
// Make newform 1323.2.a.n in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_1323_a();"
// 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_a_n();".
// 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_a_n();".
// 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
        Kf := Rationals();
    end if;
    return [Kf!elt[1] : elt in input];
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_1323_a();"
function MakeCharacter_1323_a()
    N := 1323;
    order := 1;
    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;

function MakeCharacter_1323_a_Hecke(Kf)
    return MakeCharacter_1323_a();
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], [-3], [0], [-5], [6], [6], [3], [-1], [2], [-3], [3], [9], [6], [-6], [8], [-6], [12], [-14], [7], [6], [-10], [-6], [-3], [12], [-6], [-3], [16], [9], [-16], [-2], [18], [-10], [12], [4], [12], [18], [-2], [0], [-9], [4], [-12], [17], [-18], [-8], [9], [-12], [-9], [24], [0], [4], [16], [18], [-12], [-3], [23], [21], [-24], [-1], [-26], [12], [6], [3], [18], [-6], [2], [0], [3], [-5], [12], [-21], [8], [9], [-23], [-30], [-24], [16], [12], [2], [30], [12], [-17], [-31], [-18], [-24], [25], [20], [-15], [-33], [-4], [42], [12], [30], [-1], [18], [-6], [6], [3], [3], [11], [8], [28], [-18], [4], [40], [24], [0], [-21], [5], [-30], [-12], [7], [10], [-15], [-2], [-8], [39], [-42], [-22], [-37], [-42], [-6], [51], [-7], [36], [38], [21], [6], [36], [42], [-16], [-49], [6], [-18], [6], [-18], [-9], [12], [-21], [-2], [-39], [22], [-4], [-19], [-18], [30], [-12], [9], [-45], [8], [14], [15], [-48], [6], [-48], [52], [-48], [-18], [0], [45], [-47], [-14], [32], [54], [-8], [-6], [-6], [36], [29], [-9], [20], [-54], [28], [-37], [-27], [39], [8], [16], [-33], [-24], [28], [0], [49], [42], [41], [39], [-30], [-24], [-41], [12], [-6], [4], [0], [-30], [-5], [-57], [33], [-26], [-27], [-36], [-62], [-21], [30], [36], [-54], [18], [-48], [-11], [-50], [-12], [-43], [33], [54], [-42], [24], [-48], [-42], [-57], [-17], [56], [-6], [-48], [10], [42], [48], [63], [-27], [-67], [9], [-13], [-27], [-63], [18], [-16], [27], [18], [-72], [-50], [76], [30], [-61], [45], [-45], [69], [-51], [18], [57], [6], [-14], [41], [-78], [-42], [-8], [12], [15], [-67], [-43], [33], [-30], [-42], [46], [53], [6], [18], [27], [9], [-46], [54], [4], [34], [78], [-12], [6], [42], [30], [-21], [77], [6], [-42], [-6], [78], [-40], [0], [42], [27], [-53], [9], [-58], [39], [-27], [28], [54], [-50], [48], [-19], [-45], [15], [54], [-6], [-21], [48], [-58], [52], [-59], [68], [39], [49], [66], [71], [45], [-12], [2], [20], [-66], [25], [36], [-30], [-65], [36], [26], [-72], [-27], [-81], [-54], [-20], [12], [-52], [-15], [-8], [4], [51], [68], [-15], [-18], [-70], [12], [47], [-27], [-90], [39], [53], [14], [-90], [30], [10], [76], [-6], [-18], [-66], [66], [21], [2], [-64], [51], [13], [-81], [30], [24], [14], [-4], [-26], [-15], [-8], [47], [60], [81], [69], [52], [63], [-59], [-69], [-64], [-22], [75], [-25], [-54], [-56], [96], [15], [-18], [-21], [-32], [30], [-22], [-57], [-6], [48], [12], [56], [84], [-10], [45], [-43], [-96], [40], [3], [33], [3], [22], [-34], [54], [-98], [76], [69], [-39], [15], [55], [38], [-9], [84], [0], [-74], [-44], [0], [14], [-87], [49], [48], [18], [48], [-76], [12], [41], [-102], [-96], [90], [-52], [28], [33], [-90]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_1323_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_1323_2_a_n();".
// 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_a_n(:prec:=1)
    chi := MakeCharacter_1323_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(2999) - 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_a_n();".
// 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_a_n( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_1323_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-1, 1]>,<5,R![3, 1]>,<13,R![-6, 1]>],Snew);
    return Vf;
end function;