
// Make newform 8281.2.a.bk in Magma, downloaded from the LMFDB on 25 April 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_8281_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_8281_2_a_bk();".
// 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_8281_2_a_bk();".
// 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 := [-1, -5, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0], [0, 1, 0], [-3, -1, 1]];
    Rf_basisdens := [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_8281_a();"
function MakeCharacter_8281_a()
    N := 8281;
    order := 1;
    char_gens := [1522, 3382];
    v := [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_8281_a_Hecke(Kf)
    return MakeCharacter_8281_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, 0], [1, 0, 1], [-2, 1, 0], [0, 0, 0], [1, 1, 0], [0, 0, 0], [1, 0, 1], [-2, -2, 1], [1, -3, 1], [-3, 0, 2], [0, 1, 2], [3, -1, 2], [-2, -4, 4], [3, 2, -2], [-6, 0, 1], [5, -1, -1], [-8, 1, 1], [-8, -2, 2], [4, -2, 4], [-1, -3, 2], [-4, 6, 1], [-1, 2, 2], [-8, 3, -2], [4, -4, 3], [2, -4, 1], [5, 4, -1], [-4, 4, 4], [6, -1, 3], [7, 3, -4], [3, 5, -3], [8, -1, -3], [6, -4, -2], [-7, 3, -4], [-3, 2, 1], [-3, -3, 6], [-9, 1, 4], [1, 4, -5], [-6, 0, 2], [6, 6, -3], [1, 6, -1], [-9, -4, 0], [11, -4, 3], [-6, 9, -3], [-8, 2, 0], [14, 0, 2], [-3, -10, 5], [-15, 3, 3], [4, -1, 2], [-10, -2, 0], [0, 1, -5], [7, 0, -8], [14, 4, -4], [0, 5, -4], [9, -6, -3], [19, -2, 1], [3, 0, 6], [-20, -2, 4], [0, -6, 2], [-3, 0, 0], [1, 1, -2], [2, 2, -8], [-10, -10, 5], [-6, 5, -10], [-5, 6, -7], [-25, 0, 3], [-4, 10, -4], [2, -10, 4], [-13, 3, -9], [-10, 4, -6], [2, 6, 7], [8, -5, 5], [17, -1, 2], [13, 6, -3], [-4, -11, 1], [-1, -3, -12], [-18, 8, -2], [0, 9, -7], [-2, 0, 5], [-4, -2, 0], [2, 4, -11], [1, -2, 11], [-11, -1, -4], [6, -6, 12], [-10, -4, -4], [2, 6, 0], [-13, -5, -3], [-2, -8, -2], [9, 11, -10], [0, -2, -4], [7, -5, -8], [31, -2, -1], [0, -9, -4], [8, 12, -6], [2, 3, -1], [-7, 5, 4], [-8, 2, 10], [8, 4, 9], [-8, 4, 0], [16, -2, -2], [1, -17, 10], [-15, 5, 7], [-12, 2, 10], [18, -10, -2], [7, -4, 12], [-19, -7, 5], [4, 10, -4], [-22, -5, -4], [-2, -3, -8], [9, -2, 8], [-10, -4, -4], [-5, -20, 11], [-4, -12, 4], [2, 4, 12], [0, 6, -2], [10, 4, 6], [-7, -2, 6], [-16, 0, 12], [29, -8, -1], [-32, 2, -8], [9, -7, 7], [-12, -9, -2], [9, -17, 1], [1, 14, 1], [-22, 0, -4], [-28, 12, 3], [-27, -2, 2], [-9, 9, 10], [-5, 0, -5], [-14, -18, 6], [32, -1, 2], [-35, 5, 0], [0, 12, -2], [25, -8, -6], [1, -6, -6], [4, 13, 2], [4, -11, 10], [-10, 4, -16], [-2, 13, -4], [6, 6, -10], [25, -4, -8], [6, -2, 16], [-16, 8, 8], [-20, 1, -1], [18, 0, 6], [1, -14, 15], [4, -16, 0], [10, -9, 6], [-6, -16, 18], [26, -2, -6], [-4, -4, -4], [21, -13, -4], [22, -2, 4], [-4, 11, 9], [-18, 10, -12], [11, 17, -1], [23, -2, -4], [-10, -8, -6], [-16, -1, -4], [-17, 0, 11], [6, 3, 12], [0, -2, -10], [-25, 10, 0], [11, 5, 2], [6, -12, 10], [-13, -5, 2], [8, -22, -1], [2, 11, -1], [-2, 10, 4], [13, 11, -12], [19, -12, -5], [-3, 21, -6], [-52, -2, 0], [17, 3, 1], [5, -7, 16], [30, 8, -10], [30, -8, 0], [-26, 4, -2], [0, 12, -14], [-11, 2, -7], [11, 6, -5], [-2, 6, -10], [3, -6, -9], [0, 7, 19], [-14, 20, -9], [7, 11, -2], [-27, 0, 1], [3, 3, 5], [-34, -1, 6], [-20, -2, -2], [6, 0, -1], [5, 14, -9], [-12, 16, -6], [-15, 9, 9], [6, -5, -18], [31, 13, -3], [3, -22, 15], [17, -21, 0], [-33, 6, -8], [-4, -17, 18], [-4, -22, 8], [-17, 19, -10], [-25, 20, 1], [26, -8, -11], [16, -6, -10], [-38, -2, 7], [-18, 6, -20], [-20, 3, 0], [9, -6, 24], [13, 27, -8], [-4, 12, -8], [-21, -28, 20], [-7, -10, 15], [16, 0, 0], [0, -18, 6], [-2, 13, 10], [4, 3, -4], [5, 20, -22], [6, -14, 20], [-10, -18, 4], [-21, -15, 14], [15, -9, 3], [36, -11, -7], [30, -2, 6], [-6, 2, -22], [20, 10, -16], [-1, -13, 13], [-26, 2, 6], [13, -16, 12], [3, -18, 23], [-27, -9, 12], [2, -1, 17], [-25, -4, 9], [8, -20, -2], [12, -21, 15], [4, -16, 14], [-12, 12, -9], [0, -10, 8], [-12, 12, 6], [-5, 15, -5], [36, -18, 0], [-20, -4, 4], [-2, 22, 0], [-10, 8, -8], [24, -12, 8], [42, -15, -2], [-11, -20, 19], [22, -4, -7], [-14, 1, -12], [-5, 15, -6], [-23, 13, 15], [-40, -6, 6], [-8, -14, -8], [32, -20, -4], [42, 2, 4], [32, -16, 0], [22, -4, -20], [13, 6, 0], [6, -6, -20], [-10, -8, 2], [18, -3, -2], [1, 21, -3], [-9, -24, 18], [-18, 2, 7], [-43, 10, -9], [38, 18, -10], [16, -6, -10], [40, -10, -6], [-4, 22, 5], [-11, -23, 22], [28, 7, 3], [-19, -2, -23], [-9, 11, 10], [-38, 26, 14], [43, 3, -1], [-33, 10, 1], [-18, -28, 10], [39, 3, 4], [-5, 19, -12], [-40, 12, 6], [-6, -6, 10], [16, -24, 10], [10, -10, 8], [40, 8, 0], [6, -22, 11], [4, 1, 13], [-3, 12, 6], [33, -6, -4], [-20, 18, -10], [-46, 0, 10], [13, -28, 13], [-7, 3, 23], [38, 22, -16], [-34, -26, 20], [-2, 12, -13], [19, -23, 13], [6, 6, -10], [-12, 2, -8], [-36, 14, 6], [-17, -8, -19], [-26, 19, -2], [33, -12, 9], [15, -15, 0], [-27, 3, 3], [15, -3, -21], [-43, 1, 7], [-45, -7, -8], [10, 29, -27], [33, 16, -19], [30, -6, 12], [-9, -7, 11], [-19, 34, -7], [-31, -5, -16], [-52, 7, 3], [-7, -16, 20], [29, -13, 12], [-22, 26, 2], [-18, -9, 6], [51, -3, -14], [-4, 5, -22], [19, -28, 10], [-48, 6, -8], [-6, -24, 24], [-15, -16, -5], [-2, 36, -12], [32, -14, 8], [13, -20, -13], [-12, -4, -26], [-32, -7, -4], [-54, 9, 5], [8, 25, -7], [3, 11, -16], [5, -9, 11], [-39, 12, 13], [-2, -34, 7], [-15, 13, -20], [-18, 12, -1], [-21, 17, 22], [22, 2, -2], [-16, 1, -21], [6, 4, -14], [13, -17, 4], [-6, 15, 1], [16, -13, -3], [-10, 17, 1], [-59, -1, 10], [12, -1, 1], [-33, 16, 13], [35, -5, -2], [24, 12, -15], [25, 8, -15], [26, 20, -25], [-27, 17, 4], [-11, 5, -6], [-60, -8, 8], [-22, -9, 1], [-15, -6, 15], [6, -19, -18], [-18, 13, -6], [18, 27, -27], [27, -25, 0], [36, 4, -10], [6, 18, 0], [15, -24, 12], [-3, 3, 32], [-65, 18, -1], [15, -18, 6], [24, -3, -10], [47, -7, 2], [31, 13, -1], [6, -41, 11], [-27, 4, -19], [-37, 29, -6], [-5, -2, -18], [-57, 16, -11], [35, -24, 0], [-16, -26, 2], [-58, -11, 9], [8, 11, -19], [18, 4, -30], [-4, 4, 4], [10, -5, 14], [28, 22, -10], [-67, 11, -8], [9, -11, 12], [-48, -18, 21], [12, 17, -24], [51, -9, 16], [-10, 2, -6], [-39, 3, 10], [7, -4, -15], [20, 10, 0], [-26, -21, 16], [66, -4, 6], [-14, -8, -2], [24, 24, -4], [17, -21, -15], [-68, -2, -2], [-20, 22, 10], [15, 22, 8], [-14, -6, -22], [12, 8, -7], [-1, 7, 5], [35, -25, -14], [-48, 3, -14], [18, -2, 21], [41, -6, 1], [22, 10, 4], [47, -17, 8], [-6, 16, -23], [-8, -8, 11], [-2, -38, 6], [49, -1, 25], [15, 15, -13], [-10, -1, -13], [-11, 1, -11], [-15, -18, 33], [-6, -6, -10], [22, 14, 8], [32, 26, 0], [-21, 30, -3], [-15, 7, 23], [20, -37, 9], [31, 7, -2], [-33, -16, -9], [10, 22, -2], [-8, 0, 13], [7, 14, 6], [2, 6, 16], [34, 18, -21], [0, 6, -2]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_8281_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_8281_2_a_bk();".
// 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_8281_2_a_bk(:prec:=3)
    chi := MakeCharacter_8281_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_8281_2_a_bk();".
// 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_8281_2_a_bk( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8281_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![6, -4, -2, 1]>,<3,R![14, -2, -4, 1]>,<5,R![-7, 3, 5, 1]>,<11,R![2, 0, -4, 1]>,<17,R![14, -2, -4, 1]>],Snew);
    return Vf;
end function;