
// Make newform 162.4.c.h in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_162_c();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_162_c_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_162_4_c_h();".
// 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_162_4_c_h();".
// 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_162_c();"
function MakeCharacter_162_c()
    N := 162;
    order := 3;
    char_gens := [83];
    v := [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_162_c_Hecke();"
function MakeCharacter_162_c_Hecke(Kf)
    N := 162;
    order := 3;
    char_gens := [83];
    char_values := [[0, -1]];
    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 := 4;
    raw_aps := [[2, -2], [0, 0], [0, 12], [7, -7], [60, -60], [0, 79], [108, 0], [11, 0], [0, -132], [96, -96], [0, -20], [-169, 0], [0, 192], [-488, 488], [204, -204], [-360, 0], [0, 156], [-83, 83], [0, -47], [-216, 0], [-511, 0], [529, -529], [-1128, 1128], [-36, 0], [-605, 605], [1248, -1248], [0, -965], [-1332, 0], [-1942, 0], [0, 516], [-52, 0], [0, 48], [2364, -2364], [0, -173], [0, -1608], [997, -997], [0, -614], [2693, 0], [0, 1164], [-3648, 3648], [-1800, 0], [-547, 0], [3156, -3156], [0, -1127], [-1116, 0], [-3283, 0], [0, 295], [2644, -2644], [-6024, 6024], [0, 4462], [1008, 0], [0, -5064], [-6257, 6257], [2160, 0], [0, -168], [-2424, 2424], [396, 0], [1811, 0], [3022, -3022], [-3072, 3072], [0, 8080], [0, -4668], [6752, 0], [0, 1812], [-6203, 6203], [10968, -10968], [-8165, 8165], [0, 6523], [0, 6168], [6001, -6001], [-552, 552], [5004, 0], [4291, -4291], [0, 2833], [-5137, 0], [0, 5088], [7980, -7980], [-1834, 0], [0, -1464], [0, 151], [0, -11508], [6271, -6271], [-9468, 0], [3026, 0], [-11180, 11180], [-840, 840], [-12780, 0], [15658, -15658], [-15564, 15564], [0, 4183], [-13932, 0], [-2712, 2712], [-9439, 0], [0, -11724], [0, 11968], [-8892, 0], [0, -6756], [6228, 0], [11639, 0], [17705, 0], [-3485, 3485], [19116, 0], [0, 22368], [8340, -8340], [0, 14677], [-10069, 0], [1572, -1572], [-1368, 0], [0, -23712], [11014, -11014], [0, 6415], [15851, 0], [0, 5772], [27781, -27781], [-29869, 0], [3480, -3480], [0, 7432], [12960, 0], [0, 23592], [32280, -32280], [0, -22619], [19861, -19861], [-8292, 8292], [-19728, 0], [-27272, 27272], [21996, 0], [-2009, 2009], [-26280, 0], [-20900, 20900], [0, 17638], [30080, 0], [0, 15708], [0, -20423], [-4399, 0], [0, -20652], [0, -27407], [-5976, 0], [0, -24131], [0, 912], [12888, 0], [-6856, 0], [636, -636], [0, -39827], [-38124, 0], [18965, 0], [27816, -27816], [12337, -12337], [-11352, 11352], [0, 2527], [26388, 0], [0, -6383], [-28908, 0], [36893, 0], [0, -45528], [31201, -31201], [23856, -23856], [23492, 0], [-15096, 15096], [-14965, 0], [0, 19524], [11076, -11076], [44748, 0], [0, -41519], [-28404, 0], [0, 43032], [18132, -18132], [-44467, 0], [-19550, 19550]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_162_c_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_162_4_c_h();".
// 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_162_4_c_h(:prec:=2)
    chi := MakeCharacter_162_c();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 4));
    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_162_4_c_h();".
// 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_162_4_c_h( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_162_c();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<5,R![144, -12, 1]>],Snew);
    return Vf;
end function;