Properties

Label 185.17
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([9,7]))
 
pari: [g,chi] = znchar(Mod(17,185))
 

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.z

\(\chi_{185}(17,\cdot)\) \(\chi_{185}(18,\cdot)\) \(\chi_{185}(22,\cdot)\) \(\chi_{185}(42,\cdot)\) \(\chi_{185}(72,\cdot)\) \(\chi_{185}(87,\cdot)\) \(\chi_{185}(98,\cdot)\) \(\chi_{185}(113,\cdot)\) \(\chi_{185}(143,\cdot)\) \(\chi_{185}(163,\cdot)\) \(\chi_{185}(167,\cdot)\) \(\chi_{185}(168,\cdot)\)

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: 36.36.57444765302724909954814307473256133361395843470561362005770206451416015625.2

Values on generators

\((112,76)\) → \((i,e\left(\frac{7}{36}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\(1\)\(1\)\(e\left(\frac{4}{9}\right)\)\(e\left(\frac{29}{36}\right)\)\(e\left(\frac{8}{9}\right)\)\(i\)\(e\left(\frac{17}{36}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{11}{18}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{25}{36}\right)\)\(e\left(\frac{8}{9}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 185 }(17,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 185 }(17,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 185 }(17,·),\chi_{ 185 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 185 }(17,·)) \;\) at \(\; a,b = \) e.g. 1,2