
// Make newform 2016.3.l.e in Magma, downloaded from the LMFDB on 21 April 2026.
// To make the character of type GrpDrchElt, type "MakeCharacter_2016_l();"
// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_2016_l_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_2016_3_l_e();".
// 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_2016_3_l_e();".
// 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 := [4, 0, -3, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 3, 0, -3], [0, 40, 0, -8], [-72, 0, 48, 0]];
    Rf_basisdens := [1, 1, 1, 1];
    Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf);
    Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]];
    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_2016_l();"
function MakeCharacter_2016_l()
    N := 2016;
    order := 2;
    char_gens := [127, 1765, 1793, 577];
    v := [2, 1, 2, 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;

// To make the character of type GrpDrchElt with Codomain the HeckeField, type "MakeCharacter_2016_l_Hecke();"
function MakeCharacter_2016_l_Hecke( : Kf := false)
    N := 2016;
    order := 2;
    char_gens := [127, 1765, 1793, 577];
    char_values := [[1, 0, 0, 0], [-1, 0, 0, 0], [1, 0, 0, 0], [-1, 0, 0, 0]];
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    values := ConvertToHeckeField(char_values : pass_field := ISA(Type(Kf), Fld), 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]];
    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 := 3;
    raw_aps := [[0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [7, 0, 0, 0], [0, 1, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, -1, 0], [0, 9, 0, 0], [0, 0, 0, 0], [0, 0, 0, -1], [0, 0, 0, 0], [0, 0, 0, 1], [0, 0, 0, 0], [0, 1, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 1], [0, 0, -2, 0], [0, 0, 0, 0], [-94, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 31, 0, 0], [0, 0, 0, -3], [0, 0, -1, 0], [-2, 0, 0, 0], [0, 0, 0, 0], [0, 0, -5, 0], [0, 0, 0, 0], [0, 31, 0, 0], [274, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 5], [0, 0, 0, 0], [0, 0, 0, 0], [0, 57, 0, 0], [0, 0, 0, 0], [0, 0, -5, 0], [-62, 0, 0, 0], [0, 47, 0, 0], [0, 0, 0, 0], [0, 0, 0, -5], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, -11, 0], [0, 0, -10, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 4, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 5], [0, 0, 13, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 87, 0, 0], [0, 0, 0, -3], [226, 0, 0, 0], [0, -113, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, -7, 0], [0, 0, 0, 0], [0, 0, 0, -11], [0, 0, 0, -7], [0, 0, 0, 0], [0, -111, 0, 0], [0, 0, 0, 0], [0, 0, 17, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 13], [0, 0, 20, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 81, 0, 0], [0, 0, -2, 0], [-878, 0, 0, 0], [0, 0, 0, 0], [674, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [-398, 0, 0, 0], [0, -159, 0, 0], [0, 0, 0, 15], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, -17], [0, 0, 0, -11], [0, -167, 0, 0], [0, 0, 0, 0], [0, 0, -22, 0], [0, 0, 0, 3], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 28, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, -19], [0, 0, 26, 0], [0, 0, 0, 0], [-1006, 0, 0, 0], [0, 0, 23, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, -199, 0, 0], [0, -103, 0, 0], [0, 0, 0, 0], [446, 0, 0, 0], [0, 0, 0, 0], [0, -223, 0, 0], [0, 0, 0, 0], [0, -233, 0, 0], [0, 0, 0, -5], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 13], [0, 0, 35, 0], [802, 0, 0, 0], [0, 0, 0, -9], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 38, 0], [0, 0, 0, 0], [0, 193, 0, 0], [-622, 0, 0, 0], [0, 47, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, -11, 0], [0, 0, 0, -25], [0, 0, 0, 0], [0, 0, 0, 11], [0, 0, 0, 0], [0, 0, 0, 5], [0, 0, -22, 0], [-466, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 249, 0, 0], [0, 0, -29, 0], [334, 0, 0, 0], [0, 0, 0, 0], [0, 0, 46, 0], [0, 0, 0, 0], [1406, 0, 0, 0], [0, 0, 0, 0], [2014, 0, 0, 0], [0, 0, 0, 0], [0, 111, 0, 0], [0, 0, 0, 0], [0, 0, -8, 0], [-1966, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, -33], [0, -113, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [-1922, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 29], [0, 0, 0, 0], [0, 0, 44, 0], [0, 0, 0, 0], [0, 0, 0, -11], [0, 0, 0, 0], [-1774, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 383, 0, 0], [0, 0, 0, -33], [0, 0, 0, 0], [0, -393, 0, 0], [0, 0, 0, 0], [1954, 0, 0, 0], [0, 0, 0, 31], [0, 0, 0, 0], [0, 0, 0, 0], [0, -57, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 361, 0, 0], [0, 0, -58, 0], [0, 0, 0, 0], [1438, 0, 0, 0], [0, 0, 0, 0], [2578, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [2078, 0, 0, 0], [0, 0, 0, 0], [0, 0, -56, 0], [0, -457, 0, 0], [0, 0, 0, 7], [0, 0, 0, 0], [0, 0, -62, 0], [-542, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, -45], [0, 0, 0, 0], [0, 0, -52, 0], [0, 0, 0, 0], [0, 479, 0, 0], [0, 0, 0, 21], [0, 0, 0, 0], [-2242, 0, 0, 0], [0, 0, 37, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 417, 0, 0], [0, -271, 0, 0], [0, 0, 0, 0], [0, -57, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 13], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, -5], [0, 0, -65, 0], [0, 0, 0, -35], [0, 0, 0, 0], [0, 0, 10, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 423, 0, 0], [0, 0, 0, 37], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [-3074, 0, 0, 0], [0, -233, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 551, 0, 0], [0, 0, 0, 0], [0, 0, 0, 51], [0, -337, 0, 0], [0, 0, 0, 0], [0, 0, 0, 55], [0, 0, 0, 0], [-1214, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, -367, 0, 0], [0, 0, 0, -55], [-3566, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [-2638, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, -88, 0], [3422, 0, 0, 0], [0, -607, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, -281, 0, 0], [0, 0, 0, 0], [0, 0, 44, 0], [0, 0, 0, 0], [0, 0, 0, -41], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, -249, 0, 0], [3746, 0, 0, 0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2016_l_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2016_3_l_e();".
// 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_2016_3_l_e(:prec:=4)
    chi := MakeCharacter_2016_l();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 3));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(1999) - 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_2016_3_l_e();".
// 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_2016_3_l_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2016_l();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,3,sign)));
    Vf := Kernel([<5,R![0, 1]>,<11,R![36, 0, 1]>],Snew);
    return Vf;
end function;