
// Make newform 4160.2.a.f in Magma, downloaded from the LMFDB on 08 June 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_4160_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_4160_2_a_f();".
// 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_4160_2_a_f();".
// 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_4160_a();"
function MakeCharacter_4160_a()
    N := 4160;
    order := 1;
    char_gens := [4031, 261, 2497, 1601];
    v := [1, 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_4160_a_Hecke( : Kf := Kf)
    return MakeCharacter_4160_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]];
    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 := [[0], [-2], [1], [4], [2], [1], [2], [-6], [6], [-2], [10], [2], [-6], [10], [-4], [-2], [6], [-2], [-4], [-6], [-6], [12], [-16], [2], [-2], [18], [-2], [10], [-10], [-14], [2], [20], [-2], [0], [-18], [-10], [6], [-12], [12], [6], [12], [22], [0], [-2], [6], [16], [12], [-4], [4], [22], [10], [6], [10], [24], [2], [-14], [-6], [-2], [14], [-6], [2], [-22], [8], [4], [-22], [18], [-18], [26], [-22], [30], [18], [10], [14], [-34], [10], [-12], [10], [-6], [10], [18], [-40], [-10], [-14], [10], [0], [14], [-6], [38], [-10], [24], [-10], [-30], [40], [-24], [10], [18], [6], [6], [6], [22], [-6], [26], [-22], [-26], [24], [2], [-44], [14], [4], [10], [-34], [6], [-42], [-2], [14], [-46], [-16], [-38], [6], [0], [-2], [-30], [6], [-20], [22], [26], [-10], [32], [18], [-34], [6], [12], [28], [-2], [18], [26], [2], [8], [-42], [-2], [-42], [-50], [46], [-32], [-34], [-38], [-6], [18], [40], [-12], [-18], [38], [-22], [-58], [-34], [48], [52], [-38], [-30], [-18], [24], [18], [8], [-28], [18], [-56], [-48], [22], [-22], [14], [-14], [22], [-20], [-38], [56], [58], [10], [-50], [-42], [10], [-52], [-8], [-26], [22], [24], [-42], [6], [56], [-62], [46], [-14], [24], [68], [54], [-42], [-38], [-22], [46], [18], [-30], [-50], [32], [-10], [-10], [54], [-2], [46], [46], [-54], [8], [-6], [-10], [26], [-48], [-42], [-14], [-26], [-46], [16], [-34], [38], [-70], [-46], [-20], [-58], [22], [58], [-36], [42], [10], [-26], [-8], [-62], [-54], [-2], [-48], [-14], [-22], [36], [24], [-56], [8], [66], [-34], [54], [56], [20], [10], [18], [6], [-70], [-46], [36], [10], [54], [30], [-42], [-12], [-34], [42], [-66], [18], [6], [-2], [66], [-16], [-50], [-30], [-36], [-18], [58], [52], [-26], [16], [10], [12], [-68], [38], [-30], [-24], [10], [-18], [-18], [78], [64], [52], [-30], [-74], [66], [-34], [-18], [-38], [46], [-30], [-74], [2], [-24], [-2], [72], [-64], [-38], [-34], [-40], [-42], [20], [-30], [30], [-22], [74], [86], [-6], [-18], [78], [-50], [-12], [-50], [34], [-30], [-30], [-78], [-8], [-6], [-66], [64], [-70], [78], [78], [8], [-78], [-74], [14], [-18], [84], [60], [78], [-36], [-58], [-50], [10], [50], [-10], [-78], [6], [-68], [78], [-68], [-54], [32], [-42], [-34], [62], [-82], [-2], [-2], [-22], [18], [-14], [6], [-78], [84], [42], [78], [62], [-6], [-42], [74], [-34], [24], [-86], [20], [-44], [48], [86], [56], [30], [-78], [-28], [70], [54], [26], [78], [58], [28], [18], [-58], [40], [2], [-18], [-32], [-66], [-22], [50], [-90], [-70], [6], [-70], [-34], [54], [68], [54], [14], [2], [-36], [42], [-18], [8], [2], [42], [-12], [6], [-54], [54], [6], [8], [2], [18], [-66], [62], [34], [46], [-66], [70], [-88], [-16], [-50], [-78], [10], [50], [-22], [28]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4160_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4160_2_a_f();".
// 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_4160_2_a_f(:prec:=1)
    chi := MakeCharacter_4160_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_4160_2_a_f();".
// 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_4160_2_a_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4160_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![2, 1]>,<7,R![-4, 1]>,<11,R![-2, 1]>,<17,R![-2, 1]>,<19,R![6, 1]>],Snew);
    return Vf;
end function;