Properties

Label 966.229
Modulus $966$
Conductor $161$
Order $6$
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(966, base_ring=CyclotomicField(6))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,5,3]))
 
pari: [g,chi] = znchar(Mod(229,966))
 

Basic properties

Modulus: \(966\)
Conductor: \(161\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(6\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{161}(68,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 966.k

\(\chi_{966}(229,\cdot)\) \(\chi_{966}(367,\cdot)\)

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

\((323,829,925)\) → \((1,e\left(\frac{5}{6}\right),-1)\)

Values

\(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(19\)\(25\)\(29\)\(31\)\(37\)\(41\)
\(1\)\(1\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{5}{6}\right)\)\(-1\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{3}\right)\)\(1\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{1}{6}\right)\)\(-1\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\sqrt{-3}) \)
Fixed field: 6.6.204490769.1

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 966 }(229,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{966}(229,\cdot)) = \sum_{r\in \Z/966\Z} \chi_{966}(229,r) e\left(\frac{r}{483}\right) = 4.9043074384+-11.702468481i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 966 }(229,·),\chi_{ 966 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{966}(229,\cdot),\chi_{966}(1,\cdot)) = \sum_{r\in \Z/966\Z} \chi_{966}(229,r) \chi_{966}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 966 }(229,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{966}(229,·)) = \sum_{r \in \Z/966\Z} \chi_{966}(229,r) e\left(\frac{1 r + 2 r^{-1}}{966}\right) = 8.0511282774+-13.9449632347i \)