Properties

Modulus 181
Conductor 181
Order 90
Real no
Primitive yes
Minimal yes
Parity even
Orbit label 181.q

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(181)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([59]))
 
pari: [g,chi] = znchar(Mod(33,181))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Modulus = 181
Conductor = 181
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 90
Real = no
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
 
Primitive = yes
Minimal = yes
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 
Parity = even
Orbit label = 181.q
Orbit index = 17

Galois orbit

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

\(\chi_{181}(4,\cdot)\) \(\chi_{181}(11,\cdot)\) \(\chi_{181}(12,\cdot)\) \(\chi_{181}(20,\cdot)\) \(\chi_{181}(33,\cdot)\) \(\chi_{181}(37,\cdot)\) \(\chi_{181}(52,\cdot)\) \(\chi_{181}(55,\cdot)\) \(\chi_{181}(60,\cdot)\) \(\chi_{181}(79,\cdot)\) \(\chi_{181}(94,\cdot)\) \(\chi_{181}(100,\cdot)\) \(\chi_{181}(106,\cdot)\) \(\chi_{181}(111,\cdot)\) \(\chi_{181}(136,\cdot)\) \(\chi_{181}(137,\cdot)\) \(\chi_{181}(143,\cdot)\) \(\chi_{181}(147,\cdot)\) \(\chi_{181}(165,\cdot)\) \(\chi_{181}(166,\cdot)\) \(\chi_{181}(167,\cdot)\) \(\chi_{181}(168,\cdot)\) \(\chi_{181}(172,\cdot)\) \(\chi_{181}(178,\cdot)\)

Values on generators

\(2\) → \(e\left(\frac{59}{90}\right)\)

Values

-11234567891011
\(1\)\(1\)\(e\left(\frac{59}{90}\right)\)\(e\left(\frac{32}{45}\right)\)\(e\left(\frac{14}{45}\right)\)\(e\left(\frac{4}{15}\right)\)\(e\left(\frac{11}{30}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{29}{30}\right)\)\(e\left(\frac{19}{45}\right)\)\(e\left(\frac{83}{90}\right)\)\(e\left(\frac{29}{45}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{45})\)

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 181 }(33,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{181}(33,\cdot)) = \sum_{r\in \Z/181\Z} \chi_{181}(33,r) e\left(\frac{2r}{181}\right) = 4.215794213+-12.7760353457i \)

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 181 }(33,·),\chi_{ 181 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{181}(33,\cdot),\chi_{181}(1,\cdot)) = \sum_{r\in \Z/181\Z} \chi_{181}(33,r) \chi_{181}(1,1-r) = -1 \)

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 181 }(33,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{181}(33,·)) = \sum_{r \in \Z/181\Z} \chi_{181}(33,r) e\left(\frac{1 r + 2 r^{-1}}{181}\right) = 10.8724003885+-20.4480111525i \)