
// Make newform 1323.2.h.c in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1323_h();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1323_h_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_1323_2_h_c();".
// 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_1323_2_h_c();".
// 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, 0, -1, 0, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 1, 0, 0, 0, 1], [0, 1, 1, 0, -1, 0], [0, 0, 0, 0, 1, -1], [0, 1, 0, 0, -1, -1]];
    Rf_basisdens := [1, 1, 1, 1, 1, 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_1323_h();"
function MakeCharacter_1323_h()
    N := 1323;
    order := 3;
    char_gens := [785, 1081];
    v := [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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1323_h_Hecke();"
function MakeCharacter_1323_h_Hecke(Kf)
    N := 1323;
    order := 3;
    char_gens := [785, 1081];
    char_values := [[-1, 1, 0, 0, 0, 0], [-1, 1, 0, 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 := [[-1, 0, 0, 1, 1, 0], [0, 0, 0, 0, 0, 0], [0, 1, -1, 1, 1, -1], [0, 0, 0, 0, 0, 0], [2, -2, -1, 1, 0, -1], [1, -1, -4, 4, 0, -2], [0, 2, -1, 0, 0, 0], [1, -1, 1, -1, 0, 2], [0, 4, 2, 1, 1, -1], [0, 3, -4, 5, 5, -5], [-1, 0, 0, 6, 3, 0], [1, -1, 3, -3, 0, -3], [0, 0, -1, 1, 0, 1], [0, 1, 1, 1, 1, -1], [-1, 0, 0, -5, -2, 0], [0, 2, 2, -3, -3, 3], [1, 0, 0, 5, 5, 0], [2, 0, 0, 3, 3, 0], [-4, 0, 0, 3, 0, 0], [-3, 0, 0, 3, -3, 0], [0, 7, -1, -4, -4, 4], [-7, 0, 0, 3, 0, 0], [0, -6, -1, -4, -4, 4], [4, -4, -7, 7, 0, -3], [0, 1, -8, 7, 7, -7], [-7, 7, -1, 1, 0, 2], [0, -2, 0, 3, 3, -3], [1, -1, -4, 4, 0, 0], [0, -5, 3, 0, 0, 0], [12, -12, -4, 4, 0, -2], [-4, 0, 0, -8, -13, 0], [0, -5, -10, 7, 7, -7], [5, -5, -7, 7, 0, -7], [-5, 5, 5, -5, 0, 10], [0, -2, -7, 7, 7, -7], [1, -1, -12, 12, 0, -9], [-1, 0, 0, 5, 10, 0], [-2, 2, -3, 3, 0, 0], [-3, 3, -4, 4, 0, -14], [-13, 0, 0, 10, 7, 0], [0, -7, -1, 0, 0, 0], [-7, 0, 0, -6, -3, 0], [-1, 0, 0, -8, 1, 0], [5, 0, 0, -3, -3, 0], [-15, 0, 0, 3, -3, 0], [0, -2, 0, 3, 3, -3], [1, -1, 2, -2, 0, -2], [0, -5, 6, -6, -6, 6], [-7, 7, 2, -2, 0, -1], [0, -5, 0, 12, 12, -12], [7, -7, 5, -5, 0, 9], [3, -3, 2, -2, 0, -8], [10, -10, 0, 0, 0, 3], [6, 0, 0, -15, -6, 0], [0, 16, 2, 4, 4, -4], [-1, 1, 17, -17, 0, 14], [0, -7, 2, -9, -9, 9], [-8, 8, 3, -3, 0, 3], [-2, 2, 3, -3, 0, -6], [0, 18, -7, 8, 8, -8], [5, 0, 0, -12, -15, 0], [3, -3, 2, -2, 0, 7], [2, 0, 0, 1, 8, 0], [16, 0, 0, -4, -1, 0], [-1, 0, 0, -7, -17, 0], [8, 0, 0, -8, -11, 0], [14, 0, 0, 5, 4, 0], [-5, 5, 12, -12, 0, -3], [-5, 0, 0, -7, 8, 0], [0, -8, -8, 10, 10, -10], [8, -8, 2, -2, 0, 5], [-2, 2, 5, -5, 0, -6], [-5, 5, -5, 5, 0, 5], [0, -5, 5, -1, -1, 1], [-1, 0, 0, -5, 5, 0], [0, -11, 11, 1, 1, -1], [-7, 7, -13, 13, 0, -4], [19, -19, -3, 3, 0, 3], [0, 4, -7, -11, -11, 11], [-4, 0, 0, -6, 12, 0], [-12, 12, 8, -8, 0, 13], [0, 13, 15, -6, -6, 6], [0, 5, -4, -15, -15, 15], [17, 0, 0, 4, -10, 0], [11, 0, 0, -19, -11, 0], [2, 0, 0, 10, 4, 0], [0, 0, 0, -3, -6, 0], [-19, 0, 0, -3, 12, 0], [0, 6, -7, 2, 2, -2], [4, -4, 6, -6, 0, 3], [-23, 23, 11, -11, 0, 3], [-25, 25, -7, 7, 0, -4], [0, 7, -16, -1, -1, 1], [-9, 9, -7, 7, 0, 13], [0, -2, 18, -6, -6, 6], [12, 0, 0, -18, -9, 0], [0, -8, 2, -11, -11, 11], [0, -13, 2, -6, -6, 6], [-11, 11, -12, 12, 0, -15], [-11, 11, -2, 2, 0, -10], [0, -8, -20, 1, 1, -1], [0, -1, -4, 21, 21, -21], [10, 0, 0, 8, -13, 0], [17, 0, 0, 4, 10, 0], [-13, 0, 0, -1, 13, 0], [0, 16, 0, 9, 9, -9], [0, 18, -13, -7, -7, 7], [19, -19, -7, 7, 0, 6], [-14, 0, 0, 2, 14, 0], [0, 22, 1, -5, -5, 5], [0, -17, 8, -22, -22, 22], [0, 1, 24, -6, -6, 6], [24, -24, 11, -11, 0, -5], [22, -22, 0, 0, 0, -15], [2, 0, 0, 16, -1, 0], [14, -14, -10, 10, 0, -13], [19, -19, 2, -2, 0, 13], [0, -7, -19, 18, 18, -18], [0, -8, 23, -2, -2, 2], [0, -3, 5, -28, -28, 28], [5, 0, 0, 17, -5, 0], [0, -14, 3, -9, -9, 9], [-1, 0, 0, -23, 1, 0], [0, 32, 2, 0, 0, 0], [-4, 0, 0, 6, -9, 0], [-6, 0, 0, 6, 12, 0], [2, 0, 0, 6, -6, 0], [-11, 11, -16, 16, 0, -21], [0, 13, 19, -17, -17, 17], [0, -11, -12, -6, -6, 6], [0, 31, 0, -3, -3, 3], [-6, 6, 23, -23, 0, 10], [0, -11, 11, -16, -16, 16], [14, 0, 0, -5, 8, 0], [0, 10, -16, -2, -2, 2], [-20, 20, -12, 12, 0, -9], [0, -1, -16, 12, 12, -12], [-19, 0, 0, -6, -15, 0], [24, -24, -19, 19, 0, -2], [0, 8, -7, 12, 12, -12], [-4, 0, 0, -15, 9, 0], [5, 0, 0, 25, 22, 0], [32, 0, 0, -12, 0, 0], [-3, 0, 0, -21, -3, 0], [0, -5, -1, -10, -10, 10], [0, 9, -10, 8, 8, -8], [0, -11, 15, 0, 0, 0], [-4, 4, 14, -14, 0, 17], [0, 31, 0, 12, 12, -12], [-2, 2, 11, -11, 0, 12], [0, -2, 9, 0, 0, 0], [-33, 0, 0, -6, -21, 0], [5, 0, 0, -18, 3, 0], [0, 4, 26, -17, -17, 17], [-17, 17, -6, 6, 0, 18], [0, 9, -22, 11, 11, -11], [-8, 8, 9, -9, 0, -3], [-25, 0, 0, 7, 19, 0], [2, 0, 0, 1, -16, 0], [31, 0, 0, -16, -1, 0], [8, 0, 0, -5, -17, 0], [-18, 0, 0, 12, 24, 0], [22, -22, 11, -11, 0, 28], [-26, 26, -19, 19, 0, -18], [-17, 0, 0, 8, -16, 0], [32, -32, -4, 4, 0, -13], [0, -35, -15, 0, 0, 0], [-2, 2, 24, -24, 0, 12], [14, 0, 0, -11, 14, 0], [0, 13, -19, 4, 4, -4], [41, -41, 2, -2, 0, -1], [28, -28, 3, -3, 0, -9], [0, 7, 8, -8, -8, 8], [19, -19, 4, -4, 0, 8], [5, 0, 0, 12, 6, 0], [18, -18, -31, 31, 0, -11], [0, 7, 7, 7, 7, -7], [0, -19, -1, 21, 21, -21], [-16, 0, 0, -11, -31, 0], [-37, 0, 0, -12, 3, 0], [2, 0, 0, -19, 7, 0], [0, -9, 29, -4, -4, 4], [-8, 8, -16, 16, 0, -14], [19, -19, 11, -11, 0, -18], [10, 0, 0, -7, -1, 0], [-19, 19, 23, -23, 0, 2], [0, 7, -9, 27, 27, -27], [-14, 14, -5, 5, 0, -22], [0, 16, -21, -21, -21, 21], [0, -1, -19, -12, -12, 12], [25, -25, 16, -16, 0, -1], [0, -15, 23, -25, -25, 25], [19, -19, 12, -12, 0, 27], [20, 0, 0, -2, -5, 0], [0, 17, -1, -27, -27, 27], [-32, 0, 0, -13, -1, 0], [-13, 0, 0, -31, -20, 0], [5, 0, 0, -15, -36, 0], [0, 36, -1, 8, 8, -8], [-11, 11, 20, -20, 0, 6], [25, 0, 0, 14, -16, 0], [0, -5, -32, 10, 10, -10], [0, 43, 0, -3, -3, 3], [16, -16, 15, -15, 0, 21], [-6, 0, 0, 6, -27, 0], [0, -40, 5, 3, 3, -3], [-26, 26, -24, 24, 0, -30], [0, 16, -4, -17, -17, 17], [0, 12, 8, -10, -10, 10], [14, 0, 0, -18, 9, 0], [-14, 14, -15, 15, 0, 21], [-21, 21, 5, -5, 0, -8], [0, -20, -2, -14, -14, 14], [-10, 0, 0, 16, -11, 0], [-2, 0, 0, -31, -34, 0], [17, 0, 0, 21, 27, 0], [-28, 0, 0, -13, 19, 0], [-25, 25, 17, -17, 0, -7], [40, -40, 5, -5, 0, -12], [21, -21, -4, 4, 0, 7], [0, -11, 0, 27, 27, -27], [22, -22, 8, -8, 0, 4], [0, 16, -4, -20, -20, 20], [1, -1, -17, 17, 0, -7], [6, -6, 8, -8, 0, 10], [0, -20, -29, -2, -2, 2], [-28, 0, 0, 16, -11, 0], [0, 20, 2, -3, -3, 3], [-34, 0, 0, -22, 1, 0], [5, 0, 0, -14, -23, 0], [11, 0, 0, -28, -17, 0], [0, 22, 8, -10, -10, 10], [-20, 20, -10, 10, 0, -2], [7, 0, 0, -7, -4, 0], [0, -2, -2, -20, -20, 20], [-19, 19, -31, 31, 0, -43], [0, -50, -10, 2, 2, -2], [-2, 2, -4, 4, 0, -15], [33, -33, -13, 13, 0, -17], [3, 0, 0, 3, -6, 0], [14, -14, -13, 13, 0, -7], [25, -25, 10, -10, 0, 5], [-16, 0, 0, -4, -23, 0], [7, -7, 36, -36, 0, 9], [-6, 6, -22, 22, 0, -35], [-10, 0, 0, -17, -2, 0], [-7, 0, 0, -11, -1, 0], [-2, 0, 0, 32, 29, 0], [17, 0, 0, 29, -14, 0], [3, 0, 0, -12, -18, 0], [16, -16, -1, 1, 0, -17], [10, -10, -22, 22, 0, 12], [-38, 0, 0, 11, -1, 0], [0, -2, 7, 4, 4, -4], [-19, 19, 5, -5, 0, 38], [4, -4, 11, -11, 0, 30], [0, -5, 15, -18, -18, 18], [-41, 41, -21, 21, 0, -18], [3, 0, 0, 12, 21, 0], [-17, 17, 4, -4, 0, -28], [-26, 26, -14, 14, 0, -19], [0, 18, 5, -10, -10, 10], [-16, 0, 0, -6, -21, 0], [-7, 0, 0, 34, 14, 0], [16, 0, 0, 8, -16, 0], [2, 0, 0, -6, 24, 0], [3, 0, 0, 9, 12, 0], [0, -15, -1, -31, -31, 31], [16, -16, 21, -21, 0, -15], [46, 0, 0, -7, -1, 0], [0, 28, -13, -16, -16, 16], [0, -11, 9, 6, 6, -6], [-5, 5, 6, -6, 0, 36], [0, 28, 33, -12, -12, 12], [25, -25, 24, -24, 0, 27], [31, -31, -6, 6, 0, -18], [0, 10, -10, 19, 19, -19], [1, -1, -3, 3, 0, -24], [-2, 2, -6, 6, 0, 12], [35, 0, 0, -8, 7, 0], [-17, 0, 0, 20, -4, 0], [26, 0, 0, 18, 27, 0], [0, -12, 20, -13, -13, 13], [0, 1, -32, -11, -11, 11], [0, 10, 15, -12, -12, 12], [7, -7, -19, 19, 0, 6], [0, 7, -21, 12, 12, -12], [-3, 3, 32, -32, 0, 31], [7, -7, 18, -18, 0, -9], [48, 0, 0, 9, -12, 0], [-52, 52, -10, 10, 0, 23], [0, 26, -4, -12, -12, 12], [0, 13, 5, 34, 34, -34], [3, -3, -13, 13, 0, -8], [0, 22, -26, 37, 37, -37], [37, 0, 0, 20, 17, 0], [23, 0, 0, 15, 9, 0], [0, -30, -16, -10, -10, 10], [31, -31, 17, -17, 0, -9], [0, 1, 28, -8, -8, 8], [0, 43, 12, 0, 0, 0], [-26, 26, -34, 34, 0, -51], [0, -5, -4, 5, 5, -5]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1323_h_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1323_2_h_c();".
// 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_1323_2_h_c(:prec:=6)
    chi := MakeCharacter_1323_h();
    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_1323_2_h_c();".
// 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_1323_2_h_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1323_h();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-3, 0, 3, 1]>,<5,R![9, 0, 9, -6, 9, -3, 1]>],Snew);
    return Vf;
end function;