
// Make newform 4005.2.a.a in Magma, downloaded from the LMFDB on 20 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4005_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_4005_2_a_a();".
// 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_4005_2_a_a();".
// 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_4005_a();"
function MakeCharacter_4005_a()
    N := 4005;
    order := 1;
    char_gens := [3116, 802, 181];
    v := [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_4005_a_Hecke(Kf)
    return MakeCharacter_4005_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 := [[-1], [0], [1], [4], [0], [4], [6], [2], [-8], [2], [10], [4], [10], [8], [8], [-6], [-6], [-2], [-12], [-12], [-14], [8], [0], [1], [-18], [-18], [4], [-4], [-14], [-12], [8], [16], [20], [16], [6], [-2], [-2], [-12], [-8], [22], [16], [-6], [-6], [-20], [12], [-16], [-10], [-8], [12], [-18], [-6], [6], [6], [0], [-6], [-16], [14], [20], [18], [-6], [20], [-16], [12], [-28], [20], [-6], [20], [8], [20], [-26], [16], [-30], [-16], [26], [6], [0], [30], [8], [34], [-34], [6], [-18], [2], [-36], [14], [-20], [-18], [32], [-6], [24], [12], [-24], [16], [30], [-2], [0], [18], [-26], [-4], [-14], [-8], [-44], [-4], [-10], [-46], [0], [12], [-44], [14], [-26], [32], [6], [36], [20], [28], [38], [-28], [16], [-36], [0], [38], [46], [-24], [-24], [8], [34], [22], [-6], [12], [-42], [-10], [-32], [-20], [2], [-30], [26], [28], [-20], [18], [-22], [-44], [30], [-24], [32], [-38], [30], [-4], [-8], [22], [44], [-4], [-46], [-32], [24], [28], [20], [-10], [30], [14], [-18], [4], [-36], [40], [-4], [-26], [0], [-6], [-14], [-42], [26], [-12], [14], [-2], [40], [-22], [-6], [-48], [-18], [-16], [-46], [-28], [-14], [34], [-36], [-16], [2], [-18], [-52], [18], [18], [-26], [4], [-30], [18], [-40], [30], [-50], [24], [36], [48], [-46], [-34], [-38], [22], [-6], [-44], [14], [36], [-62], [44], [-24], [30], [64], [-4], [40], [14], [32], [6], [-32], [48], [38], [20], [18], [56], [28], [-54], [-6], [-54], [32], [10], [-32], [-50], [-28], [-6], [-56], [44], [-54], [6], [50], [-12], [44], [68], [68], [-38], [54], [-18], [72], [-22], [22], [36], [-22], [-50], [-16], [-14], [-66], [-36], [58], [20], [-8], [-62], [-52], [28], [42], [-14], [-32], [76], [78], [30], [36], [26], [-42], [8], [-40], [0], [16], [-20], [8], [-2], [6], [-14], [-44], [-2], [-32], [22], [12], [-40], [78], [-6], [-80], [30], [-30], [-56], [-22], [30], [46], [18], [38], [-72], [4], [-28], [-16], [6], [18], [-44], [20], [48], [52], [-70], [-48], [20], [0], [10], [-30], [-44], [8], [-42], [-10], [52], [24], [-58], [0], [-58], [-30], [-16], [-54], [-34], [26], [-4], [-80], [80], [62], [-72], [-30], [36], [50], [-12], [50], [68], [-18], [4], [-10], [66], [-38], [-62], [-68], [-60], [-90], [48], [34], [-40], [-12], [-52], [18], [64], [6], [-54], [-60], [24], [40], [-24], [22], [-26], [8], [-30], [76], [44], [48], [80], [70], [-52], [64], [-68], [22], [-10], [52], [0], [-48], [-16], [-42], [-18], [90], [64], [-16], [-88], [-76], [4], [-64], [8], [44], [56], [-54], [24], [-50], [80], [-82], [8], [-68], [50], [2], [10], [6], [-36], [-72], [102], [-34], [20], [72], [-30], [-44], [26], [104], [-30], [20], [38], [-78], [46], [6], [56], [-42], [-8], [14], [82], [16], [-16], [-14], [-42], [96], [90], [-4], [42]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4005_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4005_2_a_a();".
// 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_4005_2_a_a(:prec:=1)
    chi := MakeCharacter_4005_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_4005_2_a_a();".
// 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_4005_2_a_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4005_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![1, 1]>,<7,R![-4, 1]>,<11,R![0, 1]>],Snew);
    return Vf;
end function;