
// Make newform 8016.2.a.e in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_8016_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_8016_2_a_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_8016_2_a_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
        Kf := Rationals();
    end if;
    return [Kf!elt[1] : elt in input];
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_8016_a();"
function MakeCharacter_8016_a()
    N := 8016;
    order := 1;
    char_gens := [3007, 2005, 5345, 673];
    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_8016_a_Hecke(Kf)
    return MakeCharacter_8016_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], [-1], [0], [0], [0], [0], [2], [8], [4], [-10], [-8], [-8], [2], [-2], [0], [-12], [6], [14], [6], [-12], [2], [4], [6], [10], [-2], [12], [-4], [4], [-8], [6], [8], [-6], [-6], [14], [-12], [-16], [-10], [10], [-1], [6], [-12], [-10], [24], [-26], [0], [-8], [-8], [-16], [-22], [-6], [6], [16], [-10], [-24], [22], [-24], [-16], [-8], [-16], [-26], [8], [14], [18], [16], [22], [6], [-2], [-14], [-26], [12], [-34], [-32], [-16], [4], [34], [0], [24], [34], [-30], [-10], [12], [-34], [0], [2], [4], [26], [2], [-26], [18], [-40], [-4], [-16], [-20], [24], [-6], [24], [18], [42], [-16], [-20], [-14], [10], [-24], [2], [-22], [-42], [2], [-2], [-32], [-34], [0], [-30], [22], [-6], [0], [2], [42], [-36], [46], [-34], [-40], [-50], [-34], [18], [30], [-38], [-16], [-12], [28], [42], [-10], [40], [40], [30], [14], [54], [32], [-34], [-12], [42], [-38], [8], [-16], [30], [4], [-16], [-38], [-34], [-4], [8], [42], [30], [-24], [48], [28], [-48], [-16], [-10], [-54], [24], [8], [-22], [-32], [6], [42], [48], [16], [-10], [46], [18], [-42], [10], [8], [-38], [-60], [-58], [8], [-4], [16], [-20], [-48], [-12], [12], [6], [-44], [50], [2], [-12], [26], [36], [34], [46], [12], [42], [40], [2], [-2], [-50], [18], [-8], [-36], [-56], [-8], [-30], [42], [62], [-36], [16], [-42], [52], [-46], [-62], [-8], [-58], [-40], [-6], [40], [-30], [32], [24], [-52], [-56], [42], [8], [22], [14], [18], [-40], [56], [12], [-64], [-6], [-56], [46], [48], [44], [46], [22], [-14], [24], [30], [12], [4], [-12], [-18], [8], [-72], [50], [24], [-12], [38], [-34], [64], [50], [-52], [-72], [56], [28], [8], [10], [44], [-22], [-52], [-32], [-62], [-8], [-40], [-18], [22], [54], [-32], [8], [26], [48], [-30], [-36], [6], [-64], [2], [36], [-12], [16], [60], [30], [66], [-40], [-70], [-12], [-56], [-50], [-6], [22], [26], [16], [-6], [34], [-56], [0], [-50], [72], [-34], [20], [16], [6], [68], [-34], [50], [-14], [48], [22], [-4], [-66], [-18], [-62], [88], [54], [-6], [-24], [58], [-42], [60], [6], [2], [8], [-30], [14], [40], [-64], [-16], [42], [-54], [-62], [56], [84], [-2], [48], [90], [-18], [82], [40], [14], [42], [-12], [-16], [-74], [-16], [-10], [12], [56], [-62], [-40], [78], [-24], [-80], [-84], [30], [24], [14], [-30], [72], [14], [-10], [-12], [60], [-42], [50], [56], [24], [-94], [-42], [22], [88], [-54], [40], [4], [2], [52], [-30], [26], [-74], [66], [54], [96], [14], [-16], [-36], [-8], [4], [-16], [40], [82], [-42], [0], [58], [-40], [38], [-88], [26], [102], [-28], [86], [-74], [84], [90], [-40], [-64], [8], [-30], [-6], [-56], [-22], [-16], [-104], [68], [42], [18], [-100], [-24], [-42], [48], [-16], [84], [-8], [24], [42], [12], [0], [42], [-16], [0]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8016_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8016_2_a_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_8016_2_a_e(:prec:=1)
    chi := MakeCharacter_8016_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_8016_2_a_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_8016_2_a_e( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8016_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![0, 1]>,<7,R![0, 1]>,<11,R![0, 1]>,<13,R![0, 1]>],Snew);
    return Vf;
end function;