
// Make newform 8281.2.a.bw in Magma, downloaded from the LMFDB on 09 May 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_bw();".
// 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_bw();".
// 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 := [2, 8, 0, -7, -1, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [2, -4, -2, 1, 0], [1, 3, -4, -2, 1], [4, 4, -5, -2, 1]];
    Rf_basisdens := [1, 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_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, 0, 0], [0, 0, 0, 0, -1], [0, 0, 1, 0, 0], [0, 0, 0, 0, 0], [-2, 0, 0, 1, 0], [0, 0, 0, 0, 0], [-1, 2, 1, 0, 1], [-2, -2, -1, -1, -1], [2, 0, 0, 0, -1], [-1, 2, 0, 0, 1], [2, 0, 2, 0, 1], [0, -2, 0, 2, 1], [-4, 2, -1, -2, 3], [0, -2, -1, -2, 1], [0, 0, 0, 1, -4], [3, 4, 0, 2, 2], [-2, 0, 0, 1, 0], [-3, 0, 0, -4, 2], [-4, -2, -3, -3, -1], [-2, -4, 0, 1, -1], [0, 0, -1, 2, 0], [0, 2, -1, 2, -2], [-2, -2, -3, 0, -1], [0, -2, -2, -2, 3], [2, -2, -1, -2, 1], [-2, 0, 3, 0, -2], [0, -8, -2, 2, -3], [12, 2, 1, 0, 2], [0, 6, 0, -2, 1], [1, -2, 0, 0, -1], [0, 0, 0, 2, 2], [-4, -2, 3, -2, 2], [-6, 4, -1, -2, 6], [-4, 0, 0, 0, 0], [4, 8, 1, -2, 2], [10, -2, 1, 3, -5], [-7, -2, -1, 2, -1], [-4, -2, -3, 3, -1], [-6, 8, 2, 3, -1], [-1, 0, 2, 4, -2], [6, 0, -2, 0, 3], [3, -6, 0, 2, -1], [6, -2, -1, -2, 0], [-4, 2, -4, 2, 1], [0, 2, -1, 0, 5], [12, -4, 0, -2, 5], [0, 2, 1, 6, 3], [-2, -10, 1, 1, -8], [2, 4, 2, -4, 1], [-8, 2, 2, 0, -7], [-1, 4, 2, 2, 4], [-8, 2, 5, -1, 4], [2, 6, 0, -4, 5], [-14, 2, -1, -4, 1], [6, 0, 1, 4, -4], [8, 0, 2, 0, 5], [-7, -6, -3, 2, -3], [4, 2, 1, 1, 1], [-5, 8, -2, -4, 2], [-2, 2, -3, 4, -7], [-10, -6, 3, 0, 0], [8, 6, 9, 4, -5], [0, -6, -3, -5, 4], [0, -4, -4, 4, -5], [0, -4, -3, -2, -4], [8, 0, 5, 6, 2], [6, 0, -4, -4, 9], [-7, 2, -6, 2, 3], [16, 4, -2, 0, 5], [2, -2, -5, 2, -3], [-8, 2, -4, 4, -1], [2, -4, -4, 8, -7], [14, 2, -3, -4, -6], [-7, -2, -5, -6, 7], [0, 10, 1, -2, 11], [0, -4, -2, 2, 3], [5, -6, 1, 4, 7], [-4, -4, -5, -6, 0], [-4, -6, -4, 0, -1], [10, -4, -1, -6, 2], [-6, 0, 2, 4, -4], [10, 0, 0, 0, 0], [12, 0, 0, 2, -5], [-7, 4, -1, -4, 2], [-20, 6, -3, -2, 4], [22, -10, 1, 2, -2], [8, -4, -4, 4, -6], [0, -2, 2, 0, 9], [8, -2, -3, 2, -11], [-2, 2, 3, -4, -3], [8, -6, 1, 2, 0], [-10, -8, -6, 7, -4], [12, -12, -6, -5, -2], [10, 4, 4, 8, -6], [6, -8, -4, 0, -3], [2, 10, 7, -2, -1], [-16, 6, 4, -2, 1], [-8, -4, -3, -10, 4], [14, -8, 6, 4, -3], [12, -4, 3, 0, -2], [-6, 14, 5, -4, 7], [2, 8, 7, 4, 0], [-8, 10, -3, 4, -8], [-17, -8, -4, 6, 2], [2, -4, -4, -6, -1], [-2, -2, 4, 6, -17], [10, 8, 6, -5, -1], [-12, -8, -1, -12, -2], [6, 6, 11, 2, -2], [13, 6, 0, -8, -5], [-14, -6, -7, 8, 0], [8, 4, 7, -4, 6], [-4, 2, 7, -4, -7], [-16, -12, -4, 6, -5], [6, -6, -1, -4, 9], [0, 0, 7, 6, -18], [10, -8, 6, 7, 3], [0, -2, 1, 10, 6], [4, 4, -1, -6, -10], [-4, -16, -8, -4, -4], [12, 2, 2, -2, -9], [-12, 8, 2, -10, 10], [1, 8, -4, -8, 14], [-28, 12, 4, 2, 3], [6, -16, -4, -5, 0], [2, 4, -2, 4, -10], [8, -8, -7, -2, -2], [-18, 8, 4, 0, -1], [6, -14, -1, -2, -5], [-14, 10, -4, 2, -1], [-10, 4, 2, -8, 1], [-6, -10, -11, -11, 4], [6, 0, 0, -10, -7], [11, 0, 7, 2, -2], [-4, 8, 3, 4, -6], [2, -4, -6, 4, 2], [4, 6, 0, 4, 3], [-26, 10, -3, 1, -3], [8, 8, 2, 10, 6], [-3, 2, -5, -4, -3], [26, 6, 5, 0, -5], [2, 4, -7, -2, 8], [-12, 8, -10, -4, 1], [-28, -4, -2, -1, 5], [1, 12, 6, -2, 10], [-20, 2, -7, -7, -2], [0, -12, -4, 2, -12], [-3, 8, 6, 12, 8], [-20, 6, 3, 4, -8], [-18, 8, -2, -16, 15], [-16, -14, 2, -2, -17], [0, -12, -4, -14, -4], [16, 2, 1, -4, -7], [-14, 0, -4, -16, 11], [2, 2, -13, 0, -4], [-16, 2, 5, -2, 3], [10, -16, 4, 8, -13], [-22, -2, 4, 2, 7], [-9, 0, -1, 12, -8], [12, 10, 6, 6, -5], [-16, 0, -2, 7, -8], [-17, 4, 1, -10, 12], [-8, 0, -8, 11, -5], [-6, 18, -5, -2, 8], [6, -8, -3, 0, 10], [0, 6, 7, 1, 5], [-8, 4, 0, -4, 7], [3, -8, 2, -14, 4], [-12, -8, 6, -4, -8], [13, -20, -10, -6, -10], [-10, -2, -1, 11, 1], [-18, 20, -10, -2, 0], [6, -4, -14, 0, 5], [-2, 14, 8, 0, 5], [12, -8, -6, 0, -7], [-29, 0, -5, -8, 12], [8, 10, 5, 7, -8], [-2, 2, 8, 4, 5], [-6, 0, 6, 2, -22], [-5, 10, 11, 0, 15], [-10, 8, 6, -5, 2], [-10, 2, -13, -8, -3], [-9, -8, -1, -2, -2], [-40, -4, -3, 4, 0], [-20, -6, 1, -3, -17], [6, -4, -7, -8, 4], [-11, -10, 3, -12, 1], [-4, 0, -10, -6, 13], [-16, -10, -6, -2, -1], [16, 6, -5, -3, 13], [-35, 10, 11, 0, 9], [-10, 14, 1, -13, 14], [2, 0, 2, 4, -19], [18, 4, -7, 0, 10], [12, 10, 9, 14, 2], [19, -8, -8, 6, -22], [-12, -4, -7, -8, 12], [20, 4, 3, 10, -6], [24, -12, 0, 6, 0], [-20, -2, 5, -6, 8], [-6, -22, -12, 4, -11], [14, 6, -7, -2, 5], [14, -10, 2, -4, -5], [-14, 24, 5, 8, 0], [-4, -2, -9, -10, -11], [2, -4, -3, -8, 12], [18, 14, 5, 1, 3], [-12, 10, -7, -2, 0], [-18, 0, 0, 12, -12], [-14, 2, 3, -12, 16], [15, -12, -2, -2, 12], [21, 0, -3, -12, 0], [-2, -12, -4, 2, -12], [-16, -22, 3, -5, -15], [-6, -16, -2, 0, -19], [-4, 18, 4, -8, 15], [-12, -14, 7, -4, -6], [1, 8, 10, 10, -10], [-8, 10, 1, 15, -13], [-10, 2, 9, 6, 19], [7, -8, 6, 2, -8], [-14, -2, 5, -16, -1], [40, -6, 4, -4, -1], [22, -8, 14, 8, -1], [-12, 10, 1, -6, -5], [15, 8, 15, 4, -4], [-8, 0, -9, 10, -6], [-8, 4, -8, 0, -13], [-20, -2, 9, 2, 4], [22, 2, 1, -11, -1], [27, -10, 5, 2, 7], [-12, 0, 4, 10, -15], [22, -12, -12, 9, -21], [-27, -2, -3, 14, -7], [-12, 0, -8, 10, -6], [-2, 4, 0, 8, 3], [14, 14, 4, -6, 3], [4, 0, 1, 2, -10], [0, 18, 1, 2, 7], [-4, -14, 1, 10, -1], [-24, 16, 6, -3, 4], [12, 0, 0, -16, 17], [-22, 12, 6, -6, 21], [12, 4, 11, -6, 0], [26, -4, 2, 10, -24], [-2, 14, 7, 8, 10], [10, 16, 6, 7, 15], [20, 4, 0, -5, 8], [-4, -6, -9, 5, -7], [-16, -22, -7, 4, -3], [-10, 4, -2, 2, -12], [10, 8, -3, -10, 4], [-14, -8, 2, -9, 8], [19, -22, -12, -12, 1], [-3, -22, 1, 14, -15], [-18, -8, 0, -4, 5], [-25, 18, 7, -4, 7], [26, 0, 0, -13, 14], [-9, -30, 0, 6, -9], [0, -16, -1, 10, 0], [4, 0, 14, 2, -9], [36, 4, 2, -10, 8], [-28, -6, -6, -4, 5], [10, -16, -4, 16, -24], [-24, -6, 12, -4, 7], [22, -22, -1, 4, -18], [-16, 0, 10, 0, 8], [-22, -24, 0, 4, -8], [14, -2, 5, -8, 7], [-5, -16, -6, -4, -2], [2, 8, -9, -4, -18], [6, 20, 2, -5, 12], [22, 0, 9, 18, 0], [-18, 0, -16, 0, -5], [10, 0, 8, -4, 12], [-14, -6, -1, 13, -7], [-28, -2, -11, 5, 1], [-20, -4, -9, -4, 18], [-10, 10, -2, 2, -3], [-14, -16, -18, -6, 15], [-34, -2, 3, 0, -12], [-14, -12, -4, 16, -13], [-2, 8, 6, -20, 8], [18, -6, -7, 2, 3], [-32, 6, -7, -17, 11], [4, 6, 1, 0, -14], [-13, 8, 0, -16, 36], [6, -14, -7, -10, 11], [4, 6, 3, 9, -3], [-21, -16, -5, -2, -14], [-5, -4, 0, 2, -14], [-8, -14, -3, 4, -8], [-18, 0, -7, -4, 10], [-26, -2, 1, -7, -12], [2, 4, -2, 12, -2], [9, -6, -15, 4, -13], [-14, 22, 15, 2, 16], [5, 18, 2, 8, 9], [28, -4, 10, 14, -23], [2, -10, -19, 8, -5], [27, -24, 8, 6, 10], [24, -8, -3, 8, -4], [-34, -8, 0, 10, -21], [18, 0, 18, 6, 6], [-6, 2, 11, -10, 16], [-12, 0, 2, -28, 12], [4, -10, -5, -2, -28], [-3, 0, -3, -16, -4], [-4, 4, 18, -3, 6], [-31, -6, -3, 0, -9], [-8, -30, -17, -18, -14], [8, 0, -3, -18, 14], [-32, 4, -3, 2, 0], [-4, -14, -23, -6, -2], [-10, -6, -13, 11, -12], [28, -8, -1, 6, -6], [-10, 10, 11, 0, 21], [26, 8, 10, 13, 6], [46, 0, 0, 0, 0], [5, 10, 2, -6, 15], [6, 14, 9, 10, 8], [-22, 0, -11, -2, -12], [-23, 12, -1, 24, -14], [30, 6, 9, 19, -14], [-10, -14, -10, -10, 1], [-3, 16, -6, -6, 2], [26, 0, 0, 20, -7], [-26, 2, 13, -5, 1], [6, 14, 11, 8, 6], [13, -2, 0, 6, -19], [-24, 18, -2, -6, -1], [36, 4, 7, 10, -6], [-10, 10, 11, -10, 23], [8, -6, 11, 7, 19], [-10, -6, -3, -13, -25], [28, 10, 9, 9, 2], [-20, 12, 4, -10, 4], [-10, -10, 0, -2, -21], [-20, 10, -5, -6, -5], [-10, 4, 2, -16, 5], [22, -10, -2, -14, 1], [31, 14, 10, -18, -3], [22, -8, -8, 8, 12], [-22, 20, 0, 14, 0], [-24, 16, 5, 12, -8], [8, -2, -1, 0, 3], [15, 10, 3, 0, -1], [-6, 20, -14, -7, 8], [-10, -10, 17, -5, -6], [12, -12, -9, -10, -14], [6, -24, 12, 4, 1], [30, 6, 8, 20, 9], [-14, 10, -7, 4, 13], [12, -20, 4, 8, -21], [15, -12, 0, -4, 20], [22, -4, 2, 8, 22], [-2, 24, -2, -13, 26], [-32, 18, 7, 9, 1], [-33, 12, -14, -14, 20], [-4, 6, 11, -13, -6], [-18, 8, 8, 10, 6], [-13, -18, -9, -12, -1], [-18, 6, 7, -6, 18], [-18, -22, -7, -17, -23], [-14, -2, 13, -8, -10], [11, -10, -9, 2, -27], [12, -14, 3, -2, 27], [20, -6, -3, -19, 11], [-21, -14, 0, 0, 17], [30, -12, -4, 2, -5], [-4, -6, 1, 8, 10], [2, 4, 0, 3, 0], [8, -24, -6, -14, -2], [-10, -4, -6, 2, -3], [34, -8, 3, 20, -14], [20, -6, 11, 13, -11], [-36, -10, -5, -8, -17], [-2, -6, 2, 16, -25], [-30, -12, -13, 0, 6], [25, -10, -12, -6, 13], [-18, -8, -13, 16, -12], [-18, 8, 4, -8, 4], [8, 18, -3, -7, 14], [-38, 10, -2, 16, -17], [-40, -2, -1, 7, 10], [2, -8, -2, 24, -3], [0, 18, 21, 7, 1], [13, 4, -6, 2, -26], [20, -8, -2, -7, -18], [6, -12, 4, 8, -14], [-30, 22, 3, -4, 19], [24, 8, -3, 12, -16], [-2, -28, -2, 20, -23], [26, 12, 0, -18, -3], [20, 12, 6, -19, 18], [-15, 0, -6, 14, 2], [-32, -16, -16, 3, 8], [-65, 18, -4, 0, 1], [-18, -18, -7, -6, -11], [34, 8, 3, 0, 0], [22, 8, 11, 0, -16], [-4, 12, 17, 10, -8], [28, 16, 20, 17, -4], [20, -14, 20, 10, -7], [-38, 12, -19, -10, 8], [20, 16, 2, -4, -1], [-50, -4, -15, -16, 18], [18, 16, 4, -6, -14], [-16, 0, -6, -1, 2], [16, -24, 10, 11, 8], [-3, 20, 0, 4, 2], [-10, 0, -1, 4, -14], [44, 10, 17, 18, 3], [-12, 0, 12, -16, 5], [27, -8, 23, 10, -14], [29, 4, 10, 0, 0], [18, -20, 4, 16, 9], [-20, -4, -20, 2, 5], [52, -8, -10, -8, 4], [-14, 24, 20, 14, 17], [18, 28, -7, 0, 16], [2, 2, 4, -12, 9], [-20, 6, -1, -6, -11], [16, 10, -11, 2, 27], [-26, -2, 11, -16, 11], [-42, 0, -21, -10, 10], [32, -18, -5, 8, -24], [14, -4, 6, 12, -38], [6, -12, -2, -12, 17], [-22, -10, -9, 16, 12]];
    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_bw();".
// 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_bw(:prec:=5)
    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_bw();".
// 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_bw( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_8281_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![3, -12, -17, -1, 4, 1]>,<3,R![4, 16, 0, -9, 0, 1]>,<5,R![48, 48, -20, -15, 2, 1]>,<11,R![-33, -45, 22, 36, 11, 1]>,<17,R![429, 93, -106, -22, 5, 1]>],Snew);
    return Vf;
end function;