Properties

Label 740.by
Modulus $740$
Conductor $185$
Order $36$
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(740, base_ring=CyclotomicField(36))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,9,26]))
 
chi.galois_orbit()
 
[g,chi] = znchar(Mod(77,740))
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(740\)
Conductor: \(185\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(36\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from 185.y
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: 36.0.1552561224397970539319305607385300901659347120825982756912708282470703125.1

Characters in Galois orbit

Character \(-1\) \(1\) \(3\) \(7\) \(9\) \(11\) \(13\) \(17\) \(19\) \(21\) \(23\) \(27\)
\(\chi_{740}(77,\cdot)\) \(-1\) \(1\) \(e\left(\frac{19}{36}\right)\) \(e\left(\frac{13}{36}\right)\) \(e\left(\frac{1}{18}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{25}{36}\right)\) \(e\left(\frac{11}{36}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{7}{12}\right)\)
\(\chi_{740}(173,\cdot)\) \(-1\) \(1\) \(e\left(\frac{17}{36}\right)\) \(e\left(\frac{23}{36}\right)\) \(e\left(\frac{17}{18}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{11}{36}\right)\) \(e\left(\frac{25}{36}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{5}{12}\right)\)
\(\chi_{740}(213,\cdot)\) \(-1\) \(1\) \(e\left(\frac{29}{36}\right)\) \(e\left(\frac{35}{36}\right)\) \(e\left(\frac{11}{18}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{23}{36}\right)\) \(e\left(\frac{13}{36}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{5}{12}\right)\)
\(\chi_{740}(317,\cdot)\) \(-1\) \(1\) \(e\left(\frac{23}{36}\right)\) \(e\left(\frac{29}{36}\right)\) \(e\left(\frac{5}{18}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{17}{36}\right)\) \(e\left(\frac{19}{36}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{11}{12}\right)\)
\(\chi_{740}(337,\cdot)\) \(-1\) \(1\) \(e\left(\frac{7}{36}\right)\) \(e\left(\frac{1}{36}\right)\) \(e\left(\frac{7}{18}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{13}{36}\right)\) \(e\left(\frac{23}{36}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{7}{12}\right)\)
\(\chi_{740}(373,\cdot)\) \(-1\) \(1\) \(e\left(\frac{1}{36}\right)\) \(e\left(\frac{31}{36}\right)\) \(e\left(\frac{1}{18}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{7}{36}\right)\) \(e\left(\frac{29}{36}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{1}{12}\right)\)
\(\chi_{740}(437,\cdot)\) \(-1\) \(1\) \(e\left(\frac{31}{36}\right)\) \(e\left(\frac{25}{36}\right)\) \(e\left(\frac{13}{18}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{1}{36}\right)\) \(e\left(\frac{35}{36}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{7}{12}\right)\) \(e\left(\frac{7}{12}\right)\)
\(\chi_{740}(613,\cdot)\) \(-1\) \(1\) \(e\left(\frac{5}{36}\right)\) \(e\left(\frac{11}{36}\right)\) \(e\left(\frac{5}{18}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{35}{36}\right)\) \(e\left(\frac{1}{36}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{5}{12}\right)\) \(e\left(\frac{5}{12}\right)\)
\(\chi_{740}(617,\cdot)\) \(-1\) \(1\) \(e\left(\frac{35}{36}\right)\) \(e\left(\frac{5}{36}\right)\) \(e\left(\frac{17}{18}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{29}{36}\right)\) \(e\left(\frac{7}{36}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{11}{12}\right)\)
\(\chi_{740}(633,\cdot)\) \(-1\) \(1\) \(e\left(\frac{25}{36}\right)\) \(e\left(\frac{19}{36}\right)\) \(e\left(\frac{7}{18}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{31}{36}\right)\) \(e\left(\frac{5}{36}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{1}{12}\right)\)
\(\chi_{740}(657,\cdot)\) \(-1\) \(1\) \(e\left(\frac{11}{36}\right)\) \(e\left(\frac{17}{36}\right)\) \(e\left(\frac{11}{18}\right)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{5}{36}\right)\) \(e\left(\frac{31}{36}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{11}{12}\right)\) \(e\left(\frac{11}{12}\right)\)
\(\chi_{740}(733,\cdot)\) \(-1\) \(1\) \(e\left(\frac{13}{36}\right)\) \(e\left(\frac{7}{36}\right)\) \(e\left(\frac{13}{18}\right)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{19}{36}\right)\) \(e\left(\frac{17}{36}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{1}{12}\right)\) \(e\left(\frac{1}{12}\right)\)