
// Make newform 4002.2.a.h in Magma, downloaded from the LMFDB on 24 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4002_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_4002_2_a_h();".
// 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_4002_2_a_h();".
// 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_4002_a();"
function MakeCharacter_4002_a()
    N := 4002;
    order := 1;
    char_gens := [2669, 3133, 553];
    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_4002_a_Hecke(Kf)
    return MakeCharacter_4002_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], [-1], [-2], [-2], [2], [-2], [0], [4], [1], [-1], [4], [8], [6], [-8], [-8], [-6], [0], [-4], [-8], [-8], [-10], [-10], [-14], [0], [10], [14], [14], [10], [4], [-16], [-16], [0], [-12], [-12], [-18], [20], [-8], [-20], [-8], [18], [-4], [-12], [-8], [-18], [-2], [10], [-12], [4], [10], [8], [10], [24], [-6], [2], [10], [12], [-18], [16], [-30], [20], [-12], [-14], [20], [0], [-10], [-30], [-4], [-18], [4], [-30], [6], [32], [-10], [-4], [4], [20], [30], [22], [-12], [-18], [34], [-20], [-36], [6], [-24], [-12], [30], [10], [14], [24], [10], [24], [24], [-24], [12], [-12], [-30], [0], [0], [30], [-28], [-2], [14], [-24], [24], [-46], [-4], [-30], [16], [-38], [40], [-8], [24], [-20], [6], [40], [40], [32], [-18], [-30], [24], [18], [-18], [-28], [-44], [42], [-40], [24], [-46], [36], [4], [12], [38], [16], [-10], [-26], [2], [48], [30], [26], [-28], [38], [-52], [-6], [-34], [12], [26], [-46], [20], [-24], [-22], [0], [-36], [-48], [-20], [12], [50], [-14], [26], [22], [24], [12], [12], [-42], [32], [-28], [-8], [46], [10], [-42], [38], [-30], [-12], [-34], [12], [64], [-44], [42], [26], [-52], [12], [-14], [-22], [54], [12], [46], [-54], [16], [-18], [0], [-34], [24], [-8], [50], [-2], [-32], [42], [-64], [16], [24], [-18], [68], [-58], [-10], [-30], [-30], [50], [16], [-54], [-4], [62], [18], [38], [-70], [-40], [-18], [12], [10], [40], [66], [-62], [-54], [-46], [-14], [-8], [-6], [24], [-40], [-42], [56], [6], [-4], [-50], [-30], [-4], [28], [42], [-66], [-52], [0], [-18], [-20], [-32], [50], [-54], [0], [44], [-14], [44], [32], [-8], [24], [52], [-26], [-14], [0], [40], [24], [14], [6], [-54], [78], [2], [-52], [46], [-4], [30], [28], [12], [-74], [-6], [16], [-26], [-34], [50], [-28], [12], [-58], [-22], [22], [72], [-10], [0], [-56], [76], [-56], [-70], [-50], [80], [70], [-42], [18], [-26], [-58], [38], [-42], [58], [-42], [-16], [28], [-34], [-50], [50], [36], [52], [22], [-12], [76], [60], [-2], [24], [-86], [-32], [-84], [44], [-10], [-60], [24], [46], [30], [-8], [-10], [-14], [8], [28], [66], [56], [-4], [-40], [54], [22], [-2], [-32], [-56], [76], [84], [-68], [-8], [-10], [-34], [74], [-60], [-70], [62], [-30], [-60], [-22], [-44], [-12], [-70], [12], [74], [94], [-6], [4], [-38], [-4], [-34], [2], [48], [-88], [78], [-24], [-22], [68], [14], [-78], [10], [6], [32], [-20], [-64], [-6], [-90], [-10], [-32], [60], [18], [0], [46], [14], [-48], [32], [6], [-92], [-16], [-32], [74], [-48], [52], [-2], [-90], [52], [44], [36], [-6], [42], [-44], [-32], [-102], [22], [90], [94], [76], [58], [52], [-28], [2], [92], [-36], [82], [-94], [74], [-72], [46], [66], [-32], [-80], [-44], [-40], [86], [72], [24], [-44], [-30], [30], [10], [14], [-84], [-24]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4002_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4002_2_a_h();".
// 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_4002_2_a_h(:prec:=1)
    chi := MakeCharacter_4002_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_4002_2_a_h();".
// 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_4002_2_a_h( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4002_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![2, 1]>,<7,R![2, 1]>,<11,R![-2, 1]>],Snew);
    return Vf;
end function;