Properties

Label 667.439
Modulus $667$
Conductor $667$
Order $154$
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(154))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([14,11]))
 
pari: [g,chi] = znchar(Mod(439,667))
 

Basic properties

Modulus: \(667\)
Conductor: \(667\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(154\)
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.u

\(\chi_{667}(4,\cdot)\) \(\chi_{667}(6,\cdot)\) \(\chi_{667}(9,\cdot)\) \(\chi_{667}(13,\cdot)\) \(\chi_{667}(35,\cdot)\) \(\chi_{667}(62,\cdot)\) \(\chi_{667}(64,\cdot)\) \(\chi_{667}(71,\cdot)\) \(\chi_{667}(96,\cdot)\) \(\chi_{667}(100,\cdot)\) \(\chi_{667}(121,\cdot)\) \(\chi_{667}(150,\cdot)\) \(\chi_{667}(151,\cdot)\) \(\chi_{667}(154,\cdot)\) \(\chi_{667}(167,\cdot)\) \(\chi_{667}(179,\cdot)\) \(\chi_{667}(187,\cdot)\) \(\chi_{667}(196,\cdot)\) \(\chi_{667}(209,\cdot)\) \(\chi_{667}(216,\cdot)\) \(\chi_{667}(225,\cdot)\) \(\chi_{667}(236,\cdot)\) \(\chi_{667}(238,\cdot)\) \(\chi_{667}(265,\cdot)\) \(\chi_{667}(266,\cdot)\) \(\chi_{667}(294,\cdot)\) \(\chi_{667}(303,\cdot)\) \(\chi_{667}(312,\cdot)\) \(\chi_{667}(324,\cdot)\) \(\chi_{667}(325,\cdot)\) ...

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: $\Q(\zeta_{77})$
Fixed field: Number field defined by a degree 154 polynomial (not computed)

Values on generators

\((465,553)\) → \((e\left(\frac{1}{11}\right),e\left(\frac{1}{14}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(1\)\(1\)\(e\left(\frac{39}{154}\right)\)\(e\left(\frac{125}{154}\right)\)\(e\left(\frac{39}{77}\right)\)\(e\left(\frac{51}{77}\right)\)\(e\left(\frac{5}{77}\right)\)\(e\left(\frac{45}{77}\right)\)\(e\left(\frac{117}{154}\right)\)\(e\left(\frac{48}{77}\right)\)\(e\left(\frac{141}{154}\right)\)\(e\left(\frac{93}{154}\right)\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 667 }(439,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{667}(439,\cdot)) = \sum_{r\in \Z/667\Z} \chi_{667}(439,r) e\left(\frac{2r}{667}\right) = 7.539616688+24.7012991601i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 667 }(439,·),\chi_{ 667 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{667}(439,\cdot),\chi_{667}(1,\cdot)) = \sum_{r\in \Z/667\Z} \chi_{667}(439,r) \chi_{667}(1,1-r) = 1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 667 }(439,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{667}(439,·)) = \sum_{r \in \Z/667\Z} \chi_{667}(439,r) e\left(\frac{1 r + 2 r^{-1}}{667}\right) = -9.8969782182+-10.1009638243i \)