Properties

Label 209.23
Modulus $209$
Conductor $19$
Order $9$
Real no
Primitive no
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(209, base_ring=CyclotomicField(18))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,2]))
 
pari: [g,chi] = znchar(Mod(23,209))
 

Basic properties

Modulus: \(209\)
Conductor: \(19\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(9\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{19}(4,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 209.j

\(\chi_{209}(23,\cdot)\) \(\chi_{209}(100,\cdot)\) \(\chi_{209}(111,\cdot)\) \(\chi_{209}(177,\cdot)\) \(\chi_{209}(188,\cdot)\) \(\chi_{209}(199,\cdot)\)

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

Values on generators

\((134,78)\) → \((1,e\left(\frac{1}{9}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\(1\)\(1\)\(e\left(\frac{1}{9}\right)\)\(e\left(\frac{4}{9}\right)\)\(e\left(\frac{2}{9}\right)\)\(e\left(\frac{7}{9}\right)\)\(e\left(\frac{5}{9}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{8}{9}\right)\)\(e\left(\frac{8}{9}\right)\)\(e\left(\frac{2}{3}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{9})\)
Fixed field: \(\Q(\zeta_{19})^+\)

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 209 }(23,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{209}(23,\cdot)) = \sum_{r\in \Z/209\Z} \chi_{209}(23,r) e\left(\frac{2r}{209}\right) = 2.6696164072+-3.4457434957i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 209 }(23,·),\chi_{ 209 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{209}(23,\cdot),\chi_{209}(1,\cdot)) = \sum_{r\in \Z/209\Z} \chi_{209}(23,r) \chi_{209}(1,1-r) = -9 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 209 }(23,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{209}(23,·)) = \sum_{r \in \Z/209\Z} \chi_{209}(23,r) e\left(\frac{1 r + 2 r^{-1}}{209}\right) = 8.0507343123+2.9302276537i \)