
// Make newform 2016.2.a.l in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_2016_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_2016_2_a_l();".
// 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_2_a_l();".
// 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_2016_a();"
function MakeCharacter_2016_a()
    N := 2016;
    order := 1;
    char_gens := [127, 1765, 1793, 577];
    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_2016_a_Hecke(Kf)
    return MakeCharacter_2016_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 := 2;
    raw_aps := [[0], [0], [2], [1], [-4], [-6], [2], [-4], [-4], [2], [-8], [-10], [2], [-8], [0], [10], [-12], [10], [8], [12], [2], [0], [12], [-6], [2], [10], [8], [-12], [-2], [14], [-8], [-20], [-18], [-20], [-6], [16], [2], [-16], [0], [-6], [-20], [2], [12], [-14], [-6], [-16], [16], [8], [20], [18], [14], [12], [-30], [-4], [18], [-12], [2], [32], [-18], [30], [12], [-6], [-4], [8], [-14], [2], [8], [-14], [-28], [18], [-6], [-12], [24], [-34], [8], [-8], [-38], [34], [-10], [-22], [12], [-34], [36], [-6], [-8], [28], [-18], [-6], [26], [-16], [36], [0], [32], [-36], [16], [0], [18], [-38], [-28], [-26], [32], [-30], [-20], [-18], [-16], [2], [12], [42], [-36], [10], [-8], [-10], [-18], [20], [0], [-26], [-28], [-16], [-14], [44], [18], [50], [18], [28], [-12], [-30], [38], [8], [40], [-14], [-32], [-28], [-16], [-26], [-30], [26], [42], [4], [42], [-10], [4], [10], [-32], [4], [-14], [-16], [42], [18], [-12], [52], [14], [42], [-16], [-16], [-48], [-12], [-8], [-22], [10], [-6], [44], [54], [-8], [28], [22], [24], [-8], [-62], [2], [-30], [36], [-30], [-20], [-38], [-32], [2], [-40], [-6], [-32], [-6], [-8], [-12], [38], [2], [-20], [-30], [-10], [4], [-6], [-8], [-22], [-12], [0], [42], [-44], [-14], [-14], [-2], [2], [-56], [-30], [0], [10], [10], [12], [66], [-64], [-36], [6], [36], [-62], [2], [-16], [12], [-40], [-46], [-56], [-46], [20], [34], [6], [-32], [-66], [16], [28], [-26], [-6], [12], [32], [-60], [14], [-28], [24], [54], [-4], [0], [34], [26], [12], [16], [-28], [-60], [72], [-26], [18], [48], [48], [52], [16], [36], [22], [-38], [28], [-62], [18], [-12], [-26], [-28], [-6], [-14], [40], [-12], [10], [-22], [-78], [28], [34], [-30], [8], [-38], [-14], [72], [34], [-80], [26], [64], [12], [-26], [-6], [-28], [-48], [80], [56], [2], [52], [-44], [34], [-6], [-40], [2], [66], [60], [-34], [-60], [-26], [74], [-24], [58], [-20], [-36], [-46], [18], [-32], [60], [32], [-14], [36], [-30], [-76], [46], [0], [-54], [54], [40], [4], [-30], [-12], [20], [-6], [-10], [36], [26], [50], [-24], [54], [-14], [-16], [28], [-12], [-38], [-10], [-62], [-40], [-84], [-56], [-28], [14], [-30], [66], [8], [78], [-66], [42], [-8], [-30], [60], [34], [88], [32], [-54], [-60], [58], [50], [0], [46], [2], [-24], [44], [30], [-44], [14], [-70], [4], [-60], [-12], [90], [-46], [56], [-70], [-84], [20], [60], [-22], [-80], [38], [76], [-36], [-14], [82], [42], [-70], [-74], [72], [-18], [-68], [48], [56], [-22], [-40], [-48], [18], [-62], [-12], [68], [-60], [26], [0], [42], [8], [42], [26], [-18], [-64], [-78], [74], [0], [38], [14], [-20], [-20], [34], [10], [-44], [80], [-38], [66], [-12], [-8], [-62], [-24], [2], [-78], [-36], [-12], [82], [-94], [4], [-18], [12], [-16]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_2016_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_2016_2_a_l();".
// 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_2_a_l(:prec:=1)
    chi := MakeCharacter_2016_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_2016_2_a_l();".
// 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_2_a_l( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2016_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![-2, 1]>,<11,R![4, 1]>,<13,R![6, 1]>,<17,R![-2, 1]>,<19,R![4, 1]>],Snew);
    return Vf;
end function;