
// Make newform 4004.2.a.a in Magma, downloaded from the LMFDB on 18 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4004_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_4004_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_4004_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_4004_a();"
function MakeCharacter_4004_a()
    N := 4004;
    order := 1;
    char_gens := [2003, 3433, 365, 925];
    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_4004_a_Hecke(Kf)
    return MakeCharacter_4004_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], [-2], [0], [1], [-1], [1], [6], [-4], [0], [6], [-4], [2], [-6], [2], [0], [6], [0], [-10], [-4], [0], [14], [-10], [12], [0], [-4], [6], [14], [-18], [8], [-6], [-10], [0], [18], [-4], [12], [8], [14], [20], [0], [-6], [-12], [26], [12], [-4], [12], [2], [26], [8], [12], [-4], [-18], [-24], [26], [6], [6], [6], [6], [8], [14], [-12], [-4], [18], [20], [6], [14], [-6], [20], [-22], [-6], [26], [24], [12], [14], [14], [20], [-36], [6], [20], [-6], [14], [30], [2], [12], [38], [8], [-36], [18], [-40], [42], [-16], [-18], [-24], [-16], [-30], [-28], [-24], [-36], [6], [-16], [8], [38], [12], [0], [-18], [2], [20], [12], [6], [12], [-22], [-4], [-28], [-6], [-28], [-40], [-18], [20], [-6], [18], [-18], [-4], [2], [-42], [-12], [8], [42], [50], [6], [14], [2], [-28], [12], [-16], [-22], [54], [-34], [36], [-28], [-6], [-18], [-52], [24], [-16], [-12], [-34], [36], [-10], [-42], [-46], [0], [-16], [-18], [32], [12], [8], [12], [-10], [0], [38], [-54], [12], [-42], [32], [-18], [18], [24], [20], [-10], [-16], [-42], [-12], [32], [54], [-4], [14], [-30], [20], [-6], [44], [-22], [-16], [24], [38], [-6], [0], [18], [-22], [56], [32], [0], [38], [36], [-4], [-48], [66], [-18], [44], [-10], [-18], [12], [-24], [44], [8], [14], [-48], [18], [56], [18], [48], [26], [38], [18], [-64], [36], [72], [-28], [-22], [-30], [-48], [-6], [-22], [8], [-42], [-16], [36], [-34], [-6], [-72], [-16], [12], [2], [56], [8], [42], [-10], [-24], [20], [-12], [60], [-30], [-60], [-16], [-46], [50], [-54], [-36], [56], [-36], [68], [-54], [32], [-42], [-48], [26], [30], [60], [2], [-28], [-30], [-70], [26], [54], [50], [-34], [-60], [-10], [6], [36], [-4], [-66], [62], [20], [-4], [50], [-22], [68], [12], [-64], [20], [48], [60], [-16], [12], [-10], [-28], [12], [-34], [24], [8], [-78], [18], [54], [24], [36], [-34], [-30], [-10], [-54], [0], [-28], [-10], [36], [74], [36], [-4], [-34], [60], [-46], [-24], [-22], [-12], [-66], [-66], [-64], [72], [50], [0], [-72], [-16], [-54], [80], [50], [-30], [-64], [-12], [2], [-4], [32], [-66], [-42], [80], [-6], [-64], [36], [32], [-12], [-10], [42], [92], [32], [26], [54], [30], [20], [18], [78], [38], [-28], [24], [-78], [-28], [-46], [6], [80], [86], [6], [-24], [-36], [-66], [-24], [-4], [78], [-84], [36], [-34], [86], [54], [20], [98], [84], [-70], [-84], [18], [-40], [2], [24], [-18], [26], [-78], [-22], [-96], [-102], [-4], [24], [-28], [24], [-64], [-82], [32], [-6], [2], [-48], [-36], [-46], [-48], [-46], [8], [-6], [44], [-48], [-22], [30], [68], [-24], [-42], [-100], [74], [36], [-40], [0], [62], [-6], [-48], [38], [-58], [18], [84], [-94], [-36], [24], [-18], [2], [24], [-60], [-16], [60], [-36], [24], [20], [-36]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4004_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4004_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_4004_2_a_a(:prec:=1)
    chi := MakeCharacter_4004_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_4004_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_4004_2_a_a( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4004_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![2, 1]>],Snew);
    return Vf;
end function;