Properties

Label 2736.he
Modulus $2736$
Conductor $2736$
Order $36$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2736, base_ring=CyclotomicField(36))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,9,30,32]))
 
chi.galois_orbit()
 
[g,chi] = znchar(Mod(5,2736))
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(2736\)
Conductor: \(2736\)
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: Number field defined by a degree 36 polynomial

Characters in Galois orbit

Character \(-1\) \(1\) \(5\) \(7\) \(11\) \(13\) \(17\) \(23\) \(25\) \(29\) \(31\) \(35\)
\(\chi_{2736}(5,\cdot)\) \(-1\) \(1\) \(e\left(\frac{23}{36}\right)\) \(e\left(\frac{1}{6}\right)\) \(-i\) \(e\left(\frac{31}{36}\right)\) \(e\left(\frac{7}{18}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{5}{18}\right)\) \(e\left(\frac{25}{36}\right)\) \(1\) \(e\left(\frac{29}{36}\right)\)
\(\chi_{2736}(101,\cdot)\) \(-1\) \(1\) \(e\left(\frac{19}{36}\right)\) \(e\left(\frac{5}{6}\right)\) \(-i\) \(e\left(\frac{35}{36}\right)\) \(e\left(\frac{5}{18}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{1}{18}\right)\) \(e\left(\frac{5}{36}\right)\) \(1\) \(e\left(\frac{13}{36}\right)\)
\(\chi_{2736}(149,\cdot)\) \(-1\) \(1\) \(e\left(\frac{35}{36}\right)\) \(e\left(\frac{1}{6}\right)\) \(-i\) \(e\left(\frac{19}{36}\right)\) \(e\left(\frac{13}{18}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{17}{18}\right)\) \(e\left(\frac{13}{36}\right)\) \(1\) \(e\left(\frac{5}{36}\right)\)
\(\chi_{2736}(389,\cdot)\) \(-1\) \(1\) \(e\left(\frac{7}{36}\right)\) \(e\left(\frac{5}{6}\right)\) \(-i\) \(e\left(\frac{11}{36}\right)\) \(e\left(\frac{17}{18}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{7}{18}\right)\) \(e\left(\frac{17}{36}\right)\) \(1\) \(e\left(\frac{1}{36}\right)\)
\(\chi_{2736}(821,\cdot)\) \(-1\) \(1\) \(e\left(\frac{31}{36}\right)\) \(e\left(\frac{5}{6}\right)\) \(-i\) \(e\left(\frac{23}{36}\right)\) \(e\left(\frac{11}{18}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{13}{18}\right)\) \(e\left(\frac{29}{36}\right)\) \(1\) \(e\left(\frac{25}{36}\right)\)
\(\chi_{2736}(1157,\cdot)\) \(-1\) \(1\) \(e\left(\frac{11}{36}\right)\) \(e\left(\frac{1}{6}\right)\) \(-i\) \(e\left(\frac{7}{36}\right)\) \(e\left(\frac{1}{18}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{11}{18}\right)\) \(e\left(\frac{1}{36}\right)\) \(1\) \(e\left(\frac{17}{36}\right)\)
\(\chi_{2736}(1373,\cdot)\) \(-1\) \(1\) \(e\left(\frac{5}{36}\right)\) \(e\left(\frac{1}{6}\right)\) \(i\) \(e\left(\frac{13}{36}\right)\) \(e\left(\frac{7}{18}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{5}{18}\right)\) \(e\left(\frac{7}{36}\right)\) \(1\) \(e\left(\frac{11}{36}\right)\)
\(\chi_{2736}(1469,\cdot)\) \(-1\) \(1\) \(e\left(\frac{1}{36}\right)\) \(e\left(\frac{5}{6}\right)\) \(i\) \(e\left(\frac{17}{36}\right)\) \(e\left(\frac{5}{18}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{1}{18}\right)\) \(e\left(\frac{23}{36}\right)\) \(1\) \(e\left(\frac{31}{36}\right)\)
\(\chi_{2736}(1517,\cdot)\) \(-1\) \(1\) \(e\left(\frac{17}{36}\right)\) \(e\left(\frac{1}{6}\right)\) \(i\) \(e\left(\frac{1}{36}\right)\) \(e\left(\frac{13}{18}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{17}{18}\right)\) \(e\left(\frac{31}{36}\right)\) \(1\) \(e\left(\frac{23}{36}\right)\)
\(\chi_{2736}(1757,\cdot)\) \(-1\) \(1\) \(e\left(\frac{25}{36}\right)\) \(e\left(\frac{5}{6}\right)\) \(i\) \(e\left(\frac{29}{36}\right)\) \(e\left(\frac{17}{18}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{7}{18}\right)\) \(e\left(\frac{35}{36}\right)\) \(1\) \(e\left(\frac{19}{36}\right)\)
\(\chi_{2736}(2189,\cdot)\) \(-1\) \(1\) \(e\left(\frac{13}{36}\right)\) \(e\left(\frac{5}{6}\right)\) \(i\) \(e\left(\frac{5}{36}\right)\) \(e\left(\frac{11}{18}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{13}{18}\right)\) \(e\left(\frac{11}{36}\right)\) \(1\) \(e\left(\frac{7}{36}\right)\)
\(\chi_{2736}(2525,\cdot)\) \(-1\) \(1\) \(e\left(\frac{29}{36}\right)\) \(e\left(\frac{1}{6}\right)\) \(i\) \(e\left(\frac{25}{36}\right)\) \(e\left(\frac{1}{18}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{11}{18}\right)\) \(e\left(\frac{19}{36}\right)\) \(1\) \(e\left(\frac{35}{36}\right)\)