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Properties

Label 163.g
Modulus $163$
Conductor $163$
Order $27$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(163, base_ring=CyclotomicField(54)) M = H._module chi = DirichletCharacter(H, M([34])) chi.galois_orbit()
 
Copy content pari:[g,chi] = znchar(Mod(6,163)) order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(163\)
Conductor: \(163\)
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: yes
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: yes
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 \(-1\) \(1\) \(2\) \(3\) \(4\) \(5\) \(6\) \(7\) \(8\) \(9\) \(10\) \(11\)
\(\chi_{163}(6,\cdot)\) \(1\) \(1\) \(e\left(\frac{17}{27}\right)\) \(e\left(\frac{16}{27}\right)\) \(e\left(\frac{7}{27}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{26}{27}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{5}{27}\right)\) \(e\left(\frac{2}{27}\right)\) \(e\left(\frac{16}{27}\right)\)
\(\chi_{163}(21,\cdot)\) \(1\) \(1\) \(e\left(\frac{2}{27}\right)\) \(e\left(\frac{13}{27}\right)\) \(e\left(\frac{4}{27}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{11}{27}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{26}{27}\right)\) \(e\left(\frac{5}{27}\right)\) \(e\left(\frac{13}{27}\right)\)
\(\chi_{163}(22,\cdot)\) \(1\) \(1\) \(e\left(\frac{8}{27}\right)\) \(e\left(\frac{25}{27}\right)\) \(e\left(\frac{16}{27}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{17}{27}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{23}{27}\right)\) \(e\left(\frac{20}{27}\right)\) \(e\left(\frac{25}{27}\right)\)
\(\chi_{163}(25,\cdot)\) \(1\) \(1\) \(e\left(\frac{5}{27}\right)\) \(e\left(\frac{19}{27}\right)\) \(e\left(\frac{10}{27}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{14}{27}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{11}{27}\right)\) \(e\left(\frac{26}{27}\right)\) \(e\left(\frac{19}{27}\right)\)
\(\chi_{163}(36,\cdot)\) \(1\) \(1\) \(e\left(\frac{7}{27}\right)\) \(e\left(\frac{5}{27}\right)\) \(e\left(\frac{14}{27}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{25}{27}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{10}{27}\right)\) \(e\left(\frac{4}{27}\right)\) \(e\left(\frac{5}{27}\right)\)
\(\chi_{163}(61,\cdot)\) \(1\) \(1\) \(e\left(\frac{13}{27}\right)\) \(e\left(\frac{17}{27}\right)\) \(e\left(\frac{26}{27}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{4}{27}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{7}{27}\right)\) \(e\left(\frac{19}{27}\right)\) \(e\left(\frac{17}{27}\right)\)
\(\chi_{163}(64,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{27}\right)\) \(e\left(\frac{20}{27}\right)\) \(e\left(\frac{2}{27}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{19}{27}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{13}{27}\right)\) \(e\left(\frac{16}{27}\right)\) \(e\left(\frac{20}{27}\right)\)
\(\chi_{163}(65,\cdot)\) \(1\) \(1\) \(e\left(\frac{11}{27}\right)\) \(e\left(\frac{4}{27}\right)\) \(e\left(\frac{22}{27}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{20}{27}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{8}{27}\right)\) \(e\left(\frac{14}{27}\right)\) \(e\left(\frac{4}{27}\right)\)
\(\chi_{163}(77,\cdot)\) \(1\) \(1\) \(e\left(\frac{20}{27}\right)\) \(e\left(\frac{22}{27}\right)\) \(e\left(\frac{13}{27}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{2}{27}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{17}{27}\right)\) \(e\left(\frac{23}{27}\right)\) \(e\left(\frac{22}{27}\right)\)
\(\chi_{163}(115,\cdot)\) \(1\) \(1\) \(e\left(\frac{4}{27}\right)\) \(e\left(\frac{26}{27}\right)\) \(e\left(\frac{8}{27}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{22}{27}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{25}{27}\right)\) \(e\left(\frac{10}{27}\right)\) \(e\left(\frac{26}{27}\right)\)
\(\chi_{163}(126,\cdot)\) \(1\) \(1\) \(e\left(\frac{19}{27}\right)\) \(e\left(\frac{2}{27}\right)\) \(e\left(\frac{11}{27}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{10}{27}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{4}{27}\right)\) \(e\left(\frac{7}{27}\right)\) \(e\left(\frac{2}{27}\right)\)
\(\chi_{163}(132,\cdot)\) \(1\) \(1\) \(e\left(\frac{25}{27}\right)\) \(e\left(\frac{14}{27}\right)\) \(e\left(\frac{23}{27}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{16}{27}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{1}{27}\right)\) \(e\left(\frac{22}{27}\right)\) \(e\left(\frac{14}{27}\right)\)
\(\chi_{163}(135,\cdot)\) \(1\) \(1\) \(e\left(\frac{26}{27}\right)\) \(e\left(\frac{7}{27}\right)\) \(e\left(\frac{25}{27}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{8}{27}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{14}{27}\right)\) \(e\left(\frac{11}{27}\right)\) \(e\left(\frac{7}{27}\right)\)
\(\chi_{163}(136,\cdot)\) \(1\) \(1\) \(e\left(\frac{10}{27}\right)\) \(e\left(\frac{11}{27}\right)\) \(e\left(\frac{20}{27}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{1}{27}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{22}{27}\right)\) \(e\left(\frac{25}{27}\right)\) \(e\left(\frac{11}{27}\right)\)
\(\chi_{163}(146,\cdot)\) \(1\) \(1\) \(e\left(\frac{23}{27}\right)\) \(e\left(\frac{1}{27}\right)\) \(e\left(\frac{19}{27}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{5}{27}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{2}{27}\right)\) \(e\left(\frac{17}{27}\right)\) \(e\left(\frac{1}{27}\right)\)
\(\chi_{163}(150,\cdot)\) \(1\) \(1\) \(e\left(\frac{22}{27}\right)\) \(e\left(\frac{8}{27}\right)\) \(e\left(\frac{17}{27}\right)\) \(e\left(\frac{2}{9}\right)\) \(e\left(\frac{1}{9}\right)\) \(e\left(\frac{13}{27}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{16}{27}\right)\) \(e\left(\frac{1}{27}\right)\) \(e\left(\frac{8}{27}\right)\)
\(\chi_{163}(155,\cdot)\) \(1\) \(1\) \(e\left(\frac{14}{27}\right)\) \(e\left(\frac{10}{27}\right)\) \(e\left(\frac{1}{27}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{23}{27}\right)\) \(e\left(\frac{5}{9}\right)\) \(e\left(\frac{20}{27}\right)\) \(e\left(\frac{8}{27}\right)\) \(e\left(\frac{10}{27}\right)\)
\(\chi_{163}(158,\cdot)\) \(1\) \(1\) \(e\left(\frac{16}{27}\right)\) \(e\left(\frac{23}{27}\right)\) \(e\left(\frac{5}{27}\right)\) \(e\left(\frac{8}{9}\right)\) \(e\left(\frac{4}{9}\right)\) \(e\left(\frac{7}{27}\right)\) \(e\left(\frac{7}{9}\right)\) \(e\left(\frac{19}{27}\right)\) \(e\left(\frac{13}{27}\right)\) \(e\left(\frac{23}{27}\right)\)