
// Make newform 8002.2.a.c in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_8002_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_8002_2_a_c();".
// 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_8002_2_a_c();".
// 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_8002_a();"
function MakeCharacter_8002_a()
    N := 8002;
    order := 1;
    char_gens := [3];
    v := [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_8002_a_Hecke(Kf)
    return MakeCharacter_8002_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], [2], [-2], [0], [-6], [2], [-2], [8], [4], [-6], [0], [0], [-2], [-6], [0], [-4], [0], [-6], [2], [-8], [10], [-16], [-12], [-2], [2], [-12], [16], [-12], [0], [2], [0], [14], [18], [-6], [6], [-24], [-22], [14], [-8], [6], [-18], [8], [-12], [6], [-18], [-20], [-16], [24], [18], [-6], [-14], [0], [2], [-2], [-14], [20], [6], [28], [-14], [2], [6], [0], [-28], [28], [14], [12], [-22], [14], [-16], [4], [-6], [-12], [0], [32], [8], [-16], [-6], [30], [-18], [38], [26], [-8], [-24], [34], [16], [4], [-2], [-22], [0], [16], [-6], [-24], [-4], [-30], [4], [8], [-18], [6], [34], [22], [-22], [-38], [18], [-6], [12], [-10], [38], [18], [-12], [-22], [40], [22], [-6], [16], [40], [-2], [-4], [24], [32], [-10], [24], [-14], [-4], [14], [-10], [34], [24], [0], [-4], [44], [-6], [-20], [-40], [-8], [6], [30], [-16], [-36], [-42], [-2], [-42], [30], [-24], [-10], [2], [0], [-30], [22], [30], [8], [8], [26], [-18], [-32], [34], [24], [32], [-18], [-42], [0], [-18], [6], [40], [-16], [-42], [-48], [20], [22], [-50], [6], [-30], [-2], [24], [26], [-24], [-30], [-28], [12], [4], [-24], [-8], [-18], [-16], [50], [20], [0], [-56], [-6], [-50], [12], [34], [6], [-42], [-36], [-60], [66], [-30], [34], [-22], [48], [40], [-32], [48], [-6], [30], [18], [-16], [42], [-42], [-10], [-50], [24], [16], [54], [32], [6], [4], [42], [-16], [48], [-40], [-8], [-54], [-32], [-14], [-46], [66], [-12], [64], [-60], [34], [-34], [-64], [58], [-42], [48], [-18], [-34], [-26], [-28], [-66], [-16], [-56], [-34], [-46], [-60], [32], [-42], [-16], [48], [8], [-6], [-24], [-38], [14], [34], [-28], [8], [56], [-2], [36], [-10], [-66], [42], [-38], [54], [-46], [-54], [36], [-54], [64], [-14], [10], [-8], [50], [-64], [0], [-46], [10], [24], [72], [24], [-16], [-46], [-26], [12], [-46], [-6], [-16], [-34], [0], [-48], [-18], [30], [28], [28], [-20], [-24], [38], [58], [-34], [46], [80], [24], [50], [78], [-38], [12], [36], [-8], [-40], [-54], [22], [68], [48], [34], [-4], [-36], [-62], [-34], [32], [14], [12], [56], [54], [-66], [-68], [-4], [72], [0], [-22], [-82], [-40], [-60], [-20], [66], [18], [-14], [-70], [48], [-46], [86], [42], [88], [66], [-38], [78], [-34], [80], [-68], [-66], [-46], [-18], [28], [2], [-26], [80], [-96], [14], [-8], [34], [50], [-16], [-34], [52], [-78], [-32], [0], [-50], [-90], [-36], [-48], [60], [-16], [2], [20], [80], [34], [-18], [82], [-46], [-54], [-72], [50], [-22], [24], [48], [-28], [70], [-12], [74], [-82], [54], [-4], [-64], [86], [20], [-30], [-52], [-94], [-48], [-74], [80], [-74], [-34], [0], [40], [30], [-26], [78], [-22], [24], [-36], [-82], [58], [-72], [-24], [-88], [62], [88], [16], [-68], [-32], [-20], [-34], [82], [-58], [10], [74], [24]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8002_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8002_2_a_c();".
// 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_8002_2_a_c(:prec:=1)
    chi := MakeCharacter_8002_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_8002_2_a_c();".
// 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_8002_2_a_c( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8002_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![-2, 1]>],Snew);
    return Vf;
end function;