5, 7, 6, 2, 3], [3, 3, 8, 3, -10, 6], [2, 2, 14, 2, -2, 4], [2, 2, 0, 0, 0, 0], [-8, 5, -11, -6, -2, -3], [13, 4, 0, -1, 0, 1], [0, 0, -12, 0, 2, 0], [-8, -2, -8, 0, 2, 0], [6, -3, 4, -4, 5, -2], [0, 0, 12, 0, -3, 0], [-1, 1, 0, 2, -2, 1], [20, 2, 0, -2, 0, 2], [19, -1, 18, -2, 2, -1], [-8, 1, 0, 0, 0, 0], [0, 6, 0, 3, 0, -3], [6, -6, 4, -4, 8, -2], [2, 2, 5, 2, -4, 4], [0, 0, 0, 0, 3, 0], [1, 1, 8, 1, 2, 2], [5, 0, 0, -4, 0, 4], [-10, -2, -8, 4, 0, 2], [18, 4, 0, -4, 0, 4], [4, -4, 0, -8, 8, -4], [-6, -6, 6, -6, 8, -12], [-9, 0, -12, -6, 3, -3], [5, 5, 0, 5, -2, 10], [14, 0, 0, 1, 0, -1], [4, 4, 6, 4, -8, 8], [4, -2, 8, 8, -2, 4], [1, 1, 4, 1, -2, 2], [6, 6, 8, 6, -4, 12], [-1, -6, 1, 4, 4, 2], [12, 8, 0, 2, 0, -2], [-2, 6, 0, -5, 0, 5], [-8, 4, -8, 0, -4, 0], [8, 4, 0, 1, 0, -1], [-4, -4, -8, -4, 9, -8], [-1, -1, 23, -1, -2, -2], [6, -10, 0, 2, 0, -2], [-6, -6, -17, -6, 4, -12], [-20, -2, 0, 0, 0, 0], [-6, 4, 0, -5, 0, 5], [-4, -4, 4, -4, 0, -8], [18, 0, 16, -4, 2, -2], [3, -8, 8, 10, 3, 5], [13, 0, 0, -3, 0, 3], [3, 3, -12, 3, 0, 6], [-14, -2, 0, 0, 0, 0], [-4, -4, 0, -4, 3, -8], [4, -2, 2, -4, 4, -2], [-5, -5, -14, -5, 20, -10], [-1, -1, -12, -1, -4, -2], [10, -2, 0, -2, 0, 2], [18, 7, 20, 4, -9, 2], [6, -9, -3, -18, 18, -9], [-4, -4, -2, -4, -2, -8], [-13, 7, -18, -10, -2, -5], [17, -4, 17, 0, 4, 0], [15, 0, 0, -1, 0, 1], [4, 4, -22, 4, -2, 8], [-24, -2, -26, -4, 4, -2], [3, -4, 0, 0, 0, 0], [-1, -1, -8, -1, 1, -2], [-22, 2, -22, 0, -2, 0], [-10, -4, 0, 4, 0, -4], [9, 9, 10, 2, -10, 1], [0, 13, 0, 0, 0, 0], [2, -12, -6, -16, 20, -8], [2, 2, 0, -5, 0, 5], [10, -8, 12, 4, 6, 2], [-2, -2, 20, -2, 9, -4], [-4, 8, 0, -6, 0, 6], [-3, -11, -4, -2, 12, -1], [-31, 0, 0, 6, 0, -6], [0, 0, -32, 0, 1, 0], [-8, -5, -13, -10, 10, -5], [1, 1, 4, 1, -14, 2], [12, -3, 0, 6, 0, -6], [-4, -4, -6, -4, 8, -8], [0, 0, -10, 0, -4, 0], [2, 11, 0, -5, 0, 5], [3, 3, 8, 3, -4, 6], [-9, 11, -8, 2, -12, 1], [2, -4, 0, -4, 0, 4], [1, 1, 4, 1, -11, 2], [-8, -14, 0, 6, 0, -6], [4, 4, 14, 4, -2, 8], [5, -8, 0, 6, 0, -6], [6, 0, 12, 12, -6, 6], [20, 2, 22, 4, -4, 2], [-4, -1, 0, -6, 0, 6], [24, -8, 20, -8, 12, -4], [-28, -6, 0, 8, 0, -8], [4, 4, 12, 4, -17, 8], [-16, 12, -16, 0, -12, 0], [21, 8, 0, 1, 0, -1], [-17, -7, -18, -2, 8, -1], [-10, -14, 0, 2, 0, -2], [-12, 0, 0, -1, 0, 1], [12, -2, 6, -12, 8, -6], [2, 2, 2, 2, -4, 4], [-4, -4, -14, -4, 0, -8], [-2, -2, 8, -2, 12, -4], [-24, -6, 0, 2, 0, -2], [-26, 0, 0, 5, 0, -5], [-8, -8, -6, -8, 6, -16], [-11, -9, -6, 10, 4, 5], [-20, -2, 0, -2, 0, 2], [-5, -5, 24, -5, 6, -10], [-6, -6, 2, -6, 4, -12], [11, 0, 0, -8, 0, 8], [-18, -1, -20, -4, 3, -2], [-2, -2, -8, -2, -4, -4], [-18, 12, 0, -4, 0, 4], [18, 3, 0, -1, 0, 1], [1, 1, 13, 1, 6, 2], [2, 4, 10, 16, -12, 8], [2, 2, -8, -20, 8, -10], [5, -8, 4, -2, 9, -1], [-23, -11, -26, -6, 14, -3], [-25, 4, 0, -2, 0, 2], [0, 0, -8, 0, -11, 0], [2, 17, 3, 2, -18, 1], [4, -2, 0, 6, 0, -6], [-2, -2, -10, -2, -6, -4], [19, 8, 23, 8, -12, 4], [13, -10, 4, -18, 19, -9], [12, 8, 0, -8, 0, 8], [16, 12, 0, 3, 0, -3], [16, 1, 8, -16, 7, -8], [-4, -8, 0, 4, 0, -4], [-3, -3, 12, -3, 8, -6], [-3, -3, -8, -3, 7, -6], [-17, -12, 0, 3, 0, -3], [24, -8, 0, 1, 0, -1], [-10, -10, -12, -10, 14, -20], [-10, -4, -10, 0, 4, 0], [4, -3, 4, 0, 3, 0], [-22, 8, -20, 4, -10, 2], [17, 9, 12, -10, -4, -5], [2, 2, -17, 2, 4, 4], [18, 6, 22, 8, -10, 4], [4, 4, -7, 4, 8, 8], [19, -12, 0, -1, 0, 1], [-2, -2, -26, -2, 18, -4], [6, 6, 0, 6, -10, 12], [-24, -7, -27, -6, 10, -3], [-16, 2, -12, 8, -6, 4], [-17, 11, -18, -2, -10, -1], [2, 2, -8, 2, -2, 4], [-22, -12, 0, 0, 0, 0], [-2, -2, -28, -2, 14, -4], [42, 3, 44, 4, -5, 2], [2, 12, 0, 3, 0, -3], [-8, -8, -30, -8, 16, -16], [-34, -16, -38, -8, 20, -4], [28, 0, 0, -10, 0, 10], [-38, 1, -36, 4, -3, 2], [-4, -4, -4, -4, 8, -8], [-7, -7, -23, -7, 22, -14], [6, 2, 0, -6, 0, 6], [-4, -6, -12, -16, 14, -8], [5, 5, 6, 5, -22, 10], [-16, -1, 0, -4, 0, 4], [44, -1, 0, 0, 0, 0], [-3, -3, -7, -3, -14, -6], [-46, 6, -42, 8, -10, 4], [-3, -3, 46, -3, 2, -6], [28, 3, 32, 8, -7, 4], [32, -12, 30, -4, 14, -2], [4, 4, 34, 4, 4, 8], [12, 6, 0, -4, 0, 4], [3, 3, 16, 3, -6, 6], [-30, -8, 0, 1, 0, -1], [-6, -6, -14, -6, 20, -12], [8, 8, -12, 8, -11, 16], [-25, -4, 0, -3, 0, 3], [16, 12, 0, -6, 0, 6], [4, 4, -8, 4, 2, 8], [42, -6, 44, 4, 4, 2], [3, 3, -20, 3, -9, 6], [4, 4, 12, 4, -8, 8], [-25, 7, -24, 2, -8, 1], [-4, -12, 0, 8, 8, 4], [-10, 7, 0, -5, 0, 5], [-6, 12, -16, -20, -2, -10], [-18, 14, 0, -5, 0, 5], [-12, -4, -2, 20, -6, 10], [2, 2, 15, 2, 8, 4], [-2, -2, -4, -2, 13, -4], [20, 8, 0, -4, 0, 4], [-22, 6, 0, -2, 0, 2], [12, 0, 4, -16, 8, -8], [-1, -1, -20, -1, -10, -2], [-43, 2, -48, -10, 3, -5], [4, 4, 2, 4, -20, 8], [-22, 10, 0, 2, 0, -2], [6, 6, 40, 6, -8, 12], [-22, 7, 0, -1, 0, 1], [24, -12, 24, 0, 12, 0], [4, -12, -8, -24, 24, -12], [0, -16, 0, 8, 0, -8], [-44, -8, 0, 5, 0, -5], [0, 0, 10, 0, -8, 0], [14, 8, 14, 0, -8, 0], [8, -4, 0, 7, 0, -7], [-24, -5, -16, 16, -3, 8], [1, 1, -57, 1, -6, 2], [31, 7, 32, 2, -8, 1], [-31, -5, -32, -2, 6, -1], [-12, -3, 0, -8, 0, 8], [-7, -7, -31, -7, 2, -14], [19, 17, 24, 10, -22, 5], [-27, 0, 0, 9, 0, -9], [-37, 3, -34, 6, -6, 3], [22, 20, 0, 1, 0, -1], [-2, 16, -6, -8, -12, -4], [-1, -1, 28, -1, -8, -2], [-36, -5, 0, 10, 0, -10], [-26, 14, 0, -6, 0, 6], [24, -4, 0, -8, 0, 8], [-26, 7, -24, 4, -9, 2], [1, 1, -20, 1, -6, 2], [-8, 6, -8, 0, -6, 0], [-21, 8, -9, 24, -20, 12], [-5, -5, -32, -5, 5, -10], [19, 0, 0, -7, 0, 7], [-8, -8, 8, -8, 4, -16], [7, 7, 25, 7, -6, 14], [-7, 3, -10, -6, 0, -3], [-7, -7, 18, -7, 10, -14], [2, 2, 8, 2, -12, 4], [0, -18, -8, -16, 26, -8], [27, -10, 20, -14, 17, -7], [-34, 10, 0, 4, 0, -4], [38, -4, 0, 5, 0, -5], [4, 4, 8, 4, -6, 8], [-4, -4, -18, -4, 18, -8], [15, -2, 21, 12, -4, 6], [-8, -21, 0, 8, 0, -8], [21, -2, 20, -2, 3, -1], [-28, -8, 0, 0, 0, 0], [48, -6, 44, -8, 10, -4], [-18, -3, -9, 18, -6, 9], [4, 4, 32, 4, 4, 8], [-76, 0, 0, 0, 0, 0], [24, -6, 0, 3, 0, -3], [-10, -1, -16, -12, 7, -6], [1, 1, 32, 1, 2, 2], [-32, 10, 0, -6, 0, 6], [8, 8, 16, 8, -4, 16], [20, -8, 0, -9, 0, 9], [0, 0, -34, 0, 8, 0], [2, 2, 32, 2, -19, 4], [-20, -9, -24, -8, 13, -4]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1344_q_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1344_2_q_y();".
// 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_1344_2_q_y(:prec:=6)
    chi := MakeCharacter_1344_q();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(1999) - 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_1344_2_q_y();".
// 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_1344_2_q_y( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1344_q();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![16, -36, 81, -8, 9, 0, 1]>,<11,R![1444, 1026, 729, 76, 27, 0, 1]>,<13,R![112, -36, -3, 1]>],Snew);
    return Vf;
end function;