
// Make newform 8001.2.a.k in Magma, downloaded from the LMFDB on 23 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_8001_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_8001_2_a_k();".
// 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_8001_2_a_k();".
// 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
        poly := [-6, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rfbasis := [Kf.1^i : i in [0..Degree(Kf)-1]];
    inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf);
    return Eltseq(inp_vec);
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_8001_a();"
function MakeCharacter_8001_a()
    N := 8001;
    order := 1;
    char_gens := [3557, 1144, 7750];
    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_8001_a_Hecke(Kf)
    return MakeCharacter_8001_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, 1], [1, 0], [0, 0], [2, 2], [3, -1], [2, -1], [-6, 0], [3, -2], [8, 0], [-7, 0], [-3, 1], [2, 0], [12, 0], [3, 2], [0, -1], [2, -1], [2, 5], [-6, -1], [-4, -3], [-1, -2], [-6, 0], [0, -4], [5, 3], [0, 0], [-4, 0], [0, 3], [2, 3], [6, 0], [1, 0], [-3, -3], [-12, -4], [-13, -3], [6, -6], [2, -3], [-4, -1], [17, 0], [-6, 7], [-6, 5], [-6, 4], [-7, -7], [6, -3], [8, 1], [-6, 1], [20, 0], [-10, 6], [5, 1], [-12, 0], [5, -1], [3, -2], [-3, 0], [-7, -5], [3, 3], [-6, -7], [6, -7], [-15, 5], [-4, 11], [20, 4], [15, 4], [17, -3], [-24, 1], [23, 1], [0, -7], [23, -1], [24, 2], [-16, 1], [-16, 3], [3, -6], [5, 9], [-12, 2], [-15, 0], [2, 0], [20, 2], [20, 0], [9, -5], [12, -9], [8, 6], [15, 6], [2, -6], [12, -8], [-16, -9], [6, 3], [8, 3], [-7, 3], [-24, -1], [-24, -2], [5, -10], [-12, -7], [-10, -4], [-12, 8], [9, 5], [-4, 11], [18, 2], [-16, -9], [-27, 1], [-15, -9], [18, 0], [-16, 5], [-4, 2], [8, 4], [-12, 4], [0, 14], [6, -10], [-28, 7], [-28, -3], [3, -1], [0, -4], [33, 0], [-13, -3], [2, -3], [14, -10], [-27, 0], [11, -3], [38, -4], [3, 2], [2, -3], [-30, 6], [-24, 1], [18, 2], [-28, -3], [29, -6], [21, -1], [6, -8], [35, 1], [0, -10], [-13, -12], [21, -7], [11, -17], [-22, 3], [-19, -6], [9, -6], [2, -17], [-1, -18], [12, -17], [35, 1], [39, -5], [38, -7], [-18, -4], [6, 7], [20, -10], [3, 4], [-19, 0], [15, -4], [8, -10], [6, -13], [-19, -15], [-18, 6], [-19, 7], [33, -2], [29, 6], [30, 5], [-10, -4], [0, 2], [23, -4], [6, -6], [5, -4], [-12, 14], [-28, -6], [-3, 13], [3, 20], [-6, -3], [-16, -11], [-18, 8], [48, -4], [15, 11], [8, -1], [32, 10], [23, -4], [6, 2], [12, -4], [17, -3], [-18, 1], [32, -8], [11, 21], [18, 12], [-19, 6], [30, 12], [-22, -6], [29, 5], [11, -2], [6, -9], [14, -19], [-18, -14], [-18, 9], [6, -14], [38, 11], [-4, 20], [35, -10], [21, -3], [17, -3], [27, -6], [14, -7], [-27, -11], [6, 23], [-3, -11], [50, -5], [29, 0], [15, -5], [36, 5], [12, -22], [-10, 14], [-16, 18], [-19, 5], [-30, -1], [-36, -11], [-16, -21], [-12, 16], [-60, 1], [-10, 20], [-10, 3], [-21, -11], [-28, 16], [63, 3], [6, 4], [20, -10], [-16, 9], [30, -15], [15, -18], [36, 1], [26, -17], [-4, 3], [39, -12], [-7, -14], [-51, -3], [23, 12], [-6, 20], [18, 13], [38, 0], [27, -2], [26, -2], [-10, 0], [-46, -6], [12, 15], [17, 9], [-12, -11], [-37, -7], [9, 16], [3, 22], [18, 8], [21, 2], [-28, -18], [-34, -15], [-25, 8], [36, 8], [-33, -5], [47, -5], [45, -5], [-40, -5], [3, -8], [47, 4], [-12, -22], [27, -6], [-19, 11], [-36, 2], [60, 4], [-34, 9], [-22, 0], [-9, 23], [-22, -20], [-4, -17], [-30, -5], [26, -12], [26, 0], [0, 20], [-28, 0], [-3, -20], [-45, 5], [-31, 2], [12, -16], [-1, 3], [-4, 18], [-58, 0], [50, 6], [-49, 3], [17, 7], [24, -9], [23, 0], [14, 26], [-12, -3], [-54, 0], [8, 3], [-15, -5], [-22, 0], [-10, 8], [-42, 16], [-46, -1], [-54, 3], [-7, 3], [0, -11], [-21, -24], [48, 8], [-6, 20], [-30, -4], [-16, -9], [30, 20], [11, -9], [-27, 3], [-39, -7], [20, -3], [-4, 3], [36, 12], [29, 2], [-30, 6], [20, -6], [50, 3], [30, 12], [-22, 9], [-60, -8], [8, 0], [-78, -4], [66, -6], [-6, -2], [20, 24], [24, -4], [-22, 14], [30, 2], [-12, 6], [-4, 32], [24, 18], [14, -6], [-4, 24], [-42, 14], [-22, -13], [18, -6], [-4, 3], [-10, -8], [8, 2], [45, 12], [-54, -8], [-16, 16], [15, 10], [-19, -19], [12, -2], [-28, 0], [12, -18], [14, -3], [-36, -3], [-22, 22], [2, 27], [-34, 25], [6, 7], [12, 10], [-1, -24], [18, 4], [-39, 16], [38, 12], [-31, 0], [24, -17], [-3, -27], [8, -4], [-10, 3], [-39, 2], [47, -15], [23, 6], [-18, 22], [-9, -11], [18, -16], [30, 5], [69, 0], [-4, 21], [42, -7], [12, -30], [-33, 18], [41, 17], [11, 20], [0, -18], [20, -10], [2, 15], [81, -4], [-7, -21], [51, -10], [24, 14], [-10, -3], [-4, -4], [60, 16], [15, 28], [-37, -21], [51, 3], [14, 12], [57, -15], [-45, -6], [11, 16], [12, -2], [35, -15], [-6, -27], [32, -20], [5, 9], [-10, 24], [78, 5], [-43, 2], [30, 14], [54, 6], [-22, 0], [48, -18], [-10, 14], [-46, -14], [-33, 27], [-34, 12], [-6, 9], [26, 20], [57, -10], [38, 18], [78, 9], [-6, -18], [26, -16], [-28, 12], [45, 12], [8, 19], [24, 12], [41, 5], [-45, -18], [12, 22], [32, -2], [-70, -3], [30, -22], [-21, 12], [-1, -13], [60, 10], [24, 7], [42, 13], [26, 12], [-36, -12], [-3, 5], [32, 28], [-57, 5], [24, -8], [-15, -16], [86, 0], [0, -13]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8001_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8001_2_a_k();".
// 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_8001_2_a_k(:prec:=2)
    chi := MakeCharacter_8001_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_8001_2_a_k();".
// 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_8001_2_a_k( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8001_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![-6, 0, 1]>,<5,R![-6, 0, 1]>],Snew);
    return Vf;
end function;