Properties

Label 185.bd
Modulus $185$
Conductor $185$
Order $36$
Real no
Primitive yes
Minimal yes
Parity odd

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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([9,32]))
 
sage: chi.galois_orbit()
 
pari: [g,chi] = znchar(Mod(7,185))
 
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: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Related number fields

Field of values: \(\Q(\zeta_{36})\)
Fixed field: 36.0.1134084166835624937413663701523229292665702790961273014545440673828125.1

Characters in Galois orbit

Character \(-1\) \(1\) \(2\) \(3\) \(4\) \(6\) \(7\) \(8\) \(9\) \(11\) \(12\) \(13\)
\(\chi_{185}(7,\cdot)\) \(-1\) \(1\) \(e\left(\frac{5}{36}\right)\) \(e\left(\frac{31}{36}\right)\) \(e\left(\frac{5}{18}\right)\) \(1\) \(e\left(\frac{25}{36}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{13}{18}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{5}{36}\right)\) \(e\left(\frac{19}{36}\right)\)
\(\chi_{185}(12,\cdot)\) \(-1\) \(1\) \(e\left(\frac{1}{36}\right)\) \(e\left(\frac{35}{36}\right)\) \(e\left(\frac{1}{18}\right)\) \(1\) \(e\left(\frac{5}{36}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{17}{18}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{1}{36}\right)\) \(e\left(\frac{11}{36}\right)\)
\(\chi_{185}(33,\cdot)\) \(-1\) \(1\) \(e\left(\frac{11}{36}\right)\) \(e\left(\frac{25}{36}\right)\) \(e\left(\frac{11}{18}\right)\) \(1\) \(e\left(\frac{19}{36}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{7}{18}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{11}{36}\right)\) \(e\left(\frac{13}{36}\right)\)
\(\chi_{185}(53,\cdot)\) \(-1\) \(1\) \(e\left(\frac{31}{36}\right)\) \(e\left(\frac{5}{36}\right)\) \(e\left(\frac{13}{18}\right)\) \(1\) \(e\left(\frac{11}{36}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{5}{18}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{31}{36}\right)\) \(e\left(\frac{17}{36}\right)\)
\(\chi_{185}(83,\cdot)\) \(-1\) \(1\) \(e\left(\frac{7}{36}\right)\) \(e\left(\frac{29}{36}\right)\) \(e\left(\frac{7}{18}\right)\) \(1\) \(e\left(\frac{35}{36}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{11}{18}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{7}{36}\right)\) \(e\left(\frac{5}{36}\right)\)
\(\chi_{185}(107,\cdot)\) \(-1\) \(1\) \(e\left(\frac{29}{36}\right)\) \(e\left(\frac{7}{36}\right)\) \(e\left(\frac{11}{18}\right)\) \(1\) \(e\left(\frac{1}{36}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{7}{18}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{29}{36}\right)\) \(e\left(\frac{31}{36}\right)\)
\(\chi_{185}(108,\cdot)\) \(-1\) \(1\) \(e\left(\frac{35}{36}\right)\) \(e\left(\frac{1}{36}\right)\) \(e\left(\frac{17}{18}\right)\) \(1\) \(e\left(\frac{31}{36}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{1}{18}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{35}{36}\right)\) \(e\left(\frac{25}{36}\right)\)
\(\chi_{185}(118,\cdot)\) \(-1\) \(1\) \(e\left(\frac{23}{36}\right)\) \(e\left(\frac{13}{36}\right)\) \(e\left(\frac{5}{18}\right)\) \(1\) \(e\left(\frac{7}{36}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{13}{18}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{23}{36}\right)\) \(e\left(\frac{1}{36}\right)\)
\(\chi_{185}(123,\cdot)\) \(-1\) \(1\) \(e\left(\frac{19}{36}\right)\) \(e\left(\frac{17}{36}\right)\) \(e\left(\frac{1}{18}\right)\) \(1\) \(e\left(\frac{23}{36}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{17}{18}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{19}{36}\right)\) \(e\left(\frac{29}{36}\right)\)
\(\chi_{185}(127,\cdot)\) \(-1\) \(1\) \(e\left(\frac{13}{36}\right)\) \(e\left(\frac{23}{36}\right)\) \(e\left(\frac{13}{18}\right)\) \(1\) \(e\left(\frac{29}{36}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{5}{18}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{13}{36}\right)\) \(e\left(\frac{35}{36}\right)\)
\(\chi_{185}(157,\cdot)\) \(-1\) \(1\) \(e\left(\frac{25}{36}\right)\) \(e\left(\frac{11}{36}\right)\) \(e\left(\frac{7}{18}\right)\) \(1\) \(e\left(\frac{17}{36}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{11}{18}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{25}{36}\right)\) \(e\left(\frac{23}{36}\right)\)
\(\chi_{185}(182,\cdot)\) \(-1\) \(1\) \(e\left(\frac{17}{36}\right)\) \(e\left(\frac{19}{36}\right)\) \(e\left(\frac{17}{18}\right)\) \(1\) \(e\left(\frac{13}{36}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{1}{18}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{17}{36}\right)\) \(e\left(\frac{7}{36}\right)\)