Properties

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

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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]))
 
sage: chi.galois_orbit()
 
pari: [g,chi] = znchar(Mod(17,185))
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

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)
 

Related number fields

Field of values: \(\Q(\zeta_{36})\)
Fixed field: 36.36.57444765302724909954814307473256133361395843470561362005770206451416015625.2

Characters in Galois orbit

Character \(-1\) \(1\) \(2\) \(3\) \(4\) \(6\) \(7\) \(8\) \(9\) \(11\) \(12\) \(13\)
\(\chi_{185}(17,\cdot)\) \(1\) \(1\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{29}{36}\right)\) \(e\left(\frac{8}{9}\right)\) \(i\) \(e\left(\frac{17}{36}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{11}{18}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{25}{36}\right)\) \(e\left(\frac{8}{9}\right)\)
\(\chi_{185}(18,\cdot)\) \(1\) \(1\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{19}{36}\right)\) \(e\left(\frac{4}{9}\right)\) \(-i\) \(e\left(\frac{31}{36}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{18}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{35}{36}\right)\) \(e\left(\frac{4}{9}\right)\)
\(\chi_{185}(22,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{5}{36}\right)\) \(e\left(\frac{2}{9}\right)\) \(i\) \(e\left(\frac{29}{36}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{5}{18}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{13}{36}\right)\) \(e\left(\frac{2}{9}\right)\)
\(\chi_{185}(42,\cdot)\) \(1\) \(1\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{13}{36}\right)\) \(e\left(\frac{7}{9}\right)\) \(i\) \(e\left(\frac{25}{36}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{13}{18}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{5}{36}\right)\) \(e\left(\frac{7}{9}\right)\)
\(\chi_{185}(72,\cdot)\) \(1\) \(1\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{17}{36}\right)\) \(e\left(\frac{5}{9}\right)\) \(i\) \(e\left(\frac{5}{36}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{17}{18}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{1}{36}\right)\) \(e\left(\frac{5}{9}\right)\)
\(\chi_{185}(87,\cdot)\) \(1\) \(1\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{25}{36}\right)\) \(e\left(\frac{1}{9}\right)\) \(i\) \(e\left(\frac{1}{36}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{7}{18}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{29}{36}\right)\) \(e\left(\frac{1}{9}\right)\)
\(\chi_{185}(98,\cdot)\) \(1\) \(1\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{7}{36}\right)\) \(e\left(\frac{1}{9}\right)\) \(-i\) \(e\left(\frac{19}{36}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{7}{18}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{11}{36}\right)\) \(e\left(\frac{1}{9}\right)\)
\(\chi_{185}(113,\cdot)\) \(1\) \(1\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{35}{36}\right)\) \(e\left(\frac{5}{9}\right)\) \(-i\) \(e\left(\frac{23}{36}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{17}{18}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{19}{36}\right)\) \(e\left(\frac{5}{9}\right)\)
\(\chi_{185}(143,\cdot)\) \(1\) \(1\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{31}{36}\right)\) \(e\left(\frac{7}{9}\right)\) \(-i\) \(e\left(\frac{7}{36}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{13}{18}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{23}{36}\right)\) \(e\left(\frac{7}{9}\right)\)
\(\chi_{185}(163,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{23}{36}\right)\) \(e\left(\frac{2}{9}\right)\) \(-i\) \(e\left(\frac{11}{36}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{5}{18}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{31}{36}\right)\) \(e\left(\frac{2}{9}\right)\)
\(\chi_{185}(167,\cdot)\) \(1\) \(1\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{1}{36}\right)\) \(e\left(\frac{4}{9}\right)\) \(i\) \(e\left(\frac{13}{36}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{18}\right)\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{17}{36}\right)\) \(e\left(\frac{4}{9}\right)\)
\(\chi_{185}(168,\cdot)\) \(1\) \(1\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{11}{36}\right)\) \(e\left(\frac{8}{9}\right)\) \(-i\) \(e\left(\frac{35}{36}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{11}{18}\right)\) \(e\left(\frac{5}{6}\right)\) \(e\left(\frac{7}{36}\right)\) \(e\left(\frac{8}{9}\right)\)