
// Make newform 162.4.c.c in Magma, downloaded from the LMFDB on 20 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_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_162_4_c_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, -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, 6], [16, -16], [12, -12], [0, -38], [126, 0], [20, 0], [0, 168], [30, -30], [0, 88], [254, 0], [0, 42], [52, -52], [-96, 96], [-198, 0], [0, -660], [538, -538], [0, -884], [-792, 0], [218, 0], [520, -520], [-492, 492], [-810, 0], [-1154, 1154], [-618, 618], [0, -128], [1476, 0], [1190, 0], [0, -462], [-2536, 0], [0, 2292], [-726, 726], [0, -380], [0, 1590], [-2432, 2432], [0, -614], [-1852, 0], [0, -2136], [1758, -1758], [540, 0], [1982, 0], [-2688, 2688], [0, 2302], [-4374, 0], [-1600, 0], [0, -3332], [-2648, 2648], [2244, -2244], [0, 5650], [-4698, 0], [0, -1200], [718, -718], [-6012, 0], [0, -2046], [-6072, 6072], [6930, 0], [1352, 0], [1186, -1186], [2442, -2442], [0, -2828], [0, 4758], [-8476, 0], [0, 4632], [4822, -4822], [-3426, 3426], [2788, -2788], [0, -434], [0, 6684], [-2630, 2630], [-7422, 7422], [10440, 0], [-10424, 10424], [0, -3278], [6140, 0], [0, -3072], [6150, -6150], [-106, 0], [0, -1758], [0, 3670], [0, -9660], [-8462, 8462], [-9792, 0], [-7342, 0], [-10640, 10640], [-17412, 17412], [1710, 0], [646, -646], [-6018, 6018], [0, 6712], [-5364, 0], [9840, -9840], [1424, 0], [0, -4548], [0, -6500], [-12168, 0], [0, -21090], [5238, 0], [8588, 0], [3062, 0], [8476, -8476], [12546, 0], [0, -12], [19290, -19290], [0, 12148], [-10366, 0], [7644, -7644], [-8658, 0], [0, 25800], [-16202, 16202], [0, 24136], [-4642, 0], [0, -6726], [21220, -21220], [29792, 0], [-10158, 10158], [0, -29828], [-1944, 0], [0, 26718], [4260, -4260], [0, -22862], [32542, -32542], [14214, -14214], [7092, 0], [13228, -13228], [-28062, 0], [27250, -27250], [14400, 0], [-17984, 17984], [0, -16598], [1460, 0], [0, -30072], [0, 18088], [24734, 0], [0, -22278], [0, -16130], [-29718, 0], [0, -9524], [0, -33906], [630, 0], [-20788, 0], [-43098, 43098], [0, 14272], [-13644, 0], [-2410, 0], [23160, -23160], [-32078, 32078], [-14406, 14406], [0, -30620], [-17568, 0], [0, 21706], [14958, 0], [-32812, 0], [0, -38856], [28276, -28276], [8112, -8112], [-26080, 0], [49170, -49170], [48314, 0], [0, 34782], [-25116, 25116], [15462, 0], [0, 736], [29268, 0], [0, 16674], [-31272, 31272], [-15928, 0], [-42014, 42014]];
    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_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_162_4_c_c(: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_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_162_4_c_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_162_c();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<5,R![36, -6, 1]>],Snew);
    return Vf;
end function;