Properties

Label 210.23
Modulus $210$
Conductor $105$
Order $12$
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(210)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([6,9,4]))
 
pari: [g,chi] = znchar(Mod(23,210))
 

Basic properties

Modulus: \(210\)
Conductor: \(105\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(12\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{105}(23,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 210.x

\(\chi_{210}(23,\cdot)\) \(\chi_{210}(53,\cdot)\) \(\chi_{210}(107,\cdot)\) \(\chi_{210}(137,\cdot)\)

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

\((71,127,31)\) → \((-1,-i,e\left(\frac{1}{3}\right))\)

Values

\(-1\)\(1\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\(1\)\(1\)\(e\left(\frac{5}{6}\right)\)\(i\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{5}{12}\right)\)\(1\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{5}{12}\right)\)\(-1\)\(i\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{12})\)
Fixed field: 12.12.8208085798828125.1

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 210 }(23,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{210}(23,\cdot)) = \sum_{r\in \Z/210\Z} \chi_{210}(23,r) e\left(\frac{r}{105}\right) = 5.4169550446+-8.6980801355i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 210 }(23,·),\chi_{ 210 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{210}(23,\cdot),\chi_{210}(1,\cdot)) = \sum_{r\in \Z/210\Z} \chi_{210}(23,r) \chi_{210}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 210 }(23,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{210}(23,·)) = \sum_{r \in \Z/210\Z} \chi_{210}(23,r) e\left(\frac{1 r + 2 r^{-1}}{210}\right) = 0.0 \)