
// Make newform 2016.2.a.i in Magma, downloaded from the LMFDB on 15 November 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_i();".
// 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_i();".
// 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]];
    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], [0], [1], [-2], [-2], [-4], [4], [-6], [2], [0], [-6], [-8], [8], [-4], [6], [0], [-14], [-4], [-2], [-2], [-4], [12], [0], [6], [-12], [8], [6], [-18], [-18], [-12], [20], [18], [-12], [-10], [-16], [2], [-4], [-12], [12], [10], [14], [-10], [-26], [-2], [-24], [24], [16], [-8], [-2], [26], [26], [22], [0], [8], [-14], [-16], [32], [10], [2], [4], [-28], [28], [-28], [-14], [14], [-16], [26], [-30], [10], [36], [-10], [-8], [14], [16], [-8], [26], [18], [-14], [14], [8], [10], [-18], [2], [24], [-22], [6], [10], [36], [4], [32], [36], [-40], [-6], [-40], [-24], [-16], [12], [28], [-14], [20], [22], [-16], [10], [-20], [10], [24], [4], [38], [18], [40], [22], [-22], [-28], [-20], [-22], [-4], [12], [-46], [6], [-30], [-34], [-12], [-30], [20], [26], [-26], [48], [8], [6], [-36], [6], [16], [-10], [-40], [-46], [12], [-28], [12], [-50], [4], [30], [4], [-42], [-50], [-4], [-30], [40], [4], [-6], [-2], [12], [56], [-12], [16], [14], [-32], [40], [-30], [-56], [14], [-26], [52], [-28], [-54], [-32], [16], [-14], [-34], [-52], [-50], [-30], [-42], [-10], [-8], [-36], [-36], [-14], [24], [26], [28], [-44], [-14], [-12], [-14], [4], [62], [28], [-10], [40], [-34], [22], [32], [-28], [-18], [60], [-2], [26], [48], [28], [-58], [8], [-18], [-50], [-20], [-12], [-24], [-34], [-26], [-28], [18], [40], [8], [-12], [-12], [10], [16], [20], [66], [-46], [-26], [16], [30], [-64], [20], [-6], [-16], [-54], [-64], [-6], [14], [20], [-32], [-18], [12], [-32], [22], [-42], [50], [-24], [74], [36], [32], [-14], [-32], [36], [-40], [-56], [28], [-54], [-2], [20], [38], [14], [-60], [30], [2], [-60], [12], [58], [68], [-6], [-10], [66], [-12], [12], [46], [-44], [-52], [50], [26], [40], [30], [56], [38], [-80], [-10], [-30], [14], [24], [-56], [12], [-48], [-62], [28], [-18], [-46], [38], [40], [-40], [54], [84], [-6], [40], [38], [-20], [-32], [0], [-48], [52], [-78], [18], [20], [70], [60], [74], [-6], [-18], [54], [26], [-72], [74], [26], [-32], [-78], [-2], [-28], [-30], [86], [-22], [60], [22], [-20], [16], [54], [78], [52], [-28], [-42], [-30], [-50], [50], [-56], [24], [44], [32], [14], [56], [38], [-40], [-34], [-14], [-32], [-32], [-62], [-74], [38], [40], [88], [-76], [-92], [-42], [38], [-24], [-50], [12], [16], [84], [14], [-18], [62], [-40], [-82], [-18], [-28], [22], [-56], [-92], [-18], [66], [-12], [-18], [-90], [80], [-94], [-28], [-22], [-38], [76], [-42], [20], [-18], [16], [-42], [-76], [80], [88], [62], [-4], [-60], [-82], [-16], [-26], [-60], [70], [-46], [-8], [-28], [12], [50], [2], [-66], [-68], [24], [-72], [-64], [-62], [-34], [20], [-104], [-94], [-6], [-6], [-32], [-42], [-56], [42], [-32], [-44], [-28], [-94], [50], [22], [-84], [-62], [36], [58], [-30], [20], [-92]];
    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_i();".
// 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_i(: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_i();".
// 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_i( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_2016_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![0, 1]>,<11,R![2, 1]>,<13,R![2, 1]>,<17,R![4, 1]>,<19,R![-4, 1]>],Snew);
    return Vf;
end function;