
// Make newform 160.4.a.a in Magma, downloaded from the LMFDB on 20 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_160_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_160_4_a_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_160_4_a_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
        Kf := Rationals();
    end if;
    return [Kf!elt[1] : elt in input];
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_160_a();"
function MakeCharacter_160_a()
    N := 160;
    order := 1;
    char_gens := [31, 101, 97];
    v := [1, 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_160_a_Hecke(Kf)
    return MakeCharacter_160_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 := 4;
    raw_aps := [[0], [-2], [-5], [6], [60], [50], [-30], [40], [178], [166], [20], [10], [-250], [142], [214], [490], [-800], [250], [-774], [100], [-230], [-1320], [982], [874], [-310], [-1498], [1402], [1194], [650], [-1510], [1246], [2660], [2770], [-560], [-2350], [580], [-1310], [1862], [726], [3250], [-1120], [-2842], [3180], [-4670], [-2990], [-4240], [-4060], [-5622], [1554], [1134], [-1710], [4440], [-850], [-660], [-7590], [762], [-150], [-6580], [4530], [6950], [-3882], [1370], [4106], [2220], [-9430], [-6470], [900], [530], [-414], [8614], [-2270], [8080], [2374], [1810], [8120], [-11782], [-4350], [-7470], [11698], [-3650], [-1120], [4850], [-12580], [13130], [8560], [-4258], [2550], [-6710], [-14482], [162], [-15974], [-10760], [-9266], [2860], [7160], [-1398], [7446], [-16438], [-7322], [10878], [16114], [3690], [-2562], [-6050], [8260], [-16870], [-966], [26290], [-11640], [-25450], [16694], [15890], [-1230], [-10840], [-14060], [-17650], [27358], [-6786], [-9030], [-15600], [16850], [-7990], [18690], [19182], [-23380], [11850], [25646], [-30280], [17446], [-16750], [-36560], [-30142], [11860], [37010], [-11718], [4706], [-28670], [20434], [3930], [-4854], [-13140], [22050], [14578], [-37054], [-6150], [-8200], [-42990], [32130], [15440], [46938], [-31230], [25550], [4318], [1766], [41906], [25140], [32920], [10150], [28530], [9678], [36986], [-3350], [43774], [8740], [-48310], [2282], [-31580], [-2790]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_160_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_160_4_a_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_160_4_a_a(:prec:=1)
    chi := MakeCharacter_160_a();
    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_160_4_a_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_160_4_a_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_160_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,4,sign)));
    Vf := Kernel([<3,R![2, 1]>],Snew);
    return Vf;
end function;