from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(177, base_ring=CyclotomicField(58))
M = H._module
chi = DirichletCharacter(H, M([0,7]))
chi.galois_orbit()
[g,chi] = znchar(Mod(10,177))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(177\) | |
| Conductor: | \(59\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(58\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 59.d | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Related number fields
| Field of values: | $\Q(\zeta_{29})$ |
| Fixed field: | Number field defined by a degree 58 polynomial |
Characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{177}(10,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{58}\right)\) | \(e\left(\frac{7}{29}\right)\) | \(e\left(\frac{21}{29}\right)\) | \(e\left(\frac{5}{29}\right)\) | \(e\left(\frac{21}{58}\right)\) | \(e\left(\frac{49}{58}\right)\) | \(e\left(\frac{1}{58}\right)\) | \(e\left(\frac{25}{58}\right)\) | \(e\left(\frac{17}{58}\right)\) | \(e\left(\frac{14}{29}\right)\) |
| \(\chi_{177}(13,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{45}{58}\right)\) | \(e\left(\frac{16}{29}\right)\) | \(e\left(\frac{19}{29}\right)\) | \(e\left(\frac{28}{29}\right)\) | \(e\left(\frac{19}{58}\right)\) | \(e\left(\frac{25}{58}\right)\) | \(e\left(\frac{23}{58}\right)\) | \(e\left(\frac{53}{58}\right)\) | \(e\left(\frac{43}{58}\right)\) | \(e\left(\frac{3}{29}\right)\) |
| \(\chi_{177}(31,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{49}{58}\right)\) | \(e\left(\frac{20}{29}\right)\) | \(e\left(\frac{2}{29}\right)\) | \(e\left(\frac{6}{29}\right)\) | \(e\left(\frac{31}{58}\right)\) | \(e\left(\frac{53}{58}\right)\) | \(e\left(\frac{7}{58}\right)\) | \(e\left(\frac{1}{58}\right)\) | \(e\left(\frac{3}{58}\right)\) | \(e\left(\frac{11}{29}\right)\) |
| \(\chi_{177}(34,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{41}{58}\right)\) | \(e\left(\frac{12}{29}\right)\) | \(e\left(\frac{7}{29}\right)\) | \(e\left(\frac{21}{29}\right)\) | \(e\left(\frac{7}{58}\right)\) | \(e\left(\frac{55}{58}\right)\) | \(e\left(\frac{39}{58}\right)\) | \(e\left(\frac{47}{58}\right)\) | \(e\left(\frac{25}{58}\right)\) | \(e\left(\frac{24}{29}\right)\) |
| \(\chi_{177}(37,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{55}{58}\right)\) | \(e\left(\frac{26}{29}\right)\) | \(e\left(\frac{20}{29}\right)\) | \(e\left(\frac{2}{29}\right)\) | \(e\left(\frac{49}{58}\right)\) | \(e\left(\frac{37}{58}\right)\) | \(e\left(\frac{41}{58}\right)\) | \(e\left(\frac{39}{58}\right)\) | \(e\left(\frac{1}{58}\right)\) | \(e\left(\frac{23}{29}\right)\) |
| \(\chi_{177}(40,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{9}{58}\right)\) | \(e\left(\frac{9}{29}\right)\) | \(e\left(\frac{27}{29}\right)\) | \(e\left(\frac{23}{29}\right)\) | \(e\left(\frac{27}{58}\right)\) | \(e\left(\frac{5}{58}\right)\) | \(e\left(\frac{51}{58}\right)\) | \(e\left(\frac{57}{58}\right)\) | \(e\left(\frac{55}{58}\right)\) | \(e\left(\frac{18}{29}\right)\) |
| \(\chi_{177}(43,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{33}{58}\right)\) | \(e\left(\frac{4}{29}\right)\) | \(e\left(\frac{12}{29}\right)\) | \(e\left(\frac{7}{29}\right)\) | \(e\left(\frac{41}{58}\right)\) | \(e\left(\frac{57}{58}\right)\) | \(e\left(\frac{13}{58}\right)\) | \(e\left(\frac{35}{58}\right)\) | \(e\left(\frac{47}{58}\right)\) | \(e\left(\frac{8}{29}\right)\) |
| \(\chi_{177}(52,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{47}{58}\right)\) | \(e\left(\frac{18}{29}\right)\) | \(e\left(\frac{25}{29}\right)\) | \(e\left(\frac{17}{29}\right)\) | \(e\left(\frac{25}{58}\right)\) | \(e\left(\frac{39}{58}\right)\) | \(e\left(\frac{15}{58}\right)\) | \(e\left(\frac{27}{58}\right)\) | \(e\left(\frac{23}{58}\right)\) | \(e\left(\frac{7}{29}\right)\) |
| \(\chi_{177}(55,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{58}\right)\) | \(e\left(\frac{2}{29}\right)\) | \(e\left(\frac{6}{29}\right)\) | \(e\left(\frac{18}{29}\right)\) | \(e\left(\frac{35}{58}\right)\) | \(e\left(\frac{43}{58}\right)\) | \(e\left(\frac{21}{58}\right)\) | \(e\left(\frac{3}{58}\right)\) | \(e\left(\frac{9}{58}\right)\) | \(e\left(\frac{4}{29}\right)\) |
| \(\chi_{177}(61,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{58}\right)\) | \(e\left(\frac{1}{29}\right)\) | \(e\left(\frac{3}{29}\right)\) | \(e\left(\frac{9}{29}\right)\) | \(e\left(\frac{3}{58}\right)\) | \(e\left(\frac{7}{58}\right)\) | \(e\left(\frac{25}{58}\right)\) | \(e\left(\frac{45}{58}\right)\) | \(e\left(\frac{19}{58}\right)\) | \(e\left(\frac{2}{29}\right)\) |
| \(\chi_{177}(67,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{58}\right)\) | \(e\left(\frac{3}{29}\right)\) | \(e\left(\frac{9}{29}\right)\) | \(e\left(\frac{27}{29}\right)\) | \(e\left(\frac{9}{58}\right)\) | \(e\left(\frac{21}{58}\right)\) | \(e\left(\frac{17}{58}\right)\) | \(e\left(\frac{19}{58}\right)\) | \(e\left(\frac{57}{58}\right)\) | \(e\left(\frac{6}{29}\right)\) |
| \(\chi_{177}(70,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{25}{58}\right)\) | \(e\left(\frac{25}{29}\right)\) | \(e\left(\frac{17}{29}\right)\) | \(e\left(\frac{22}{29}\right)\) | \(e\left(\frac{17}{58}\right)\) | \(e\left(\frac{1}{58}\right)\) | \(e\left(\frac{45}{58}\right)\) | \(e\left(\frac{23}{58}\right)\) | \(e\left(\frac{11}{58}\right)\) | \(e\left(\frac{21}{29}\right)\) |
| \(\chi_{177}(73,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{58}\right)\) | \(e\left(\frac{19}{29}\right)\) | \(e\left(\frac{28}{29}\right)\) | \(e\left(\frac{26}{29}\right)\) | \(e\left(\frac{57}{58}\right)\) | \(e\left(\frac{17}{58}\right)\) | \(e\left(\frac{11}{58}\right)\) | \(e\left(\frac{43}{58}\right)\) | \(e\left(\frac{13}{58}\right)\) | \(e\left(\frac{9}{29}\right)\) |
| \(\chi_{177}(82,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{15}{58}\right)\) | \(e\left(\frac{15}{29}\right)\) | \(e\left(\frac{16}{29}\right)\) | \(e\left(\frac{19}{29}\right)\) | \(e\left(\frac{45}{58}\right)\) | \(e\left(\frac{47}{58}\right)\) | \(e\left(\frac{27}{58}\right)\) | \(e\left(\frac{37}{58}\right)\) | \(e\left(\frac{53}{58}\right)\) | \(e\left(\frac{1}{29}\right)\) |
| \(\chi_{177}(91,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{58}\right)\) | \(e\left(\frac{5}{29}\right)\) | \(e\left(\frac{15}{29}\right)\) | \(e\left(\frac{16}{29}\right)\) | \(e\left(\frac{15}{58}\right)\) | \(e\left(\frac{35}{58}\right)\) | \(e\left(\frac{9}{58}\right)\) | \(e\left(\frac{51}{58}\right)\) | \(e\left(\frac{37}{58}\right)\) | \(e\left(\frac{10}{29}\right)\) |
| \(\chi_{177}(97,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{39}{58}\right)\) | \(e\left(\frac{10}{29}\right)\) | \(e\left(\frac{1}{29}\right)\) | \(e\left(\frac{3}{29}\right)\) | \(e\left(\frac{1}{58}\right)\) | \(e\left(\frac{41}{58}\right)\) | \(e\left(\frac{47}{58}\right)\) | \(e\left(\frac{15}{58}\right)\) | \(e\left(\frac{45}{58}\right)\) | \(e\left(\frac{20}{29}\right)\) |
| \(\chi_{177}(103,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{27}{58}\right)\) | \(e\left(\frac{27}{29}\right)\) | \(e\left(\frac{23}{29}\right)\) | \(e\left(\frac{11}{29}\right)\) | \(e\left(\frac{23}{58}\right)\) | \(e\left(\frac{15}{58}\right)\) | \(e\left(\frac{37}{58}\right)\) | \(e\left(\frac{55}{58}\right)\) | \(e\left(\frac{49}{58}\right)\) | \(e\left(\frac{25}{29}\right)\) |
| \(\chi_{177}(106,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{58}\right)\) | \(e\left(\frac{23}{29}\right)\) | \(e\left(\frac{11}{29}\right)\) | \(e\left(\frac{4}{29}\right)\) | \(e\left(\frac{11}{58}\right)\) | \(e\left(\frac{45}{58}\right)\) | \(e\left(\frac{53}{58}\right)\) | \(e\left(\frac{49}{58}\right)\) | \(e\left(\frac{31}{58}\right)\) | \(e\left(\frac{17}{29}\right)\) |
| \(\chi_{177}(109,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{58}\right)\) | \(e\left(\frac{13}{29}\right)\) | \(e\left(\frac{10}{29}\right)\) | \(e\left(\frac{1}{29}\right)\) | \(e\left(\frac{39}{58}\right)\) | \(e\left(\frac{33}{58}\right)\) | \(e\left(\frac{35}{58}\right)\) | \(e\left(\frac{5}{58}\right)\) | \(e\left(\frac{15}{58}\right)\) | \(e\left(\frac{26}{29}\right)\) |
| \(\chi_{177}(115,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{21}{58}\right)\) | \(e\left(\frac{21}{29}\right)\) | \(e\left(\frac{5}{29}\right)\) | \(e\left(\frac{15}{29}\right)\) | \(e\left(\frac{5}{58}\right)\) | \(e\left(\frac{31}{58}\right)\) | \(e\left(\frac{3}{58}\right)\) | \(e\left(\frac{17}{58}\right)\) | \(e\left(\frac{51}{58}\right)\) | \(e\left(\frac{13}{29}\right)\) |
| \(\chi_{177}(124,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{51}{58}\right)\) | \(e\left(\frac{22}{29}\right)\) | \(e\left(\frac{8}{29}\right)\) | \(e\left(\frac{24}{29}\right)\) | \(e\left(\frac{37}{58}\right)\) | \(e\left(\frac{9}{58}\right)\) | \(e\left(\frac{57}{58}\right)\) | \(e\left(\frac{33}{58}\right)\) | \(e\left(\frac{41}{58}\right)\) | \(e\left(\frac{15}{29}\right)\) |
| \(\chi_{177}(136,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{58}\right)\) | \(e\left(\frac{14}{29}\right)\) | \(e\left(\frac{13}{29}\right)\) | \(e\left(\frac{10}{29}\right)\) | \(e\left(\frac{13}{58}\right)\) | \(e\left(\frac{11}{58}\right)\) | \(e\left(\frac{31}{58}\right)\) | \(e\left(\frac{21}{58}\right)\) | \(e\left(\frac{5}{58}\right)\) | \(e\left(\frac{28}{29}\right)\) |
| \(\chi_{177}(142,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{53}{58}\right)\) | \(e\left(\frac{24}{29}\right)\) | \(e\left(\frac{14}{29}\right)\) | \(e\left(\frac{13}{29}\right)\) | \(e\left(\frac{43}{58}\right)\) | \(e\left(\frac{23}{58}\right)\) | \(e\left(\frac{49}{58}\right)\) | \(e\left(\frac{7}{58}\right)\) | \(e\left(\frac{21}{58}\right)\) | \(e\left(\frac{19}{29}\right)\) |
| \(\chi_{177}(148,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{57}{58}\right)\) | \(e\left(\frac{28}{29}\right)\) | \(e\left(\frac{26}{29}\right)\) | \(e\left(\frac{20}{29}\right)\) | \(e\left(\frac{55}{58}\right)\) | \(e\left(\frac{51}{58}\right)\) | \(e\left(\frac{33}{58}\right)\) | \(e\left(\frac{13}{58}\right)\) | \(e\left(\frac{39}{58}\right)\) | \(e\left(\frac{27}{29}\right)\) |
| \(\chi_{177}(151,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{58}\right)\) | \(e\left(\frac{17}{29}\right)\) | \(e\left(\frac{22}{29}\right)\) | \(e\left(\frac{8}{29}\right)\) | \(e\left(\frac{51}{58}\right)\) | \(e\left(\frac{3}{58}\right)\) | \(e\left(\frac{19}{58}\right)\) | \(e\left(\frac{11}{58}\right)\) | \(e\left(\frac{33}{58}\right)\) | \(e\left(\frac{5}{29}\right)\) |
| \(\chi_{177}(157,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{37}{58}\right)\) | \(e\left(\frac{8}{29}\right)\) | \(e\left(\frac{24}{29}\right)\) | \(e\left(\frac{14}{29}\right)\) | \(e\left(\frac{53}{58}\right)\) | \(e\left(\frac{27}{58}\right)\) | \(e\left(\frac{55}{58}\right)\) | \(e\left(\frac{41}{58}\right)\) | \(e\left(\frac{7}{58}\right)\) | \(e\left(\frac{16}{29}\right)\) |
| \(\chi_{177}(160,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{58}\right)\) | \(e\left(\frac{11}{29}\right)\) | \(e\left(\frac{4}{29}\right)\) | \(e\left(\frac{12}{29}\right)\) | \(e\left(\frac{33}{58}\right)\) | \(e\left(\frac{19}{58}\right)\) | \(e\left(\frac{43}{58}\right)\) | \(e\left(\frac{31}{58}\right)\) | \(e\left(\frac{35}{58}\right)\) | \(e\left(\frac{22}{29}\right)\) |
| \(\chi_{177}(172,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{35}{58}\right)\) | \(e\left(\frac{6}{29}\right)\) | \(e\left(\frac{18}{29}\right)\) | \(e\left(\frac{25}{29}\right)\) | \(e\left(\frac{47}{58}\right)\) | \(e\left(\frac{13}{58}\right)\) | \(e\left(\frac{5}{58}\right)\) | \(e\left(\frac{9}{58}\right)\) | \(e\left(\frac{27}{58}\right)\) | \(e\left(\frac{12}{29}\right)\) |