Properties

Label 185.2
Modulus $185$
Conductor $185$
Order $36$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(185, base_ring=CyclotomicField(36))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([9,1]))
 
pari: [g,chi] = znchar(Mod(2,185))
 

Basic properties

Modulus: \(185\)
Conductor: \(185\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(36\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 185.bc

\(\chi_{185}(2,\cdot)\) \(\chi_{185}(13,\cdot)\) \(\chi_{185}(32,\cdot)\) \(\chi_{185}(52,\cdot)\) \(\chi_{185}(57,\cdot)\) \(\chi_{185}(92,\cdot)\) \(\chi_{185}(93,\cdot)\) \(\chi_{185}(128,\cdot)\) \(\chi_{185}(133,\cdot)\) \(\chi_{185}(153,\cdot)\) \(\chi_{185}(172,\cdot)\) \(\chi_{185}(183,\cdot)\)

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: \(\Q(\zeta_{36})\)
Fixed field: 36.36.57444765302724909954814307473256133361395843470561362005770206451416015625.1

Values on generators

\((112,76)\) → \((i,e\left(\frac{1}{36}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\(1\)\(1\)\(e\left(\frac{5}{18}\right)\)\(e\left(\frac{17}{36}\right)\)\(e\left(\frac{5}{9}\right)\)\(-i\)\(e\left(\frac{5}{36}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{17}{18}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{1}{36}\right)\)\(e\left(\frac{1}{18}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 185 }(2,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 185 }(2,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 185 }(2,·),\chi_{ 185 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 185 }(2,·)) \;\) at \(\; a,b = \) e.g. 1,2