
// Make newform 4000.2.a.f in Magma, downloaded from the LMFDB on 27 December 2025.
// To make the character of type GrpDrchElt, type "MakeCharacter_4000_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_4000_2_a_f();".
// 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_4000_2_a_f();".
// 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 := [5, 0, -5, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [-3, 0, 1, 0], [0, -3, 0, 1]];
    Rf_basisdens := [1, 1, 1, 1];
    Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf);
    Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]];
    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_4000_a();"
function MakeCharacter_4000_a()
    N := 4000;
    order := 1;
    char_gens := [2751, 2501, 1377];
    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_4000_a_Hecke( : Kf := Kf)
    return MakeCharacter_4000_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]];
    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, 0, 0, 0], [0, 1, 0, 0], [0, 0, 0, 0], [0, 0, 0, 1], [0, 0, 0, 2], [2, 0, 2, 0], [2, 0, 0, 0], [0, 2, 0, -2], [0, 0, 0, -1], [2, 0, -1, 0], [0, 4, 0, 2], [0, 0, -6, 0], [-4, 0, -3, 0], [0, -3, 0, 4], [0, 5, 0, -2], [4, 0, 0, 0], [0, 0, 0, -2], [3, 0, 1, 0], [0, -2, 0, -2], [0, 2, 0, 6], [10, 0, 2, 0], [0, -2, 0, -8], [0, 2, 0, -1], [5, 0, 5, 0], [-2, 0, -10, 0], [3, 0, 5, 0], [0, -2, 0, -6], [0, -1, 0, -4], [6, 0, -5, 0], [14, 0, 6, 0], [0, 5, 0, 6], [0, 10, 0, -4], [2, 0, -12, 0], [0, -2, 0, 8], [1, 0, -1, 0], [0, -6, 0, -4], [10, 0, -4, 0], [0, 2, 0, 11], [0, -3, 0, 6], [14, 0, 0, 0], [0, -12, 0, -2], [14, 0, 9, 0], [0, -2, 0, -2], [16, 0, 10, 0], [14, 0, -8, 0], [0, -2, 0, -6], [0, -14, 0, 4], [0, 3, 0, 10], [0, -6, 0, -11], [1, 0, -13, 0], [-2, 0, 4, 0], [0, -2, 0, 12], [-21, 0, 3, 0], [0, -12, 0, -4], [16, 0, 8, 0], [0, -5, 0, -10], [14, 0, 0, 0], [0, -4, 0, 0], [-10, 0, -4, 0], [-9, 0, -13, 0], [0, -10, 0, -6], [22, 0, 2, 0], [0, 14, 0, -3], [0, -12, 0, 6], [4, 0, 16, 0], [16, 0, 18, 0], [0, 2, 0, 4], [-4, 0, -14, 0], [0, -6, 0, 5], [25, 0, 15, 0], [-10, 0, -18, 0], [0, -8, 0, 2], [0, 0, 0, -19], [4, 0, -10, 0], [0, 0, 0, -16], [0, 3, 0, 10], [-10, 0, -17, 0], [-18, 0, -22, 0], [-9, 0, 3, 0], [5, 0, 17, 0], [0, -4, 0, 8], [6, 0, 17, 0], [0, -2, 0, 12], [-20, 0, 2, 0], [0, 2, 0, 2], [0, -6, 0, 7], [-22, 0, -16, 0], [-4, 0, 8, 0], [3, 0, 5, 0], [0, -8, 0, -5], [0, 14, 0, 9], [0, -10, 0, -8], [0, -19, 0, -2], [0, 10, 0, -2], [0, -2, 0, -14], [0, -24, 0, 3], [-2, 0, -24, 0], [4, 0, -23, 0], [0, 9, 0, -16], [18, 0, -15, 0], [0, -6, 0, 5], [14, 0, 32, 0], [0, 22, 0, -6], [-4, 0, -5, 0], [0, 8, 0, 0], [-20, 0, -4, 0], [0, 10, 0, 18], [16, 0, -6, 0], [0, 2, 0, -20], [8, 0, 21, 0], [0, -10, 0, -6], [8, 0, 24, 0], [22, 0, 40, 0], [0, -10, 0, 6], [0, 16, 0, -10], [-24, 0, -3, 0], [0, 6, 0, 3], [0, 10, 0, -18], [-24, 0, -26, 0], [0, 6, 0, -2], [-26, 0, -7, 0], [-18, 0, -34, 0], [0, 0, 24, 0], [0, -3, 0, -16], [0, 6, 0, 16], [10, 0, -20, 0], [1, 0, -13, 0], [0, 14, 0, -8], [0, -7, 0, 6], [-42, 0, -2, 0], [0, -10, 0, -12], [0, -18, 0, 6], [0, -14, 0, 20], [14, 0, 14, 0], [39, 0, -1, 0], [-4, 0, -33, 0], [4, 0, -24, 0], [0, 23, 0, -4], [2, 0, -10, 0], [25, 0, 45, 0], [0, 12, 0, -2], [-5, 0, 25, 0], [0, -14, 0, -2], [0, 2, 0, -22], [14, 0, 15, 0], [0, 10, 0, -16], [-40, 0, -12, 0], [30, 0, 34, 0], [0, 20, 0, 0], [0, -12, 0, 11], [-44, 0, -2, 0], [-17, 0, 11, 0], [0, 14, 0, -9], [0, -16, 0, 1], [0, -2, 0, -11], [0, 12, 0, -6], [0, -26, 0, 10], [-15, 0, -3, 0], [-8, 0, -20, 0], [10, 0, -4, 0], [0, 15, 0, -4], [-16, 0, -44, 0], [0, -3, 0, 22], [0, -18, 0, -16], [18, 0, 40, 0], [0, 26, 0, 6], [0, 0, 0, 12], [-14, 0, 8, 0], [-12, 0, -9, 0], [-8, 0, -18, 0], [0, -10, 0, -14], [-26, 0, -31, 0], [0, -24, 0, 2], [-50, 0, -18, 0], [0, -18, 0, -10], [36, 0, -13, 0], [0, 0, 0, -4], [-42, 0, -3, 0], [0, 3, 0, -6], [22, 0, 31, 0], [0, 5, 0, 14], [0, -28, 0, 10], [2, 0, 6, 0], [0, 0, -6, 0], [0, 3, 0, -18], [-34, 0, -33, 0], [-2, 0, 30, 0], [0, -19, 0, 8], [-26, 0, -4, 0], [0, 10, 0, -16], [-8, 0, 12, 0], [0, -7, 0, 8], [0, -6, 0, -26], [10, 0, 5, 0], [0, -10, 0, -31], [12, 0, -18, 0], [-9, 0, 27, 0], [12, 0, 36, 0], [26, 0, 26, 0], [0, -8, 0, -13], [26, 0, 32, 0], [0, -4, 0, -24], [-12, 0, -28, 0], [-32, 0, -49, 0], [0, 12, 0, -2], [46, 0, 28, 0], [0, 0, 0, 30], [0, -19, 0, 24], [37, 0, 9, 0], [0, -8, 0, 4], [-34, 0, -22, 0], [-58, 0, -11, 0], [0, -9, 0, 18], [0, -18, 0, -27], [0, -8, 0, -14], [14, 0, -12, 0], [0, 25, 0, 2], [20, 0, 9, 0], [0, 28, 0, -19], [14, 0, -24, 0], [14, 0, 8, 0], [0, 26, 0, 8], [8, 0, -9, 0], [0, -16, 0, -9], [0, 11, 0, 0], [-7, 0, -1, 0], [32, 0, 26, 0], [0, -12, 0, -20], [0, 4, 0, -11], [0, -4, 0, 16], [28, 0, 24, 0], [0, 2, 0, 16], [0, -18, 0, -6], [7, 0, -21, 0], [0, -14, 0, 19], [0, 6, 0, 34], [-28, 0, -13, 0], [48, 0, 8, 0], [0, 20, 0, 14], [0, 24, 0, -16], [0, 1, 0, -16], [0, -10, 0, -18], [0, -4, 0, 7], [33, 0, 51, 0], [6, 0, -6, 0], [0, 36, 0, -4], [0, 22, 0, -2], [0, -4, 0, 16], [0, -6, 0, 10], [0, -9, 0, -34], [-24, 0, -14, 0], [32, 0, 25, 0], [0, 4, 0, 13], [-22, 0, 28, 0], [18, 0, -2, 0], [0, -4, 0, 16], [3, 0, 25, 0], [0, -10, 0, -7], [16, 0, -12, 0], [20, 0, -34, 0], [0, -17, 0, -14], [0, -14, 0, -2], [-2, 0, 31, 0], [22, 0, 22, 0], [52, 0, -10, 0], [0, 14, 0, 24], [1, 0, -13, 0], [48, 0, 5, 0], [0, 18, 0, 35], [8, 0, -6, 0], [-25, 0, -59, 0], [0, 15, 0, 28], [2, 0, 32, 0], [0, 34, 0, -14], [-16, 0, -26, 0], [0, -2, 0, 30], [0, -18, 0, 25], [-26, 0, 3, 0], [-26, 0, 8, 0], [0, 42, 0, -10], [0, 12, 0, 15], [0, 6, 0, 30], [0, -19, 0, 10], [-22, 0, -43, 0], [0, -1, 0, -20], [0, 20, 0, -18], [50, 0, 28, 0], [42, 0, 28, 0], [0, 30, 0, -18], [-6, 0, 0, 0], [-18, 0, -31, 0], [0, 11, 0, 44], [6, 0, 14, 0], [0, 18, 0, -10], [16, 0, -36, 0], [-19, 0, 15, 0], [0, 16, 0, -26], [34, 0, 36, 0], [0, 22, 0, 24], [0, -22, 0, -15], [24, 0, 40, 0], [64, 0, 24, 0], [0, -10, 0, -14], [0, 2, 0, -29], [0, 2, 0, 32], [56, 0, 6, 0], [0, 26, 0, -7], [21, 0, 27, 0], [0, -32, 0, -2], [46, 0, 60, 0], [0, -21, 0, -2], [-47, 0, -9, 0], [6, 0, 28, 0], [0, 26, 0, -5], [0, 5, 0, -34], [36, 0, 27, 0], [0, 6, 0, 4], [0, 12, 0, 50], [-4, 0, 20, 0], [-27, 0, 13, 0], [0, 6, 0, 12], [40, 0, -16, 0], [51, 0, 17, 0], [0, -17, 0, -2], [10, 0, -18, 0], [31, 0, 47, 0], [0, -10, 0, -42], [0, -7, 0, -16], [0, -8, 0, -27], [24, 0, -30, 0], [-10, 0, -35, 0], [14, 0, 4, 0], [0, 36, 0, -6], [0, -14, 0, -21], [0, -10, 0, -34], [0, 11, 0, -24], [41, 0, 75, 0], [-40, 0, 18, 0], [-66, 0, -32, 0], [0, 12, 0, 29], [-14, 0, 0, 0], [54, 0, 24, 0], [-27, 0, -17, 0], [0, 26, 0, 2], [48, 0, 14, 0], [0, 26, 0, 4], [22, 0, 9, 0], [0, 35, 0, -8], [0, -8, 0, 0], [22, 0, 8, 0], [0, 10, 0, -36], [-20, 0, -24, 0], [54, 0, -7, 0], [0, -37, 0, -14], [57, 0, -1, 0], [22, 0, -4, 0], [0, 36, 0, 2], [0, -20, 0, -6], [10, 0, -36, 0], [0, -34, 0, 10], [42, 0, 50, 0], [-58, 0, -52, 0], [0, -32, 0, -15], [0, -36, 0, 22], [0, -22, 0, 9], [-30, 0, -42, 0], [30, 0, -6, 0], [0, -4, 0, 39], [32, 0, 49, 0], [0, -2, 0, 18], [0, 10, 0, -2], [0, 16, 0, 31], [10, 0, 27, 0], [0, -36, 0, 8], [-6, 0, -38, 0], [0, -32, 0, 0], [0, -14, 0, 6], [-6, 0, 10, 0], [20, 0, 43, 0], [-86, 0, -8, 0], [-41, 0, 9, 0], [50, 0, 12, 0], [0, 0, 0, -23], [-10, 0, 4, 0], [0, 30, 0, 26], [0, 28, 0, 31], [0, -30, 0, 20], [-6, 0, -28, 0], [0, 1, 0, -12], [0, 33, 0, 6], [-23, 0, -3, 0], [-56, 0, -10, 0], [0, -24, 0, -20], [0, 23, 0, 40], [0, -6, 0, -2], [36, 0, -6, 0], [0, -8, 0, 32], [-46, 0, 8, 0], [0, -32, 0, -8], [-13, 0, -11, 0], [6, 0, 15, 0], [28, 0, 14, 0], [0, -32, 0, 21], [48, 0, 10, 0], [-58, 0, -53, 0], [0, 18, 0, -36], [-38, 0, 30, 0], [-4, 0, -39, 0], [0, -14, 0, -41], [0, 24, 0, 44], [-32, 0, -32, 0], [-4, 0, 6, 0], [0, 21, 0, 44], [0, -32, 0, -28], [34, 0, -18, 0], [-33, 0, 33, 0], [0, -16, 0, -12], [0, -4, 0, -15], [8, 0, 30, 0], [0, 16, 0, 27], [-39, 0, -81, 0], [-32, 0, -30, 0], [0, 1, 0, 26], [0, 10, 0, 60], [56, 0, 70, 0], [-28, 0, 20, 0], [0, -3, 0, -24], [-32, 0, -37, 0], [0, -8, 0, 8], [0, 16, 0, 2]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4000_a_Hecke( : Kf := Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4000_2_a_f();".
// 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_4000_2_a_f(:prec:=4)
    chi := MakeCharacter_4000_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_4000_2_a_f();".
// 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_4000_2_a_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4000_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<3,R![5, 0, -5, 0, 1]>,<7,R![5, 0, -5, 0, 1]>,<11,R![80, 0, -20, 0, 1]>,<13,R![-4, -2, 1]>],Snew);
    return Vf;
end function;