
// Make newform 4026.2.a.s in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_4026_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_4026_2_a_s();".
// 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_4026_2_a_s();".
// 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, 5, -5, -2, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0], [0, 1, 0, 0], [-3, -1, 1, 0], [3, -3, -2, 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_4026_a();"
function MakeCharacter_4026_a()
    N := 4026;
    order := 1;
    char_gens := [1343, 1465, 3235];
    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_4026_a_Hecke(Kf)
    return MakeCharacter_4026_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, 0, 0, 0], [-1, 0, 0, 0], [1, 0, 0, -1], [0, -1, 1, 0], [-1, 0, 0, 0], [3, 1, 1, -1], [0, 2, -1, -1], [1, -1, 0, 0], [1, -1, 0, 0], [0, 1, 1, 0], [0, -1, -2, 1], [-3, 3, 2, 0], [2, 0, 1, 1], [-5, 0, 1, 0], [5, 0, -2, -1], [3, -2, 1, 0], [4, -3, 2, 3], [-1, 0, 0, 0], [-2, 3, -2, -1], [-2, 3, 0, 1], [-2, 3, -3, 0], [3, 3, -5, 1], [2, 1, -1, 2], [3, 2, 0, 3], [-2, 3, -2, 1], [5, -2, 4, 1], [-1, 0, 0, -1], [3, 1, -5, -1], [5, -5, -1, 1], [13, 2, 1, -2], [0, 4, -3, -3], [-9, 4, -1, -2], [-4, 6, 1, 1], [-3, 7, -3, 1], [2, -4, 1, 1], [8, 0, -2, 2], [2, -3, 3, 0], [-2, -1, 5, 2], [5, -6, 3, 2], [4, 1, -1, 2], [-3, 0, 6, -1], [2, -2, -1, 1], [7, -3, -1, 3], [8, -1, 0, 1], [4, 1, 5, 0], [0, 2, 2, -4], [4, 2, -2, 0], [-3, 4, 7, 0], [-4, -1, 5, 0], [-7, -3, -4, 0], [4, -8, -2, 0], [1, 0, -10, -1], [1, 1, -4, 0], [-2, 7, -1, 4], [-2, -7, 0, 1], [-2, 9, -3, 0], [2, 0, 1, 1], [-3, -9, -2, 2], [-1, -3, 3, -7], [8, -7, 1, -4], [13, 0, 3, 4], [-5, -6, 1, 0], [-9, 5, 0, 4], [7, -3, -2, -6], [15, -5, -2, 2], [1, -8, 5, 4], [-4, 9, -2, -5], [6, -8, 8, 4], [7, -6, 6, 3], [-10, 2, 0, 6], [11, -9, 1, -3], [-4, 4, -3, 5], [22, -2, -3, 1], [-6, -1, -2, -7], [-2, -14, 5, 1], [-18, 4, 3, 1], [2, 11, -7, 0], [-8, 1, 4, 3], [-2, -2, 8, 2], [-2, 5, 4, 1], [13, -3, -1, -7], [-3, -7, 4, 0], [2, 2, -1, 3], [-4, 0, 13, -1], [-7, 3, -9, -5], [6, 5, -2, -7], [-2, 0, -7, 5], [-10, 1, 2, 3], [8, 0, 2, -4], [-27, 3, 5, 1], [-11, -7, 2, 0], [11, 2, 1, -2], [-2, 1, -1, 2], [-5, -2, 3, -8], [-9, -7, 7, -1], [2, -5, 4, 5], [1, 5, -6, 2], [4, 7, -1, 4], [-9, -8, 1, 4], [-1, -2, -5, -2], [13, -1, -3, 1], [-16, 3, -6, 3], [4, 8, -1, -9], [12, 1, -2, -3], [-1, -4, 2, 1], [5, -9, 0, -2], [-14, 5, -6, 1], [9, 2, -5, 4], [-3, -13, 5, -3], [18, -2, 7, -3], [-13, 14, 2, -1], [-15, -5, 0, 6], [-14, -7, -1, 4], [11, 0, -12, 3], [2, 13, -2, 1], [-24, 4, -4, -2], [3, -8, 0, 7], [9, -12, 3, 4], [14, 2, -1, -3], [-4, -2, 4, -6], [-9, 3, -10, 6], [-13, 4, -8, -11], [13, 2, -10, -1], [-18, 2, -10, 2], [-22, 15, 4, -3], [-9, -5, -2, 6], [20, -2, 8, 4], [9, -5, 3, 7], [-14, 7, -8, 5], [-16, -13, 2, -1], [-15, 5, 3, 1], [-18, 2, 10, 2], [-11, 2, 6, 5], [-8, 3, 3, 4], [-7, -1, 9, 3], [-9, 1, 0, 10], [-15, 3, 9, 7], [-9, 6, 6, -1], [23, -1, -5, 3], [32, 0, 8, -2], [-10, 3, 1, 2], [-22, 8, -11, 1], [-5, -12, -4, 3], [13, -11, -5, 5], [28, -11, -4, -5], [-10, -2, -2, -2], [9, 13, -8, 0], [-1, -8, -2, 11], [-9, -17, 3, 1], [3, 0, -16, -1], [9, -8, -8, -1], [1, 6, 0, -3], [-7, -1, -20, 4], [-1, -13, 7, 5], [17, -11, -5, -7], [-7, 8, 7, 4], [2, -3, 13, 4], [2, 6, 5, 3], [-1, 3, 1, 5], [0, -8, 5, -1], [-15, -5, 5, 1], [8, -9, 12, 5], [10, 1, 5, 0], [-2, -1, -14, 7], [-5, 24, 0, -7], [-29, 15, 7, 1], [-1, -21, 1, 7], [-8, -13, 12, -7], [15, -8, 2, -1], [-17, 19, -9, -5], [3, 5, -4, -12], [-24, 2, 7, -5], [7, -11, 7, -9], [3, 6, -2, -11], [15, 14, 0, -11], [38, -7, -2, -1], [11, -7, -11, 1], [-19, 10, 4, 9], [7, -19, -7, 5], [41, -5, 2, 0], [-20, 10, -16, -2], [21, -11, 0, 2], [0, -15, 3, 2], [23, -3, -3, -5], [24, 3, 6, -7], [6, -9, -7, 8], [7, -7, -1, 1], [7, 3, -7, -5], [25, 1, 9, -11], [-5, 9, -16, 0], [-37, 7, -3, -7], [9, 1, 11, -3], [-8, -2, 15, -7], [6, -14, -9, 5], [1, 2, -23, 0], [14, -10, 1, -5], [13, -12, -7, 0], [-7, 5, 9, 5], [21, -20, -7, 0], [22, -9, -4, 5], [-7, 14, 7, 0], [0, 12, 0, -8], [15, 0, -5, 4], [18, 7, 4, -1], [26, -12, 12, 14], [-10, -7, 3, 4], [-4, 13, -4, 9], [-47, 4, 9, 2], [19, -8, -16, 1], [-10, -11, -11, 8], [-14, 24, -3, -11], [10, -8, 9, 9], [12, -9, 20, 3], [-31, 13, 10, 0], [27, 13, -1, -5], [17, -10, -1, -4], [11, -18, 0, 5], [17, -4, 6, 1], [37, -9, 6, 2], [36, 0, -5, 5], [-19, 19, 8, 2], [32, -4, -8, 0], [-12, -18, 7, -1], [-13, 7, -17, 5], [28, -15, -10, -13], [22, 7, -10, 5], [17, -3, -3, 5], [30, 0, 8, 2], [4, 2, -3, 15], [9, -9, 4, 4], [-14, -21, 7, 2], [-14, 1, 16, 3], [-17, -9, -6, 0], [-13, 7, -5, -9], [3, -20, -8, 3], [-37, 19, 18, 4], [11, 2, -8, 11], [34, -16, 12, 8], [19, 16, 1, 0], [-18, 16, 5, -1], [1, 12, -10, -9], [8, -7, 19, 4], [27, 5, -2, -14], [-19, 11, -19, -5], [-8, 5, -3, 12], [8, -3, 22, -9], [13, 4, 10, 3], [10, 26, -2, -2], [-8, 8, 6, -6], [2, -12, -11, 11], [-15, -5, -27, 11], [34, -2, -2, -12], [48, -16, -10, -6], [7, -20, 10, 3], [-37, 13, -21, -7], [22, -4, 1, 7], [-11, 10, 6, -9], [40, -4, -6, -6], [35, 0, 11, -4], [-11, 14, -8, 1], [16, 19, -1, -6], [-3, -7, -4, -6], [15, 4, 4, -11], [-38, -5, 6, 5], [36, -16, -3, -1], [3, -7, 18, -10], [7, 3, 6, 2], [-2, 7, -2, -3], [-26, -15, -4, 3], [28, -17, -3, 6], [-13, 5, -16, -2], [3, -3, -12, -4], [29, -8, 10, 5], [14, -11, 1, 12], [-5, 4, -18, 11], [-24, -6, 0, 10], [-45, 17, 4, 0], [20, 7, 2, 15], [9, -4, -4, 7], [-38, 1, 10, 5], [-10, -25, 5, -2], [48, -7, -1, -4], [-24, 1, 6, 11], [14, 11, -8, 3], [-11, 3, 17, -7], [0, -7, -9, 8], [9, -20, -10, 9], [-11, -6, -12, -3], [-28, -15, 8, -7], [30, 10, -3, 3], [-26, 7, -7, -8], [-14, -7, 8, 11], [-42, 1, -15, 2], [-31, -2, -16, 3], [3, 7, -23, 5], [-1, 17, -5, 1], [-7, 15, 10, -2], [-21, -16, 16, -7], [23, 0, 10, -11], [-17, -7, 12, 4], [10, 18, -18, 0], [8, -29, 10, -5], [24, 14, 2, -4], [1, -7, -5, -19], [10, -9, 3, 4], [10, -7, 9, -2], [-6, -7, 12, -9], [20, 11, -1, -4], [-17, 10, -2, 3], [-20, 9, 7, 14], [-9, 21, -11, -3], [7, 2, 16, 7], [14, -15, -5, -4], [-8, -11, 3, 8], [11, -14, -10, -13], [7, 8, 17, -2], [-14, -1, 1, 18], [-51, 5, 1, 7], [4, 17, 11, -14], [8, 2, -4, -20], [-10, -5, 5, -18], [-40, 17, 4, -1], [23, 0, -29, 4], [-8, 3, 1, 16], [11, -8, 3, 4], [17, -17, -9, 1], [-39, 4, -7, 10], [-20, 18, -16, -14], [-31, -5, 2, 10], [10, -1, 8, 7], [29, -14, 24, 13], [-16, 0, 2, 8], [68, 3, -6, 1], [-53, 2, -10, -1], [-3, 2, 12, 15], [-15, 4, 24, -7], [-2, 30, 4, -14], [35, -21, -5, 3], [-29, -2, 17, 8], [44, -14, 2, -4], [2, -15, 10, 11], [33, -2, -8, -3], [-1, -2, 0, 15], [-14, 2, -24, -4], [45, 14, 0, -7], [18, 15, 2, -13], [18, -3, -11, -6], [-45, 22, 8, 5], [37, -20, 14, 7], [31, 8, -9, 4], [-1, 9, 6, -2], [-2, 1, -17, 8], [-9, -9, 12, -12], [38, 7, 0, -11], [0, 17, -11, 0], [-24, 14, 13, 9], [-52, 2, 0, -6], [-21, 2, 4, -11], [-15, 10, -23, 2], [1, -5, 9, -11], [-16, 21, -15, 0], [38, 11, -10, 1], [-28, -17, 20, -1], [-18, 8, -7, 19], [-7, -2, 21, 8], [-16, 25, -4, 7], [-17, 12, -7, -18], [18, -19, 4, -5], [23, -33, 3, 3], [1, -11, 12, -6], [3, -21, -5, 17], [11, 19, -11, -1], [24, 22, -8, 0], [2, -13, 29, -6], [-3, -7, -1, -5], [-22, -15, 11, -2], [33, -24, 15, 8], [48, -12, -12, -10], [35, 11, 10, -6], [-4, -30, 3, 15], [-63, -5, -2, 4], [3, 19, -19, -3], [34, -2, 9, -11], [-27, 22, 0, -15], [27, -3, -14, -6], [-11, -2, -1, 6], [26, -20, 17, 15], [40, -2, 8, -16], [-28, 5, -26, 7], [-6, 13, -20, -5], [16, 2, -16, 8], [-8, 18, 3, -13], [14, -8, -20, 6], [1, -19, 1, -9], [-47, -12, -2, 7], [6, -11, 28, 1], [46, -30, -13, -1], [33, 5, 2, -14], [30, -32, -6, 4], [3, 18, -17, -10], [38, -6, -4, -14], [13, 13, -4, -8], [-12, -9, 4, -1], [0, -41, 5, 6], [39, -8, 10, -13], [-13, -1, 10, 20], [-25, 8, -30, 3], [-13, 9, 12, -14], [-32, 26, 6, 4], [-7, 5, -13, 9], [10, 6, 5, 3], [-42, 30, -14, -8], [-30, 11, -7, -2], [17, -12, -2, -1], [-41, 10, 18, 1], [-34, 14, -16, -14], [-25, -27, -12, 20], [-2, -3, -4, 19], [-20, 10, 18, -10], [42, -17, -9, -20], [-12, 17, -30, 1], [30, 6, 3, -19], [25, 24, -8, 13], [15, -23, 4, 20], [-19, -5, -20, -2], [-9, -4, 0, 9], [30, -47, -1, 6], [36, -6, 27, -1], [-4, -3, 3, 12], [40, -26, 15, 5]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_4026_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_4026_2_a_s();".
// 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_4026_2_a_s(:prec:=4)
    chi := MakeCharacter_4026_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_4026_2_a_s();".
// 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_4026_2_a_s( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_4026_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<5,R![38, 25, -12, -3, 1]>,<7,R![-4, -17, -11, 2, 1]>,<13,R![-514, 133, 34, -13, 1]>],Snew);
    return Vf;
end function;