Properties

Label 667.10
Modulus $667$
Conductor $667$
Order $308$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(667, base_ring=CyclotomicField(308))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([42,253]))
 
pari: [g,chi] = znchar(Mod(10,667))
 

Basic properties

Modulus: \(667\)
Conductor: \(667\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(308\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 667.x

\(\chi_{667}(10,\cdot)\) \(\chi_{667}(11,\cdot)\) \(\chi_{667}(14,\cdot)\) \(\chi_{667}(15,\cdot)\) \(\chi_{667}(19,\cdot)\) \(\chi_{667}(21,\cdot)\) \(\chi_{667}(37,\cdot)\) \(\chi_{667}(40,\cdot)\) \(\chi_{667}(43,\cdot)\) \(\chi_{667}(44,\cdot)\) \(\chi_{667}(56,\cdot)\) \(\chi_{667}(60,\cdot)\) \(\chi_{667}(61,\cdot)\) \(\chi_{667}(66,\cdot)\) \(\chi_{667}(76,\cdot)\) \(\chi_{667}(79,\cdot)\) \(\chi_{667}(84,\cdot)\) \(\chi_{667}(89,\cdot)\) \(\chi_{667}(90,\cdot)\) \(\chi_{667}(97,\cdot)\) \(\chi_{667}(102,\cdot)\) \(\chi_{667}(106,\cdot)\) \(\chi_{667}(113,\cdot)\) \(\chi_{667}(126,\cdot)\) \(\chi_{667}(130,\cdot)\) \(\chi_{667}(134,\cdot)\) \(\chi_{667}(135,\cdot)\) \(\chi_{667}(143,\cdot)\) \(\chi_{667}(148,\cdot)\) \(\chi_{667}(153,\cdot)\) ...

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: $\Q(\zeta_{308})$
Fixed field: Number field defined by a degree 308 polynomial (not computed)

Values on generators

\((465,553)\) → \((e\left(\frac{3}{22}\right),e\left(\frac{23}{28}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(1\)\(1\)\(e\left(\frac{29}{308}\right)\)\(e\left(\frac{89}{308}\right)\)\(e\left(\frac{29}{154}\right)\)\(e\left(\frac{16}{77}\right)\)\(e\left(\frac{59}{154}\right)\)\(e\left(\frac{69}{154}\right)\)\(e\left(\frac{87}{308}\right)\)\(e\left(\frac{89}{154}\right)\)\(e\left(\frac{93}{308}\right)\)\(e\left(\frac{235}{308}\right)\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 667 }(10,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{667}(10,\cdot)) = \sum_{r\in \Z/667\Z} \chi_{667}(10,r) e\left(\frac{2r}{667}\right) = -24.2292862795+-8.9410114857i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 667 }(10,·),\chi_{ 667 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{667}(10,\cdot),\chi_{667}(1,\cdot)) = \sum_{r\in \Z/667\Z} \chi_{667}(10,r) \chi_{667}(1,1-r) = 1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 667 }(10,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{667}(10,·)) = \sum_{r \in \Z/667\Z} \chi_{667}(10,r) e\left(\frac{1 r + 2 r^{-1}}{667}\right) = -22.5455977824+-6.8705351944i \)