
// Make newform 1323.2.i.a in Magma, downloaded from the LMFDB on 04 September 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_1323_i();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_1323_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_1323_2_i_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_1323_2_i_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, -1, 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_1323_i();"
function MakeCharacter_1323_i()
    N := 1323;
    order := 6;
    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_i_Hecke();"
function MakeCharacter_1323_i_Hecke( : Kf := false)
    N := 1323;
    order := 6;
    char_gens := [785, 1081];
    char_values := [[0, 1], [0, 1]];
    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, 2], [0, 0], [-3, 3], [0, 0], [-2, 1], [-2, 1], [-3, 3], [6, -3], [-3, -3], [3, 3], [-2, 4], [0, -7], [0, -3], [-1, 1], [0, 0], [-5, -5], [0, 0], [-8, 16], [-4, 0], [2, -4], [3, 3], [8, 0], [15, -15], [0, -3], [-1, -1], [0, -3], [-7, -7], [-10, 5], [-19, 19], [-2, 1], [20, 0], [-9, 9], [14, -7], [-10, 5], [-1, -1], [0, 17], [0, 0], [0, -11], [0, 9], [-6, 0], [9, 9], [0, 0], [-10, 20], [-2, 0], [-8, 16], [5, 5], [0, 5], [-3, -3], [0, 21], [-5, -5], [6, -3], [-2, 1], [-26, 13], [-12, 0], [-3, 3], [-26, 13], [-15, 15], [14, -7], [0, 1], [11, 11], [2, -4], [0, 9], [-14, 28], [-24, 0], [12, -24], [0, 0], [-8, 0], [0, -19], [2, -4], [7, 7], [0, 21], [-26, 13], [6, -3], [37, -37], [-20, 0], [-9, 9], [-42, 21], [-10, 5], [19, 19], [-4, 8], [0, 33], [-11, 11], [9, 9], [8, -16], [-18, 36], [18, -36], [20, -40], [-26, 0], [-15, 15], [0, 1], [0, -3], [0, -27], [23, -23], [30, -15], [-25, 25], [24, 0], [33, -33], [45, -45], [22, -11], [0, 13], [19, -19], [7, 7], [36, 0], [4, -8], [32, 0], [23, 23], [-21, 21], [0, -39], [14, -28], [-17, -17], [9, 9], [-47, 47], [6, -3], [22, -11], [8, 0], [14, -7], [14, -7], [3, -3], [23, 23], [-7, -7], [24, -48], [29, -29], [18, 0], [5, 5], [2, -4], [20, -40], [10, 0], [0, 9], [-7, -7], [-25, -25], [7, -7], [14, -7], [-37, 37], [10, 0], [45, -45], [-18, 9], [-51, 51], [-22, 44], [0, 45], [-1, -1], [-6, 12], [4, -8], [-16, 0], [-14, 28], [-21, -21], [39, -39], [15, 15], [0, -27], [29, 29], [-50, 25], [-23, 23], [54, 0], [4, 0], [15, -15], [0, -19], [-3, -3], [0, 29], [-30, 0], [8, -16], [18, 0], [-30, 60], [12, -24], [0, 25], [0, 57], [24, -48], [0, -39], [47, -47], [-10, 5], [-34, 0], [-51, 51], [-18, 9], [6, -3], [-3, -3], [0, -47], [-26, 52], [0, -27], [-41, 41], [23, 23], [-10, 20], [16, -32], [-16, 0], [-33, 33], [0, -59], [0, -27], [14, -28], [0, -3], [-7, 7], [-66, 33], [13, -13], [-21, 21], [30, -15], [-23, -23], [0, -55], [-6, 0], [-15, -15], [-6, 0], [-26, 0], [26, 0], [-15, 15], [0, 45], [-24, 48], [5, 5], [-17, -17], [38, -19], [12, 0], [33, -33], [-50, 25], [25, 25], [31, 31], [22, -44], [0, -11], [0, 69], [-49, 49], [-12, 0], [-24, 0], [-4, 8], [28, 0], [0, -3], [-2, 1], [-50, 25], [41, -41], [-2, 1], [-9, -9], [0, 41], [0, -3], [5, -5], [54, 0], [21, 21], [-18, 36], [-14, 28], [-50, 0], [21, 21], [0, -43], [4, -8], [37, 37], [0, 9], [39, 39], [54, -27], [-58, 29], [-48, 0], [6, -3], [54, -27], [6, -12], [0, -43], [0, -39], [36, 0], [10, -20], [36, 0], [44, 0], [-6, 12], [0, 77], [0, 45], [-18, 36], [11, 11], [0, -15], [-18, 9], [61, -61], [-50, 25], [-54, 0], [-42, 21], [0, 25], [5, 5], [-8, 16], [-32, 64], [-66, 0], [34, -68], [-32, 64], [-3, 3], [0, 13], [16, -32], [3, 3], [-25, 25], [14, -7], [-13, 13], [-90, 45], [-2, 1], [-19, -19], [0, -19], [0, 29], [-72, 0], [72, 0], [-4, 0], [39, -39], [7, 7], [37, 37], [46, -23], [-19, 19], [62, -31], [14, -7], [54, 0], [-10, 5], [-9, 9], [11, 11], [0, -3], [-71, 71], [-30, 0], [10, -20], [-63, 63], [0, 9], [-35, -35], [-25, -25], [-18, 9], [31, -31]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1323_i_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1323_2_i_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_1323_2_i_a(:prec:=2)
    chi := MakeCharacter_1323_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(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_i_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_1323_2_i_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1323_i();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![3, 0, 1]>],Snew);
    return Vf;
end function;