Properties

Label 199.i
Modulus $199$
Conductor $199$
Order $33$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(199, base_ring=CyclotomicField(66))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([46]))
 
sage: chi.galois_orbit()
 
pari: [g,chi] = znchar(Mod(5,199))
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Basic properties

Modulus: \(199\)
Conductor: \(199\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(33\)
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: \(\Q(\zeta_{33})\)
Fixed field: 33.33.36584611296554742180833097810429342639777502523008874222975105176339833601.1

Characters in Galois orbit

Character \(-1\) \(1\) \(2\) \(3\) \(4\) \(5\) \(6\) \(7\) \(8\) \(9\) \(10\) \(11\)
\(\chi_{199}(5,\cdot)\) \(1\) \(1\) \(e\left(\frac{29}{33}\right)\) \(e\left(\frac{23}{33}\right)\) \(e\left(\frac{25}{33}\right)\) \(e\left(\frac{2}{11}\right)\) \(e\left(\frac{19}{33}\right)\) \(e\left(\frac{32}{33}\right)\) \(e\left(\frac{7}{11}\right)\) \(e\left(\frac{13}{33}\right)\) \(e\left(\frac{2}{33}\right)\) \(e\left(\frac{8}{11}\right)\)
\(\chi_{199}(8,\cdot)\) \(1\) \(1\) \(e\left(\frac{8}{33}\right)\) \(e\left(\frac{20}{33}\right)\) \(e\left(\frac{16}{33}\right)\) \(e\left(\frac{7}{11}\right)\) \(e\left(\frac{28}{33}\right)\) \(e\left(\frac{2}{33}\right)\) \(e\left(\frac{8}{11}\right)\) \(e\left(\frac{7}{33}\right)\) \(e\left(\frac{29}{33}\right)\) \(e\left(\frac{6}{11}\right)\)
\(\chi_{199}(25,\cdot)\) \(1\) \(1\) \(e\left(\frac{25}{33}\right)\) \(e\left(\frac{13}{33}\right)\) \(e\left(\frac{17}{33}\right)\) \(e\left(\frac{4}{11}\right)\) \(e\left(\frac{5}{33}\right)\) \(e\left(\frac{31}{33}\right)\) \(e\left(\frac{3}{11}\right)\) \(e\left(\frac{26}{33}\right)\) \(e\left(\frac{4}{33}\right)\) \(e\left(\frac{5}{11}\right)\)
\(\chi_{199}(28,\cdot)\) \(1\) \(1\) \(e\left(\frac{17}{33}\right)\) \(e\left(\frac{26}{33}\right)\) \(e\left(\frac{1}{33}\right)\) \(e\left(\frac{8}{11}\right)\) \(e\left(\frac{10}{33}\right)\) \(e\left(\frac{29}{33}\right)\) \(e\left(\frac{6}{11}\right)\) \(e\left(\frac{19}{33}\right)\) \(e\left(\frac{8}{33}\right)\) \(e\left(\frac{10}{11}\right)\)
\(\chi_{199}(40,\cdot)\) \(1\) \(1\) \(e\left(\frac{4}{33}\right)\) \(e\left(\frac{10}{33}\right)\) \(e\left(\frac{8}{33}\right)\) \(e\left(\frac{9}{11}\right)\) \(e\left(\frac{14}{33}\right)\) \(e\left(\frac{1}{33}\right)\) \(e\left(\frac{4}{11}\right)\) \(e\left(\frac{20}{33}\right)\) \(e\left(\frac{31}{33}\right)\) \(e\left(\frac{3}{11}\right)\)
\(\chi_{199}(52,\cdot)\) \(1\) \(1\) \(e\left(\frac{19}{33}\right)\) \(e\left(\frac{31}{33}\right)\) \(e\left(\frac{5}{33}\right)\) \(e\left(\frac{7}{11}\right)\) \(e\left(\frac{17}{33}\right)\) \(e\left(\frac{13}{33}\right)\) \(e\left(\frac{8}{11}\right)\) \(e\left(\frac{29}{33}\right)\) \(e\left(\frac{7}{33}\right)\) \(e\left(\frac{6}{11}\right)\)
\(\chi_{199}(64,\cdot)\) \(1\) \(1\) \(e\left(\frac{16}{33}\right)\) \(e\left(\frac{7}{33}\right)\) \(e\left(\frac{32}{33}\right)\) \(e\left(\frac{3}{11}\right)\) \(e\left(\frac{23}{33}\right)\) \(e\left(\frac{4}{33}\right)\) \(e\left(\frac{5}{11}\right)\) \(e\left(\frac{14}{33}\right)\) \(e\left(\frac{25}{33}\right)\) \(e\left(\frac{1}{11}\right)\)
\(\chi_{199}(90,\cdot)\) \(1\) \(1\) \(e\left(\frac{23}{33}\right)\) \(e\left(\frac{8}{33}\right)\) \(e\left(\frac{13}{33}\right)\) \(e\left(\frac{5}{11}\right)\) \(e\left(\frac{31}{33}\right)\) \(e\left(\frac{14}{33}\right)\) \(e\left(\frac{1}{11}\right)\) \(e\left(\frac{16}{33}\right)\) \(e\left(\frac{5}{33}\right)\) \(e\left(\frac{9}{11}\right)\)
\(\chi_{199}(98,\cdot)\) \(1\) \(1\) \(e\left(\frac{26}{33}\right)\) \(e\left(\frac{32}{33}\right)\) \(e\left(\frac{19}{33}\right)\) \(e\left(\frac{9}{11}\right)\) \(e\left(\frac{25}{33}\right)\) \(e\left(\frac{23}{33}\right)\) \(e\left(\frac{4}{11}\right)\) \(e\left(\frac{31}{33}\right)\) \(e\left(\frac{20}{33}\right)\) \(e\left(\frac{3}{11}\right)\)
\(\chi_{199}(111,\cdot)\) \(1\) \(1\) \(e\left(\frac{14}{33}\right)\) \(e\left(\frac{2}{33}\right)\) \(e\left(\frac{28}{33}\right)\) \(e\left(\frac{4}{11}\right)\) \(e\left(\frac{16}{33}\right)\) \(e\left(\frac{20}{33}\right)\) \(e\left(\frac{3}{11}\right)\) \(e\left(\frac{4}{33}\right)\) \(e\left(\frac{26}{33}\right)\) \(e\left(\frac{5}{11}\right)\)
\(\chi_{199}(116,\cdot)\) \(1\) \(1\) \(e\left(\frac{32}{33}\right)\) \(e\left(\frac{14}{33}\right)\) \(e\left(\frac{31}{33}\right)\) \(e\left(\frac{6}{11}\right)\) \(e\left(\frac{13}{33}\right)\) \(e\left(\frac{8}{33}\right)\) \(e\left(\frac{10}{11}\right)\) \(e\left(\frac{28}{33}\right)\) \(e\left(\frac{17}{33}\right)\) \(e\left(\frac{2}{11}\right)\)
\(\chi_{199}(117,\cdot)\) \(1\) \(1\) \(e\left(\frac{5}{33}\right)\) \(e\left(\frac{29}{33}\right)\) \(e\left(\frac{10}{33}\right)\) \(e\left(\frac{3}{11}\right)\) \(e\left(\frac{1}{33}\right)\) \(e\left(\frac{26}{33}\right)\) \(e\left(\frac{5}{11}\right)\) \(e\left(\frac{25}{33}\right)\) \(e\left(\frac{14}{33}\right)\) \(e\left(\frac{1}{11}\right)\)
\(\chi_{199}(123,\cdot)\) \(1\) \(1\) \(e\left(\frac{31}{33}\right)\) \(e\left(\frac{28}{33}\right)\) \(e\left(\frac{29}{33}\right)\) \(e\left(\frac{1}{11}\right)\) \(e\left(\frac{26}{33}\right)\) \(e\left(\frac{16}{33}\right)\) \(e\left(\frac{9}{11}\right)\) \(e\left(\frac{23}{33}\right)\) \(e\left(\frac{1}{33}\right)\) \(e\left(\frac{4}{11}\right)\)
\(\chi_{199}(132,\cdot)\) \(1\) \(1\) \(e\left(\frac{7}{33}\right)\) \(e\left(\frac{1}{33}\right)\) \(e\left(\frac{14}{33}\right)\) \(e\left(\frac{2}{11}\right)\) \(e\left(\frac{8}{33}\right)\) \(e\left(\frac{10}{33}\right)\) \(e\left(\frac{7}{11}\right)\) \(e\left(\frac{2}{33}\right)\) \(e\left(\frac{13}{33}\right)\) \(e\left(\frac{8}{11}\right)\)
\(\chi_{199}(140,\cdot)\) \(1\) \(1\) \(e\left(\frac{13}{33}\right)\) \(e\left(\frac{16}{33}\right)\) \(e\left(\frac{26}{33}\right)\) \(e\left(\frac{10}{11}\right)\) \(e\left(\frac{29}{33}\right)\) \(e\left(\frac{28}{33}\right)\) \(e\left(\frac{2}{11}\right)\) \(e\left(\frac{32}{33}\right)\) \(e\left(\frac{10}{33}\right)\) \(e\left(\frac{7}{11}\right)\)
\(\chi_{199}(144,\cdot)\) \(1\) \(1\) \(e\left(\frac{2}{33}\right)\) \(e\left(\frac{5}{33}\right)\) \(e\left(\frac{4}{33}\right)\) \(e\left(\frac{10}{11}\right)\) \(e\left(\frac{7}{33}\right)\) \(e\left(\frac{17}{33}\right)\) \(e\left(\frac{2}{11}\right)\) \(e\left(\frac{10}{33}\right)\) \(e\left(\frac{32}{33}\right)\) \(e\left(\frac{7}{11}\right)\)
\(\chi_{199}(157,\cdot)\) \(1\) \(1\) \(e\left(\frac{10}{33}\right)\) \(e\left(\frac{25}{33}\right)\) \(e\left(\frac{20}{33}\right)\) \(e\left(\frac{6}{11}\right)\) \(e\left(\frac{2}{33}\right)\) \(e\left(\frac{19}{33}\right)\) \(e\left(\frac{10}{11}\right)\) \(e\left(\frac{17}{33}\right)\) \(e\left(\frac{28}{33}\right)\) \(e\left(\frac{2}{11}\right)\)
\(\chi_{199}(172,\cdot)\) \(1\) \(1\) \(e\left(\frac{20}{33}\right)\) \(e\left(\frac{17}{33}\right)\) \(e\left(\frac{7}{33}\right)\) \(e\left(\frac{1}{11}\right)\) \(e\left(\frac{4}{33}\right)\) \(e\left(\frac{5}{33}\right)\) \(e\left(\frac{9}{11}\right)\) \(e\left(\frac{1}{33}\right)\) \(e\left(\frac{23}{33}\right)\) \(e\left(\frac{4}{11}\right)\)
\(\chi_{199}(182,\cdot)\) \(1\) \(1\) \(e\left(\frac{28}{33}\right)\) \(e\left(\frac{4}{33}\right)\) \(e\left(\frac{23}{33}\right)\) \(e\left(\frac{8}{11}\right)\) \(e\left(\frac{32}{33}\right)\) \(e\left(\frac{7}{33}\right)\) \(e\left(\frac{6}{11}\right)\) \(e\left(\frac{8}{33}\right)\) \(e\left(\frac{19}{33}\right)\) \(e\left(\frac{10}{11}\right)\)
\(\chi_{199}(187,\cdot)\) \(1\) \(1\) \(e\left(\frac{1}{33}\right)\) \(e\left(\frac{19}{33}\right)\) \(e\left(\frac{2}{33}\right)\) \(e\left(\frac{5}{11}\right)\) \(e\left(\frac{20}{33}\right)\) \(e\left(\frac{25}{33}\right)\) \(e\left(\frac{1}{11}\right)\) \(e\left(\frac{5}{33}\right)\) \(e\left(\frac{16}{33}\right)\) \(e\left(\frac{9}{11}\right)\)