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Properties

Label 729.g
Modulus $729$
Conductor $81$
Order $27$
Real no
Primitive no
Minimal no
Parity even

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(729, base_ring=CyclotomicField(54)) M = H._module chi = DirichletCharacter(H, M([44])) chi.galois_orbit()
 
Copy content pari:[g,chi] = znchar(Mod(28,729)) order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(729\)
Conductor: \(81\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(27\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: no, induced from 81.g
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Related number fields

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

Characters in Galois orbit

Character \(-1\) \(1\) \(2\) \(4\) \(5\) \(7\) \(8\) \(10\) \(11\) \(13\) \(14\) \(16\)
\(\chi_{729}(28,\cdot)\) \(1\) \(1\) \(e\left(\frac{22}{27}\right)\) \(e\left(\frac{17}{27}\right)\) \(e\left(\frac{20}{27}\right)\) \(e\left(\frac{1}{27}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{16}{27}\right)\) \(e\left(\frac{14}{27}\right)\) \(e\left(\frac{23}{27}\right)\) \(e\left(\frac{7}{27}\right)\)
\(\chi_{729}(55,\cdot)\) \(1\) \(1\) \(e\left(\frac{17}{27}\right)\) \(e\left(\frac{7}{27}\right)\) \(e\left(\frac{13}{27}\right)\) \(e\left(\frac{2}{27}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{5}{27}\right)\) \(e\left(\frac{1}{27}\right)\) \(e\left(\frac{19}{27}\right)\) \(e\left(\frac{14}{27}\right)\)
\(\chi_{729}(109,\cdot)\) \(1\) \(1\) \(e\left(\frac{7}{27}\right)\) \(e\left(\frac{14}{27}\right)\) \(e\left(\frac{26}{27}\right)\) \(e\left(\frac{4}{27}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{10}{27}\right)\) \(e\left(\frac{2}{27}\right)\) \(e\left(\frac{11}{27}\right)\) \(e\left(\frac{1}{27}\right)\)
\(\chi_{729}(136,\cdot)\) \(1\) \(1\) \(e\left(\frac{2}{27}\right)\) \(e\left(\frac{4}{27}\right)\) \(e\left(\frac{19}{27}\right)\) \(e\left(\frac{5}{27}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{26}{27}\right)\) \(e\left(\frac{16}{27}\right)\) \(e\left(\frac{7}{27}\right)\) \(e\left(\frac{8}{27}\right)\)
\(\chi_{729}(190,\cdot)\) \(1\) \(1\) \(e\left(\frac{19}{27}\right)\) \(e\left(\frac{11}{27}\right)\) \(e\left(\frac{5}{27}\right)\) \(e\left(\frac{7}{27}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{4}{27}\right)\) \(e\left(\frac{17}{27}\right)\) \(e\left(\frac{26}{27}\right)\) \(e\left(\frac{22}{27}\right)\)
\(\chi_{729}(217,\cdot)\) \(1\) \(1\) \(e\left(\frac{14}{27}\right)\) \(e\left(\frac{1}{27}\right)\) \(e\left(\frac{25}{27}\right)\) \(e\left(\frac{8}{27}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{20}{27}\right)\) \(e\left(\frac{4}{27}\right)\) \(e\left(\frac{22}{27}\right)\) \(e\left(\frac{2}{27}\right)\)
\(\chi_{729}(271,\cdot)\) \(1\) \(1\) \(e\left(\frac{4}{27}\right)\) \(e\left(\frac{8}{27}\right)\) \(e\left(\frac{11}{27}\right)\) \(e\left(\frac{10}{27}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{25}{27}\right)\) \(e\left(\frac{5}{27}\right)\) \(e\left(\frac{14}{27}\right)\) \(e\left(\frac{16}{27}\right)\)
\(\chi_{729}(298,\cdot)\) \(1\) \(1\) \(e\left(\frac{26}{27}\right)\) \(e\left(\frac{25}{27}\right)\) \(e\left(\frac{4}{27}\right)\) \(e\left(\frac{11}{27}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{14}{27}\right)\) \(e\left(\frac{19}{27}\right)\) \(e\left(\frac{10}{27}\right)\) \(e\left(\frac{23}{27}\right)\)
\(\chi_{729}(352,\cdot)\) \(1\) \(1\) \(e\left(\frac{16}{27}\right)\) \(e\left(\frac{5}{27}\right)\) \(e\left(\frac{17}{27}\right)\) \(e\left(\frac{13}{27}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{19}{27}\right)\) \(e\left(\frac{20}{27}\right)\) \(e\left(\frac{2}{27}\right)\) \(e\left(\frac{10}{27}\right)\)
\(\chi_{729}(379,\cdot)\) \(1\) \(1\) \(e\left(\frac{11}{27}\right)\) \(e\left(\frac{22}{27}\right)\) \(e\left(\frac{10}{27}\right)\) \(e\left(\frac{14}{27}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{8}{27}\right)\) \(e\left(\frac{7}{27}\right)\) \(e\left(\frac{25}{27}\right)\) \(e\left(\frac{17}{27}\right)\)
\(\chi_{729}(433,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{27}\right)\) \(e\left(\frac{2}{27}\right)\) \(e\left(\frac{23}{27}\right)\) \(e\left(\frac{16}{27}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{13}{27}\right)\) \(e\left(\frac{8}{27}\right)\) \(e\left(\frac{17}{27}\right)\) \(e\left(\frac{4}{27}\right)\)
\(\chi_{729}(460,\cdot)\) \(1\) \(1\) \(e\left(\frac{23}{27}\right)\) \(e\left(\frac{19}{27}\right)\) \(e\left(\frac{16}{27}\right)\) \(e\left(\frac{17}{27}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{2}{27}\right)\) \(e\left(\frac{22}{27}\right)\) \(e\left(\frac{13}{27}\right)\) \(e\left(\frac{11}{27}\right)\)
\(\chi_{729}(514,\cdot)\) \(1\) \(1\) \(e\left(\frac{13}{27}\right)\) \(e\left(\frac{26}{27}\right)\) \(e\left(\frac{2}{27}\right)\) \(e\left(\frac{19}{27}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{7}{27}\right)\) \(e\left(\frac{23}{27}\right)\) \(e\left(\frac{5}{27}\right)\) \(e\left(\frac{25}{27}\right)\)
\(\chi_{729}(541,\cdot)\) \(1\) \(1\) \(e\left(\frac{8}{27}\right)\) \(e\left(\frac{16}{27}\right)\) \(e\left(\frac{22}{27}\right)\) \(e\left(\frac{20}{27}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{23}{27}\right)\) \(e\left(\frac{10}{27}\right)\) \(e\left(\frac{1}{27}\right)\) \(e\left(\frac{5}{27}\right)\)
\(\chi_{729}(595,\cdot)\) \(1\) \(1\) \(e\left(\frac{25}{27}\right)\) \(e\left(\frac{23}{27}\right)\) \(e\left(\frac{8}{27}\right)\) \(e\left(\frac{22}{27}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{1}{27}\right)\) \(e\left(\frac{11}{27}\right)\) \(e\left(\frac{20}{27}\right)\) \(e\left(\frac{19}{27}\right)\)
\(\chi_{729}(622,\cdot)\) \(1\) \(1\) \(e\left(\frac{20}{27}\right)\) \(e\left(\frac{13}{27}\right)\) \(e\left(\frac{1}{27}\right)\) \(e\left(\frac{23}{27}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{17}{27}\right)\) \(e\left(\frac{25}{27}\right)\) \(e\left(\frac{16}{27}\right)\) \(e\left(\frac{26}{27}\right)\)
\(\chi_{729}(676,\cdot)\) \(1\) \(1\) \(e\left(\frac{10}{27}\right)\) \(e\left(\frac{20}{27}\right)\) \(e\left(\frac{14}{27}\right)\) \(e\left(\frac{25}{27}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{22}{27}\right)\) \(e\left(\frac{26}{27}\right)\) \(e\left(\frac{8}{27}\right)\) \(e\left(\frac{13}{27}\right)\)
\(\chi_{729}(703,\cdot)\) \(1\) \(1\) \(e\left(\frac{5}{27}\right)\) \(e\left(\frac{10}{27}\right)\) \(e\left(\frac{7}{27}\right)\) \(e\left(\frac{26}{27}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{11}{27}\right)\) \(e\left(\frac{13}{27}\right)\) \(e\left(\frac{4}{27}\right)\) \(e\left(\frac{20}{27}\right)\)