from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(705, base_ring=CyclotomicField(92))
M = H._module
chi = DirichletCharacter(H, M([0,69,22]))
chi.galois_orbit()
[g,chi] = znchar(Mod(13,705))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(705\) | |
| Conductor: | \(235\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(92\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 235.l | 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: | $\Q(\zeta_{92})$ |
| Fixed field: | Number field defined by a degree 92 polynomial |
First 31 of 44 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{705}(13,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{92}\right)\) | \(e\left(\frac{5}{46}\right)\) | \(e\left(\frac{37}{92}\right)\) | \(e\left(\frac{15}{92}\right)\) | \(e\left(\frac{31}{46}\right)\) | \(e\left(\frac{81}{92}\right)\) | \(e\left(\frac{21}{46}\right)\) | \(e\left(\frac{5}{23}\right)\) | \(e\left(\frac{53}{92}\right)\) | \(e\left(\frac{6}{23}\right)\) |
| \(\chi_{705}(22,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{92}\right)\) | \(e\left(\frac{3}{46}\right)\) | \(e\left(\frac{59}{92}\right)\) | \(e\left(\frac{9}{92}\right)\) | \(e\left(\frac{37}{46}\right)\) | \(e\left(\frac{67}{92}\right)\) | \(e\left(\frac{31}{46}\right)\) | \(e\left(\frac{3}{23}\right)\) | \(e\left(\frac{87}{92}\right)\) | \(e\left(\frac{22}{23}\right)\) |
| \(\chi_{705}(43,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{77}{92}\right)\) | \(e\left(\frac{31}{46}\right)\) | \(e\left(\frac{73}{92}\right)\) | \(e\left(\frac{47}{92}\right)\) | \(e\left(\frac{45}{46}\right)\) | \(e\left(\frac{33}{92}\right)\) | \(e\left(\frac{29}{46}\right)\) | \(e\left(\frac{8}{23}\right)\) | \(e\left(\frac{25}{92}\right)\) | \(e\left(\frac{5}{23}\right)\) |
| \(\chi_{705}(52,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{59}{92}\right)\) | \(e\left(\frac{13}{46}\right)\) | \(e\left(\frac{87}{92}\right)\) | \(e\left(\frac{85}{92}\right)\) | \(e\left(\frac{7}{46}\right)\) | \(e\left(\frac{91}{92}\right)\) | \(e\left(\frac{27}{46}\right)\) | \(e\left(\frac{13}{23}\right)\) | \(e\left(\frac{55}{92}\right)\) | \(e\left(\frac{11}{23}\right)\) |
| \(\chi_{705}(58,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{45}{92}\right)\) | \(e\left(\frac{45}{46}\right)\) | \(e\left(\frac{57}{92}\right)\) | \(e\left(\frac{43}{92}\right)\) | \(e\left(\frac{3}{46}\right)\) | \(e\left(\frac{85}{92}\right)\) | \(e\left(\frac{5}{46}\right)\) | \(e\left(\frac{22}{23}\right)\) | \(e\left(\frac{17}{92}\right)\) | \(e\left(\frac{8}{23}\right)\) |
| \(\chi_{705}(67,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{67}{92}\right)\) | \(e\left(\frac{21}{46}\right)\) | \(e\left(\frac{91}{92}\right)\) | \(e\left(\frac{17}{92}\right)\) | \(e\left(\frac{29}{46}\right)\) | \(e\left(\frac{55}{92}\right)\) | \(e\left(\frac{33}{46}\right)\) | \(e\left(\frac{21}{23}\right)\) | \(e\left(\frac{11}{92}\right)\) | \(e\left(\frac{16}{23}\right)\) |
| \(\chi_{705}(73,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{92}\right)\) | \(e\left(\frac{9}{46}\right)\) | \(e\left(\frac{85}{92}\right)\) | \(e\left(\frac{27}{92}\right)\) | \(e\left(\frac{19}{46}\right)\) | \(e\left(\frac{17}{92}\right)\) | \(e\left(\frac{1}{46}\right)\) | \(e\left(\frac{9}{23}\right)\) | \(e\left(\frac{77}{92}\right)\) | \(e\left(\frac{20}{23}\right)\) |
| \(\chi_{705}(82,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{15}{92}\right)\) | \(e\left(\frac{15}{46}\right)\) | \(e\left(\frac{19}{92}\right)\) | \(e\left(\frac{45}{92}\right)\) | \(e\left(\frac{1}{46}\right)\) | \(e\left(\frac{59}{92}\right)\) | \(e\left(\frac{17}{46}\right)\) | \(e\left(\frac{15}{23}\right)\) | \(e\left(\frac{67}{92}\right)\) | \(e\left(\frac{18}{23}\right)\) |
| \(\chi_{705}(88,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{57}{92}\right)\) | \(e\left(\frac{11}{46}\right)\) | \(e\left(\frac{17}{92}\right)\) | \(e\left(\frac{79}{92}\right)\) | \(e\left(\frac{13}{46}\right)\) | \(e\left(\frac{77}{92}\right)\) | \(e\left(\frac{37}{46}\right)\) | \(e\left(\frac{11}{23}\right)\) | \(e\left(\frac{89}{92}\right)\) | \(e\left(\frac{4}{23}\right)\) |
| \(\chi_{705}(127,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{75}{92}\right)\) | \(e\left(\frac{29}{46}\right)\) | \(e\left(\frac{3}{92}\right)\) | \(e\left(\frac{41}{92}\right)\) | \(e\left(\frac{5}{46}\right)\) | \(e\left(\frac{19}{92}\right)\) | \(e\left(\frac{39}{46}\right)\) | \(e\left(\frac{6}{23}\right)\) | \(e\left(\frac{59}{92}\right)\) | \(e\left(\frac{21}{23}\right)\) |
| \(\chi_{705}(133,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{81}{92}\right)\) | \(e\left(\frac{35}{46}\right)\) | \(e\left(\frac{29}{92}\right)\) | \(e\left(\frac{59}{92}\right)\) | \(e\left(\frac{33}{46}\right)\) | \(e\left(\frac{61}{92}\right)\) | \(e\left(\frac{9}{46}\right)\) | \(e\left(\frac{12}{23}\right)\) | \(e\left(\frac{49}{92}\right)\) | \(e\left(\frac{19}{23}\right)\) |
| \(\chi_{705}(163,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{49}{92}\right)\) | \(e\left(\frac{3}{46}\right)\) | \(e\left(\frac{13}{92}\right)\) | \(e\left(\frac{55}{92}\right)\) | \(e\left(\frac{37}{46}\right)\) | \(e\left(\frac{21}{92}\right)\) | \(e\left(\frac{31}{46}\right)\) | \(e\left(\frac{3}{23}\right)\) | \(e\left(\frac{41}{92}\right)\) | \(e\left(\frac{22}{23}\right)\) |
| \(\chi_{705}(172,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{39}{92}\right)\) | \(e\left(\frac{39}{46}\right)\) | \(e\left(\frac{31}{92}\right)\) | \(e\left(\frac{25}{92}\right)\) | \(e\left(\frac{21}{46}\right)\) | \(e\left(\frac{43}{92}\right)\) | \(e\left(\frac{35}{46}\right)\) | \(e\left(\frac{16}{23}\right)\) | \(e\left(\frac{27}{92}\right)\) | \(e\left(\frac{10}{23}\right)\) |
| \(\chi_{705}(193,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{92}\right)\) | \(e\left(\frac{13}{46}\right)\) | \(e\left(\frac{41}{92}\right)\) | \(e\left(\frac{39}{92}\right)\) | \(e\left(\frac{7}{46}\right)\) | \(e\left(\frac{45}{92}\right)\) | \(e\left(\frac{27}{46}\right)\) | \(e\left(\frac{13}{23}\right)\) | \(e\left(\frac{9}{92}\right)\) | \(e\left(\frac{11}{23}\right)\) |
| \(\chi_{705}(208,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{21}{92}\right)\) | \(e\left(\frac{21}{46}\right)\) | \(e\left(\frac{45}{92}\right)\) | \(e\left(\frac{63}{92}\right)\) | \(e\left(\frac{29}{46}\right)\) | \(e\left(\frac{9}{92}\right)\) | \(e\left(\frac{33}{46}\right)\) | \(e\left(\frac{21}{23}\right)\) | \(e\left(\frac{57}{92}\right)\) | \(e\left(\frac{16}{23}\right)\) |
| \(\chi_{705}(217,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{87}{92}\right)\) | \(e\left(\frac{41}{46}\right)\) | \(e\left(\frac{55}{92}\right)\) | \(e\left(\frac{77}{92}\right)\) | \(e\left(\frac{15}{46}\right)\) | \(e\left(\frac{11}{92}\right)\) | \(e\left(\frac{25}{46}\right)\) | \(e\left(\frac{18}{23}\right)\) | \(e\left(\frac{39}{92}\right)\) | \(e\left(\frac{17}{23}\right)\) |
| \(\chi_{705}(223,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{61}{92}\right)\) | \(e\left(\frac{15}{46}\right)\) | \(e\left(\frac{65}{92}\right)\) | \(e\left(\frac{91}{92}\right)\) | \(e\left(\frac{1}{46}\right)\) | \(e\left(\frac{13}{92}\right)\) | \(e\left(\frac{17}{46}\right)\) | \(e\left(\frac{15}{23}\right)\) | \(e\left(\frac{21}{92}\right)\) | \(e\left(\frac{18}{23}\right)\) |
| \(\chi_{705}(232,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{92}\right)\) | \(e\left(\frac{7}{46}\right)\) | \(e\left(\frac{15}{92}\right)\) | \(e\left(\frac{21}{92}\right)\) | \(e\left(\frac{25}{46}\right)\) | \(e\left(\frac{3}{92}\right)\) | \(e\left(\frac{11}{46}\right)\) | \(e\left(\frac{7}{23}\right)\) | \(e\left(\frac{19}{92}\right)\) | \(e\left(\frac{13}{23}\right)\) |
| \(\chi_{705}(268,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{92}\right)\) | \(e\left(\frac{29}{46}\right)\) | \(e\left(\frac{49}{92}\right)\) | \(e\left(\frac{87}{92}\right)\) | \(e\left(\frac{5}{46}\right)\) | \(e\left(\frac{65}{92}\right)\) | \(e\left(\frac{39}{46}\right)\) | \(e\left(\frac{6}{23}\right)\) | \(e\left(\frac{13}{92}\right)\) | \(e\left(\frac{21}{23}\right)\) |
| \(\chi_{705}(292,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{63}{92}\right)\) | \(e\left(\frac{17}{46}\right)\) | \(e\left(\frac{43}{92}\right)\) | \(e\left(\frac{5}{92}\right)\) | \(e\left(\frac{41}{46}\right)\) | \(e\left(\frac{27}{92}\right)\) | \(e\left(\frac{7}{46}\right)\) | \(e\left(\frac{17}{23}\right)\) | \(e\left(\frac{79}{92}\right)\) | \(e\left(\frac{2}{23}\right)\) |
| \(\chi_{705}(313,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{85}{92}\right)\) | \(e\left(\frac{39}{46}\right)\) | \(e\left(\frac{77}{92}\right)\) | \(e\left(\frac{71}{92}\right)\) | \(e\left(\frac{21}{46}\right)\) | \(e\left(\frac{89}{92}\right)\) | \(e\left(\frac{35}{46}\right)\) | \(e\left(\frac{16}{23}\right)\) | \(e\left(\frac{73}{92}\right)\) | \(e\left(\frac{10}{23}\right)\) |
| \(\chi_{705}(322,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{71}{92}\right)\) | \(e\left(\frac{25}{46}\right)\) | \(e\left(\frac{47}{92}\right)\) | \(e\left(\frac{29}{92}\right)\) | \(e\left(\frac{17}{46}\right)\) | \(e\left(\frac{83}{92}\right)\) | \(e\left(\frac{13}{46}\right)\) | \(e\left(\frac{2}{23}\right)\) | \(e\left(\frac{35}{92}\right)\) | \(e\left(\frac{7}{23}\right)\) |
| \(\chi_{705}(352,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{92}\right)\) | \(e\left(\frac{19}{46}\right)\) | \(e\left(\frac{67}{92}\right)\) | \(e\left(\frac{57}{92}\right)\) | \(e\left(\frac{35}{46}\right)\) | \(e\left(\frac{87}{92}\right)\) | \(e\left(\frac{43}{46}\right)\) | \(e\left(\frac{19}{23}\right)\) | \(e\left(\frac{91}{92}\right)\) | \(e\left(\frac{9}{23}\right)\) |
| \(\chi_{705}(358,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{92}\right)\) | \(e\left(\frac{41}{46}\right)\) | \(e\left(\frac{9}{92}\right)\) | \(e\left(\frac{31}{92}\right)\) | \(e\left(\frac{15}{46}\right)\) | \(e\left(\frac{57}{92}\right)\) | \(e\left(\frac{25}{46}\right)\) | \(e\left(\frac{18}{23}\right)\) | \(e\left(\frac{85}{92}\right)\) | \(e\left(\frac{17}{23}\right)\) |
| \(\chi_{705}(367,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{83}{92}\right)\) | \(e\left(\frac{37}{46}\right)\) | \(e\left(\frac{7}{92}\right)\) | \(e\left(\frac{65}{92}\right)\) | \(e\left(\frac{27}{46}\right)\) | \(e\left(\frac{75}{92}\right)\) | \(e\left(\frac{45}{46}\right)\) | \(e\left(\frac{14}{23}\right)\) | \(e\left(\frac{15}{92}\right)\) | \(e\left(\frac{3}{23}\right)\) |
| \(\chi_{705}(373,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{53}{92}\right)\) | \(e\left(\frac{7}{46}\right)\) | \(e\left(\frac{61}{92}\right)\) | \(e\left(\frac{67}{92}\right)\) | \(e\left(\frac{25}{46}\right)\) | \(e\left(\frac{49}{92}\right)\) | \(e\left(\frac{11}{46}\right)\) | \(e\left(\frac{7}{23}\right)\) | \(e\left(\frac{65}{92}\right)\) | \(e\left(\frac{13}{23}\right)\) |
| \(\chi_{705}(433,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{92}\right)\) | \(e\left(\frac{17}{46}\right)\) | \(e\left(\frac{89}{92}\right)\) | \(e\left(\frac{51}{92}\right)\) | \(e\left(\frac{41}{46}\right)\) | \(e\left(\frac{73}{92}\right)\) | \(e\left(\frac{7}{46}\right)\) | \(e\left(\frac{17}{23}\right)\) | \(e\left(\frac{33}{92}\right)\) | \(e\left(\frac{2}{23}\right)\) |
| \(\chi_{705}(442,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{79}{92}\right)\) | \(e\left(\frac{33}{46}\right)\) | \(e\left(\frac{51}{92}\right)\) | \(e\left(\frac{53}{92}\right)\) | \(e\left(\frac{39}{46}\right)\) | \(e\left(\frac{47}{92}\right)\) | \(e\left(\frac{19}{46}\right)\) | \(e\left(\frac{10}{23}\right)\) | \(e\left(\frac{83}{92}\right)\) | \(e\left(\frac{12}{23}\right)\) |
| \(\chi_{705}(463,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{92}\right)\) | \(e\left(\frac{25}{46}\right)\) | \(e\left(\frac{1}{92}\right)\) | \(e\left(\frac{75}{92}\right)\) | \(e\left(\frac{17}{46}\right)\) | \(e\left(\frac{37}{92}\right)\) | \(e\left(\frac{13}{46}\right)\) | \(e\left(\frac{2}{23}\right)\) | \(e\left(\frac{81}{92}\right)\) | \(e\left(\frac{7}{23}\right)\) |
| \(\chi_{705}(493,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{65}{92}\right)\) | \(e\left(\frac{19}{46}\right)\) | \(e\left(\frac{21}{92}\right)\) | \(e\left(\frac{11}{92}\right)\) | \(e\left(\frac{35}{46}\right)\) | \(e\left(\frac{41}{92}\right)\) | \(e\left(\frac{43}{46}\right)\) | \(e\left(\frac{19}{23}\right)\) | \(e\left(\frac{45}{92}\right)\) | \(e\left(\frac{9}{23}\right)\) |
| \(\chi_{705}(508,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{92}\right)\) | \(e\left(\frac{37}{46}\right)\) | \(e\left(\frac{53}{92}\right)\) | \(e\left(\frac{19}{92}\right)\) | \(e\left(\frac{27}{46}\right)\) | \(e\left(\frac{29}{92}\right)\) | \(e\left(\frac{45}{46}\right)\) | \(e\left(\frac{14}{23}\right)\) | \(e\left(\frac{61}{92}\right)\) | \(e\left(\frac{3}{23}\right)\) |