Properties

Label 230.9
Modulus $230$
Conductor $115$
Order $22$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(230, base_ring=CyclotomicField(22))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([11,10]))
 
pari: [g,chi] = znchar(Mod(9,230))
 

Basic properties

Modulus: \(230\)
Conductor: \(115\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(22\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{115}(9,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 230.j

\(\chi_{230}(9,\cdot)\) \(\chi_{230}(29,\cdot)\) \(\chi_{230}(39,\cdot)\) \(\chi_{230}(49,\cdot)\) \(\chi_{230}(59,\cdot)\) \(\chi_{230}(119,\cdot)\) \(\chi_{230}(169,\cdot)\) \(\chi_{230}(179,\cdot)\) \(\chi_{230}(209,\cdot)\) \(\chi_{230}(219,\cdot)\)

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: 22.22.83796671451884098775580820361328125.1

Values on generators

\((47,51)\) → \((-1,e\left(\frac{5}{11}\right))\)

Values

\(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(27\)\(29\)
\(1\)\(1\)\(e\left(\frac{17}{22}\right)\)\(e\left(\frac{3}{22}\right)\)\(e\left(\frac{6}{11}\right)\)\(e\left(\frac{1}{11}\right)\)\(e\left(\frac{19}{22}\right)\)\(e\left(\frac{15}{22}\right)\)\(e\left(\frac{9}{11}\right)\)\(e\left(\frac{10}{11}\right)\)\(e\left(\frac{7}{22}\right)\)\(e\left(\frac{2}{11}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 230 }(9,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 230 }(9,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 230 }(9,·),\chi_{ 230 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 230 }(9,·)) \;\) at \(\; a,b = \) e.g. 1,2