Properties

Label 184.9
Modulus $184$
Conductor $23$
Order $11$
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(184, base_ring=CyclotomicField(22))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,0,10]))
 
pari: [g,chi] = znchar(Mod(9,184))
 

Basic properties

Modulus: \(184\)
Conductor: \(23\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(11\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{23}(9,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 184.i

\(\chi_{184}(9,\cdot)\) \(\chi_{184}(25,\cdot)\) \(\chi_{184}(41,\cdot)\) \(\chi_{184}(49,\cdot)\) \(\chi_{184}(73,\cdot)\) \(\chi_{184}(81,\cdot)\) \(\chi_{184}(105,\cdot)\) \(\chi_{184}(121,\cdot)\) \(\chi_{184}(169,\cdot)\) \(\chi_{184}(177,\cdot)\)

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: \(\Q(\zeta_{11})\)
Fixed field: \(\Q(\zeta_{23})^+\)

Values on generators

\((47,93,97)\) → \((1,1,e\left(\frac{5}{11}\right))\)

Values

\(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\(1\)\(1\)\(e\left(\frac{3}{11}\right)\)\(e\left(\frac{5}{11}\right)\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{6}{11}\right)\)\(e\left(\frac{1}{11}\right)\)\(e\left(\frac{4}{11}\right)\)\(e\left(\frac{8}{11}\right)\)\(e\left(\frac{2}{11}\right)\)\(e\left(\frac{9}{11}\right)\)\(e\left(\frac{10}{11}\right)\)
value at e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 184 }(9,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{184}(9,\cdot)) = \sum_{r\in \Z/184\Z} \chi_{184}(9,r) e\left(\frac{r}{92}\right) = 0.0 \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 184 }(9,·),\chi_{ 184 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{184}(9,\cdot),\chi_{184}(1,\cdot)) = \sum_{r\in \Z/184\Z} \chi_{184}(9,r) \chi_{184}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 184 }(9,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{184}(9,·)) = \sum_{r \in \Z/184\Z} \chi_{184}(9,r) e\left(\frac{1 r + 2 r^{-1}}{184}\right) = -0.0 \)