Properties

Label 729.h
Modulus $729$
Conductor $81$
Order $54$
Real no
Primitive no
Minimal no
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(729, base_ring=CyclotomicField(54))
 
M = H._module
 
chi = DirichletCharacter(H, M([37]))
 
chi.galois_orbit()
 
[g,chi] = znchar(Mod(26,729))
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(729\)
Conductor: \(81\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(54\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from 81.h
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Related number fields

Field of values: \(\Q(\zeta_{27})\)
Fixed field: Number field defined by a degree 54 polynomial

Characters in Galois orbit

Character \(-1\) \(1\) \(2\) \(4\) \(5\) \(7\) \(8\) \(10\) \(11\) \(13\) \(14\) \(16\)
\(\chi_{729}(26,\cdot)\) \(-1\) \(1\) \(e\left(\frac{37}{54}\right)\) \(e\left(\frac{10}{27}\right)\) \(e\left(\frac{41}{54}\right)\) \(e\left(\frac{26}{27}\right)\) \(e\left(\frac{1}{18}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{49}{54}\right)\) \(e\left(\frac{13}{27}\right)\) \(e\left(\frac{35}{54}\right)\) \(e\left(\frac{20}{27}\right)\)
\(\chi_{729}(53,\cdot)\) \(-1\) \(1\) \(e\left(\frac{47}{54}\right)\) \(e\left(\frac{20}{27}\right)\) \(e\left(\frac{1}{54}\right)\) \(e\left(\frac{25}{27}\right)\) \(e\left(\frac{11}{18}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{17}{54}\right)\) \(e\left(\frac{26}{27}\right)\) \(e\left(\frac{43}{54}\right)\) \(e\left(\frac{13}{27}\right)\)
\(\chi_{729}(107,\cdot)\) \(-1\) \(1\) \(e\left(\frac{13}{54}\right)\) \(e\left(\frac{13}{27}\right)\) \(e\left(\frac{29}{54}\right)\) \(e\left(\frac{23}{27}\right)\) \(e\left(\frac{13}{18}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{7}{54}\right)\) \(e\left(\frac{25}{27}\right)\) \(e\left(\frac{5}{54}\right)\) \(e\left(\frac{26}{27}\right)\)
\(\chi_{729}(134,\cdot)\) \(-1\) \(1\) \(e\left(\frac{23}{54}\right)\) \(e\left(\frac{23}{27}\right)\) \(e\left(\frac{43}{54}\right)\) \(e\left(\frac{22}{27}\right)\) \(e\left(\frac{5}{18}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{29}{54}\right)\) \(e\left(\frac{11}{27}\right)\) \(e\left(\frac{13}{54}\right)\) \(e\left(\frac{19}{27}\right)\)
\(\chi_{729}(188,\cdot)\) \(-1\) \(1\) \(e\left(\frac{43}{54}\right)\) \(e\left(\frac{16}{27}\right)\) \(e\left(\frac{17}{54}\right)\) \(e\left(\frac{20}{27}\right)\) \(e\left(\frac{7}{18}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{19}{54}\right)\) \(e\left(\frac{10}{27}\right)\) \(e\left(\frac{29}{54}\right)\) \(e\left(\frac{5}{27}\right)\)
\(\chi_{729}(215,\cdot)\) \(-1\) \(1\) \(e\left(\frac{53}{54}\right)\) \(e\left(\frac{26}{27}\right)\) \(e\left(\frac{31}{54}\right)\) \(e\left(\frac{19}{27}\right)\) \(e\left(\frac{17}{18}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{41}{54}\right)\) \(e\left(\frac{23}{27}\right)\) \(e\left(\frac{37}{54}\right)\) \(e\left(\frac{25}{27}\right)\)
\(\chi_{729}(269,\cdot)\) \(-1\) \(1\) \(e\left(\frac{19}{54}\right)\) \(e\left(\frac{19}{27}\right)\) \(e\left(\frac{5}{54}\right)\) \(e\left(\frac{17}{27}\right)\) \(e\left(\frac{1}{18}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{31}{54}\right)\) \(e\left(\frac{22}{27}\right)\) \(e\left(\frac{53}{54}\right)\) \(e\left(\frac{11}{27}\right)\)
\(\chi_{729}(296,\cdot)\) \(-1\) \(1\) \(e\left(\frac{29}{54}\right)\) \(e\left(\frac{2}{27}\right)\) \(e\left(\frac{19}{54}\right)\) \(e\left(\frac{16}{27}\right)\) \(e\left(\frac{11}{18}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{53}{54}\right)\) \(e\left(\frac{8}{27}\right)\) \(e\left(\frac{7}{54}\right)\) \(e\left(\frac{4}{27}\right)\)
\(\chi_{729}(350,\cdot)\) \(-1\) \(1\) \(e\left(\frac{49}{54}\right)\) \(e\left(\frac{22}{27}\right)\) \(e\left(\frac{47}{54}\right)\) \(e\left(\frac{14}{27}\right)\) \(e\left(\frac{13}{18}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{43}{54}\right)\) \(e\left(\frac{7}{27}\right)\) \(e\left(\frac{23}{54}\right)\) \(e\left(\frac{17}{27}\right)\)
\(\chi_{729}(377,\cdot)\) \(-1\) \(1\) \(e\left(\frac{5}{54}\right)\) \(e\left(\frac{5}{27}\right)\) \(e\left(\frac{7}{54}\right)\) \(e\left(\frac{13}{27}\right)\) \(e\left(\frac{5}{18}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{11}{54}\right)\) \(e\left(\frac{20}{27}\right)\) \(e\left(\frac{31}{54}\right)\) \(e\left(\frac{10}{27}\right)\)
\(\chi_{729}(431,\cdot)\) \(-1\) \(1\) \(e\left(\frac{25}{54}\right)\) \(e\left(\frac{25}{27}\right)\) \(e\left(\frac{35}{54}\right)\) \(e\left(\frac{11}{27}\right)\) \(e\left(\frac{7}{18}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{1}{54}\right)\) \(e\left(\frac{19}{27}\right)\) \(e\left(\frac{47}{54}\right)\) \(e\left(\frac{23}{27}\right)\)
\(\chi_{729}(458,\cdot)\) \(-1\) \(1\) \(e\left(\frac{35}{54}\right)\) \(e\left(\frac{8}{27}\right)\) \(e\left(\frac{49}{54}\right)\) \(e\left(\frac{10}{27}\right)\) \(e\left(\frac{17}{18}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{23}{54}\right)\) \(e\left(\frac{5}{27}\right)\) \(e\left(\frac{1}{54}\right)\) \(e\left(\frac{16}{27}\right)\)
\(\chi_{729}(512,\cdot)\) \(-1\) \(1\) \(e\left(\frac{1}{54}\right)\) \(e\left(\frac{1}{27}\right)\) \(e\left(\frac{23}{54}\right)\) \(e\left(\frac{8}{27}\right)\) \(e\left(\frac{1}{18}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{13}{54}\right)\) \(e\left(\frac{4}{27}\right)\) \(e\left(\frac{17}{54}\right)\) \(e\left(\frac{2}{27}\right)\)
\(\chi_{729}(539,\cdot)\) \(-1\) \(1\) \(e\left(\frac{11}{54}\right)\) \(e\left(\frac{11}{27}\right)\) \(e\left(\frac{37}{54}\right)\) \(e\left(\frac{7}{27}\right)\) \(e\left(\frac{11}{18}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{35}{54}\right)\) \(e\left(\frac{17}{27}\right)\) \(e\left(\frac{25}{54}\right)\) \(e\left(\frac{22}{27}\right)\)
\(\chi_{729}(593,\cdot)\) \(-1\) \(1\) \(e\left(\frac{31}{54}\right)\) \(e\left(\frac{4}{27}\right)\) \(e\left(\frac{11}{54}\right)\) \(e\left(\frac{5}{27}\right)\) \(e\left(\frac{13}{18}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{25}{54}\right)\) \(e\left(\frac{16}{27}\right)\) \(e\left(\frac{41}{54}\right)\) \(e\left(\frac{8}{27}\right)\)
\(\chi_{729}(620,\cdot)\) \(-1\) \(1\) \(e\left(\frac{41}{54}\right)\) \(e\left(\frac{14}{27}\right)\) \(e\left(\frac{25}{54}\right)\) \(e\left(\frac{4}{27}\right)\) \(e\left(\frac{5}{18}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{47}{54}\right)\) \(e\left(\frac{2}{27}\right)\) \(e\left(\frac{49}{54}\right)\) \(e\left(\frac{1}{27}\right)\)
\(\chi_{729}(674,\cdot)\) \(-1\) \(1\) \(e\left(\frac{7}{54}\right)\) \(e\left(\frac{7}{27}\right)\) \(e\left(\frac{53}{54}\right)\) \(e\left(\frac{2}{27}\right)\) \(e\left(\frac{7}{18}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{37}{54}\right)\) \(e\left(\frac{1}{27}\right)\) \(e\left(\frac{11}{54}\right)\) \(e\left(\frac{14}{27}\right)\)
\(\chi_{729}(701,\cdot)\) \(-1\) \(1\) \(e\left(\frac{17}{54}\right)\) \(e\left(\frac{17}{27}\right)\) \(e\left(\frac{13}{54}\right)\) \(e\left(\frac{1}{27}\right)\) \(e\left(\frac{17}{18}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{5}{54}\right)\) \(e\left(\frac{14}{27}\right)\) \(e\left(\frac{19}{54}\right)\) \(e\left(\frac{7}{27}\right)\)