Properties

Label 243.i
Modulus $243$
Conductor $243$
Order $81$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(243, base_ring=CyclotomicField(162))
 
M = H._module
 
chi = DirichletCharacter(H, M([2]))
 
chi.galois_orbit()
 
[g,chi] = znchar(Mod(4,243))
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(243\)
Conductor: \(243\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(81\)
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: $\Q(\zeta_{81})$
Fixed field: Number field defined by a degree 81 polynomial

First 31 of 54 characters in Galois orbit

Character \(-1\) \(1\) \(2\) \(4\) \(5\) \(7\) \(8\) \(10\) \(11\) \(13\) \(14\) \(16\)
\(\chi_{243}(4,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{81}\right)\) \(e\left(\frac{2}{81}\right)\) \(e\left(\frac{23}{81}\right)\) \(e\left(\frac{70}{81}\right)\) \(e\left(\frac{1}{27}\right)\) \(e\left(\frac{8}{27}\right)\) \(e\left(\frac{40}{81}\right)\) \(e\left(\frac{8}{81}\right)\) \(e\left(\frac{71}{81}\right)\) \(e\left(\frac{4}{81}\right)\)
\(\chi_{243}(7,\cdot)\) \(1\) \(1\) \(e\left(\frac{35}{81}\right)\) \(e\left(\frac{70}{81}\right)\) \(e\left(\frac{76}{81}\right)\) \(e\left(\frac{20}{81}\right)\) \(e\left(\frac{8}{27}\right)\) \(e\left(\frac{10}{27}\right)\) \(e\left(\frac{23}{81}\right)\) \(e\left(\frac{37}{81}\right)\) \(e\left(\frac{55}{81}\right)\) \(e\left(\frac{59}{81}\right)\)
\(\chi_{243}(13,\cdot)\) \(1\) \(1\) \(e\left(\frac{4}{81}\right)\) \(e\left(\frac{8}{81}\right)\) \(e\left(\frac{11}{81}\right)\) \(e\left(\frac{37}{81}\right)\) \(e\left(\frac{4}{27}\right)\) \(e\left(\frac{5}{27}\right)\) \(e\left(\frac{79}{81}\right)\) \(e\left(\frac{32}{81}\right)\) \(e\left(\frac{41}{81}\right)\) \(e\left(\frac{16}{81}\right)\)
\(\chi_{243}(16,\cdot)\) \(1\) \(1\) \(e\left(\frac{2}{81}\right)\) \(e\left(\frac{4}{81}\right)\) \(e\left(\frac{46}{81}\right)\) \(e\left(\frac{59}{81}\right)\) \(e\left(\frac{2}{27}\right)\) \(e\left(\frac{16}{27}\right)\) \(e\left(\frac{80}{81}\right)\) \(e\left(\frac{16}{81}\right)\) \(e\left(\frac{61}{81}\right)\) \(e\left(\frac{8}{81}\right)\)
\(\chi_{243}(22,\cdot)\) \(1\) \(1\) \(e\left(\frac{61}{81}\right)\) \(e\left(\frac{41}{81}\right)\) \(e\left(\frac{26}{81}\right)\) \(e\left(\frac{58}{81}\right)\) \(e\left(\frac{7}{27}\right)\) \(e\left(\frac{2}{27}\right)\) \(e\left(\frac{10}{81}\right)\) \(e\left(\frac{2}{81}\right)\) \(e\left(\frac{38}{81}\right)\) \(e\left(\frac{1}{81}\right)\)
\(\chi_{243}(25,\cdot)\) \(1\) \(1\) \(e\left(\frac{23}{81}\right)\) \(e\left(\frac{46}{81}\right)\) \(e\left(\frac{43}{81}\right)\) \(e\left(\frac{71}{81}\right)\) \(e\left(\frac{23}{27}\right)\) \(e\left(\frac{22}{27}\right)\) \(e\left(\frac{29}{81}\right)\) \(e\left(\frac{22}{81}\right)\) \(e\left(\frac{13}{81}\right)\) \(e\left(\frac{11}{81}\right)\)
\(\chi_{243}(31,\cdot)\) \(1\) \(1\) \(e\left(\frac{10}{81}\right)\) \(e\left(\frac{20}{81}\right)\) \(e\left(\frac{68}{81}\right)\) \(e\left(\frac{52}{81}\right)\) \(e\left(\frac{10}{27}\right)\) \(e\left(\frac{26}{27}\right)\) \(e\left(\frac{76}{81}\right)\) \(e\left(\frac{80}{81}\right)\) \(e\left(\frac{62}{81}\right)\) \(e\left(\frac{40}{81}\right)\)
\(\chi_{243}(34,\cdot)\) \(1\) \(1\) \(e\left(\frac{17}{81}\right)\) \(e\left(\frac{34}{81}\right)\) \(e\left(\frac{67}{81}\right)\) \(e\left(\frac{56}{81}\right)\) \(e\left(\frac{17}{27}\right)\) \(e\left(\frac{1}{27}\right)\) \(e\left(\frac{32}{81}\right)\) \(e\left(\frac{55}{81}\right)\) \(e\left(\frac{73}{81}\right)\) \(e\left(\frac{68}{81}\right)\)
\(\chi_{243}(40,\cdot)\) \(1\) \(1\) \(e\left(\frac{13}{81}\right)\) \(e\left(\frac{26}{81}\right)\) \(e\left(\frac{56}{81}\right)\) \(e\left(\frac{19}{81}\right)\) \(e\left(\frac{13}{27}\right)\) \(e\left(\frac{23}{27}\right)\) \(e\left(\frac{34}{81}\right)\) \(e\left(\frac{23}{81}\right)\) \(e\left(\frac{32}{81}\right)\) \(e\left(\frac{52}{81}\right)\)
\(\chi_{243}(43,\cdot)\) \(1\) \(1\) \(e\left(\frac{65}{81}\right)\) \(e\left(\frac{49}{81}\right)\) \(e\left(\frac{37}{81}\right)\) \(e\left(\frac{14}{81}\right)\) \(e\left(\frac{11}{27}\right)\) \(e\left(\frac{7}{27}\right)\) \(e\left(\frac{8}{81}\right)\) \(e\left(\frac{34}{81}\right)\) \(e\left(\frac{79}{81}\right)\) \(e\left(\frac{17}{81}\right)\)
\(\chi_{243}(49,\cdot)\) \(1\) \(1\) \(e\left(\frac{70}{81}\right)\) \(e\left(\frac{59}{81}\right)\) \(e\left(\frac{71}{81}\right)\) \(e\left(\frac{40}{81}\right)\) \(e\left(\frac{16}{27}\right)\) \(e\left(\frac{20}{27}\right)\) \(e\left(\frac{46}{81}\right)\) \(e\left(\frac{74}{81}\right)\) \(e\left(\frac{29}{81}\right)\) \(e\left(\frac{37}{81}\right)\)
\(\chi_{243}(52,\cdot)\) \(1\) \(1\) \(e\left(\frac{5}{81}\right)\) \(e\left(\frac{10}{81}\right)\) \(e\left(\frac{34}{81}\right)\) \(e\left(\frac{26}{81}\right)\) \(e\left(\frac{5}{27}\right)\) \(e\left(\frac{13}{27}\right)\) \(e\left(\frac{38}{81}\right)\) \(e\left(\frac{40}{81}\right)\) \(e\left(\frac{31}{81}\right)\) \(e\left(\frac{20}{81}\right)\)
\(\chi_{243}(58,\cdot)\) \(1\) \(1\) \(e\left(\frac{19}{81}\right)\) \(e\left(\frac{38}{81}\right)\) \(e\left(\frac{32}{81}\right)\) \(e\left(\frac{34}{81}\right)\) \(e\left(\frac{19}{27}\right)\) \(e\left(\frac{17}{27}\right)\) \(e\left(\frac{31}{81}\right)\) \(e\left(\frac{71}{81}\right)\) \(e\left(\frac{53}{81}\right)\) \(e\left(\frac{76}{81}\right)\)
\(\chi_{243}(61,\cdot)\) \(1\) \(1\) \(e\left(\frac{80}{81}\right)\) \(e\left(\frac{79}{81}\right)\) \(e\left(\frac{58}{81}\right)\) \(e\left(\frac{11}{81}\right)\) \(e\left(\frac{26}{27}\right)\) \(e\left(\frac{19}{27}\right)\) \(e\left(\frac{41}{81}\right)\) \(e\left(\frac{73}{81}\right)\) \(e\left(\frac{10}{81}\right)\) \(e\left(\frac{77}{81}\right)\)
\(\chi_{243}(67,\cdot)\) \(1\) \(1\) \(e\left(\frac{22}{81}\right)\) \(e\left(\frac{44}{81}\right)\) \(e\left(\frac{20}{81}\right)\) \(e\left(\frac{1}{81}\right)\) \(e\left(\frac{22}{27}\right)\) \(e\left(\frac{14}{27}\right)\) \(e\left(\frac{70}{81}\right)\) \(e\left(\frac{14}{81}\right)\) \(e\left(\frac{23}{81}\right)\) \(e\left(\frac{7}{81}\right)\)
\(\chi_{243}(70,\cdot)\) \(1\) \(1\) \(e\left(\frac{47}{81}\right)\) \(e\left(\frac{13}{81}\right)\) \(e\left(\frac{28}{81}\right)\) \(e\left(\frac{50}{81}\right)\) \(e\left(\frac{20}{27}\right)\) \(e\left(\frac{25}{27}\right)\) \(e\left(\frac{17}{81}\right)\) \(e\left(\frac{52}{81}\right)\) \(e\left(\frac{16}{81}\right)\) \(e\left(\frac{26}{81}\right)\)
\(\chi_{243}(76,\cdot)\) \(1\) \(1\) \(e\left(\frac{79}{81}\right)\) \(e\left(\frac{77}{81}\right)\) \(e\left(\frac{35}{81}\right)\) \(e\left(\frac{22}{81}\right)\) \(e\left(\frac{25}{27}\right)\) \(e\left(\frac{11}{27}\right)\) \(e\left(\frac{1}{81}\right)\) \(e\left(\frac{65}{81}\right)\) \(e\left(\frac{20}{81}\right)\) \(e\left(\frac{73}{81}\right)\)
\(\chi_{243}(79,\cdot)\) \(1\) \(1\) \(e\left(\frac{68}{81}\right)\) \(e\left(\frac{55}{81}\right)\) \(e\left(\frac{25}{81}\right)\) \(e\left(\frac{62}{81}\right)\) \(e\left(\frac{14}{27}\right)\) \(e\left(\frac{4}{27}\right)\) \(e\left(\frac{47}{81}\right)\) \(e\left(\frac{58}{81}\right)\) \(e\left(\frac{49}{81}\right)\) \(e\left(\frac{29}{81}\right)\)
\(\chi_{243}(85,\cdot)\) \(1\) \(1\) \(e\left(\frac{28}{81}\right)\) \(e\left(\frac{56}{81}\right)\) \(e\left(\frac{77}{81}\right)\) \(e\left(\frac{16}{81}\right)\) \(e\left(\frac{1}{27}\right)\) \(e\left(\frac{8}{27}\right)\) \(e\left(\frac{67}{81}\right)\) \(e\left(\frac{62}{81}\right)\) \(e\left(\frac{44}{81}\right)\) \(e\left(\frac{31}{81}\right)\)
\(\chi_{243}(88,\cdot)\) \(1\) \(1\) \(e\left(\frac{62}{81}\right)\) \(e\left(\frac{43}{81}\right)\) \(e\left(\frac{49}{81}\right)\) \(e\left(\frac{47}{81}\right)\) \(e\left(\frac{8}{27}\right)\) \(e\left(\frac{10}{27}\right)\) \(e\left(\frac{50}{81}\right)\) \(e\left(\frac{10}{81}\right)\) \(e\left(\frac{28}{81}\right)\) \(e\left(\frac{5}{81}\right)\)
\(\chi_{243}(94,\cdot)\) \(1\) \(1\) \(e\left(\frac{31}{81}\right)\) \(e\left(\frac{62}{81}\right)\) \(e\left(\frac{65}{81}\right)\) \(e\left(\frac{64}{81}\right)\) \(e\left(\frac{4}{27}\right)\) \(e\left(\frac{5}{27}\right)\) \(e\left(\frac{25}{81}\right)\) \(e\left(\frac{5}{81}\right)\) \(e\left(\frac{14}{81}\right)\) \(e\left(\frac{43}{81}\right)\)
\(\chi_{243}(97,\cdot)\) \(1\) \(1\) \(e\left(\frac{29}{81}\right)\) \(e\left(\frac{58}{81}\right)\) \(e\left(\frac{19}{81}\right)\) \(e\left(\frac{5}{81}\right)\) \(e\left(\frac{2}{27}\right)\) \(e\left(\frac{16}{27}\right)\) \(e\left(\frac{26}{81}\right)\) \(e\left(\frac{70}{81}\right)\) \(e\left(\frac{34}{81}\right)\) \(e\left(\frac{35}{81}\right)\)
\(\chi_{243}(103,\cdot)\) \(1\) \(1\) \(e\left(\frac{7}{81}\right)\) \(e\left(\frac{14}{81}\right)\) \(e\left(\frac{80}{81}\right)\) \(e\left(\frac{4}{81}\right)\) \(e\left(\frac{7}{27}\right)\) \(e\left(\frac{2}{27}\right)\) \(e\left(\frac{37}{81}\right)\) \(e\left(\frac{56}{81}\right)\) \(e\left(\frac{11}{81}\right)\) \(e\left(\frac{28}{81}\right)\)
\(\chi_{243}(106,\cdot)\) \(1\) \(1\) \(e\left(\frac{50}{81}\right)\) \(e\left(\frac{19}{81}\right)\) \(e\left(\frac{16}{81}\right)\) \(e\left(\frac{17}{81}\right)\) \(e\left(\frac{23}{27}\right)\) \(e\left(\frac{22}{27}\right)\) \(e\left(\frac{56}{81}\right)\) \(e\left(\frac{76}{81}\right)\) \(e\left(\frac{67}{81}\right)\) \(e\left(\frac{38}{81}\right)\)
\(\chi_{243}(112,\cdot)\) \(1\) \(1\) \(e\left(\frac{37}{81}\right)\) \(e\left(\frac{74}{81}\right)\) \(e\left(\frac{41}{81}\right)\) \(e\left(\frac{79}{81}\right)\) \(e\left(\frac{10}{27}\right)\) \(e\left(\frac{26}{27}\right)\) \(e\left(\frac{22}{81}\right)\) \(e\left(\frac{53}{81}\right)\) \(e\left(\frac{35}{81}\right)\) \(e\left(\frac{67}{81}\right)\)
\(\chi_{243}(115,\cdot)\) \(1\) \(1\) \(e\left(\frac{44}{81}\right)\) \(e\left(\frac{7}{81}\right)\) \(e\left(\frac{40}{81}\right)\) \(e\left(\frac{2}{81}\right)\) \(e\left(\frac{17}{27}\right)\) \(e\left(\frac{1}{27}\right)\) \(e\left(\frac{59}{81}\right)\) \(e\left(\frac{28}{81}\right)\) \(e\left(\frac{46}{81}\right)\) \(e\left(\frac{14}{81}\right)\)
\(\chi_{243}(121,\cdot)\) \(1\) \(1\) \(e\left(\frac{40}{81}\right)\) \(e\left(\frac{80}{81}\right)\) \(e\left(\frac{29}{81}\right)\) \(e\left(\frac{46}{81}\right)\) \(e\left(\frac{13}{27}\right)\) \(e\left(\frac{23}{27}\right)\) \(e\left(\frac{61}{81}\right)\) \(e\left(\frac{77}{81}\right)\) \(e\left(\frac{5}{81}\right)\) \(e\left(\frac{79}{81}\right)\)
\(\chi_{243}(124,\cdot)\) \(1\) \(1\) \(e\left(\frac{11}{81}\right)\) \(e\left(\frac{22}{81}\right)\) \(e\left(\frac{10}{81}\right)\) \(e\left(\frac{41}{81}\right)\) \(e\left(\frac{11}{27}\right)\) \(e\left(\frac{7}{27}\right)\) \(e\left(\frac{35}{81}\right)\) \(e\left(\frac{7}{81}\right)\) \(e\left(\frac{52}{81}\right)\) \(e\left(\frac{44}{81}\right)\)
\(\chi_{243}(130,\cdot)\) \(1\) \(1\) \(e\left(\frac{16}{81}\right)\) \(e\left(\frac{32}{81}\right)\) \(e\left(\frac{44}{81}\right)\) \(e\left(\frac{67}{81}\right)\) \(e\left(\frac{16}{27}\right)\) \(e\left(\frac{20}{27}\right)\) \(e\left(\frac{73}{81}\right)\) \(e\left(\frac{47}{81}\right)\) \(e\left(\frac{2}{81}\right)\) \(e\left(\frac{64}{81}\right)\)
\(\chi_{243}(133,\cdot)\) \(1\) \(1\) \(e\left(\frac{32}{81}\right)\) \(e\left(\frac{64}{81}\right)\) \(e\left(\frac{7}{81}\right)\) \(e\left(\frac{53}{81}\right)\) \(e\left(\frac{5}{27}\right)\) \(e\left(\frac{13}{27}\right)\) \(e\left(\frac{65}{81}\right)\) \(e\left(\frac{13}{81}\right)\) \(e\left(\frac{4}{81}\right)\) \(e\left(\frac{47}{81}\right)\)
\(\chi_{243}(139,\cdot)\) \(1\) \(1\) \(e\left(\frac{46}{81}\right)\) \(e\left(\frac{11}{81}\right)\) \(e\left(\frac{5}{81}\right)\) \(e\left(\frac{61}{81}\right)\) \(e\left(\frac{19}{27}\right)\) \(e\left(\frac{17}{27}\right)\) \(e\left(\frac{58}{81}\right)\) \(e\left(\frac{44}{81}\right)\) \(e\left(\frac{26}{81}\right)\) \(e\left(\frac{22}{81}\right)\)