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Properties

Label 460.181
Modulus $460$
Conductor $23$
Order $22$
Real no
Primitive no
Minimal yes
Parity odd

Related objects

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(460, base_ring=CyclotomicField(22)) M = H._module chi = DirichletCharacter(H, M([0,0,5]))
 
Copy content pari:[g,chi] = znchar(Mod(181,460))
 

Basic properties

Modulus: \(460\)
Conductor: \(23\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(22\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{23}(20,\cdot)\)
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Galois orbit 460.p

\(\chi_{460}(21,\cdot)\) \(\chi_{460}(61,\cdot)\) \(\chi_{460}(181,\cdot)\) \(\chi_{460}(201,\cdot)\) \(\chi_{460}(221,\cdot)\) \(\chi_{460}(241,\cdot)\) \(\chi_{460}(281,\cdot)\) \(\chi_{460}(341,\cdot)\) \(\chi_{460}(401,\cdot)\) \(\chi_{460}(421,\cdot)\)

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

Related number fields

Field of values: \(\Q(\zeta_{11})\)
Fixed field: Number field defined by a degree 22 polynomial

Values on generators

\((231,277,281)\) → \((1,1,e\left(\frac{5}{22}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(27\)\(29\)
\( \chi_{ 460 }(181, a) \) \(-1\)\(1\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{7}{22}\right)\)\(e\left(\frac{3}{11}\right)\)\(e\left(\frac{1}{22}\right)\)\(e\left(\frac{2}{11}\right)\)\(e\left(\frac{13}{22}\right)\)\(e\left(\frac{9}{22}\right)\)\(e\left(\frac{21}{22}\right)\)\(e\left(\frac{10}{11}\right)\)\(e\left(\frac{1}{11}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 460 }(181,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

Copy content sage:chi.gauss_sum(a)
 
Copy content pari:znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 460 }(181,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

Copy content sage:chi.jacobi_sum(n)
 
\( J(\chi_{ 460 }(181,·),\chi_{ 460 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

Copy content sage:chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 460 }(181,·)) \;\) at \(\; a,b = \) e.g. 1,2