from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(289, base_ring=CyclotomicField(68))
M = H._module
chi = DirichletCharacter(H, M([27]))
chi.galois_orbit()
[g,chi] = znchar(Mod(4,289))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(289\) | |
| Conductor: | \(289\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(68\) | 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: | $\Q(\zeta_{68})$ |
| Fixed field: | Number field defined by a degree 68 polynomial |
First 31 of 32 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{289}(4,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{15}{34}\right)\) | \(e\left(\frac{27}{68}\right)\) | \(e\left(\frac{15}{17}\right)\) | \(e\left(\frac{63}{68}\right)\) | \(e\left(\frac{57}{68}\right)\) | \(e\left(\frac{37}{68}\right)\) | \(e\left(\frac{11}{34}\right)\) | \(e\left(\frac{27}{34}\right)\) | \(e\left(\frac{25}{68}\right)\) | \(e\left(\frac{9}{68}\right)\) |
| \(\chi_{289}(13,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{34}\right)\) | \(e\left(\frac{49}{68}\right)\) | \(e\left(\frac{14}{17}\right)\) | \(e\left(\frac{1}{68}\right)\) | \(e\left(\frac{43}{68}\right)\) | \(e\left(\frac{47}{68}\right)\) | \(e\left(\frac{25}{34}\right)\) | \(e\left(\frac{15}{34}\right)\) | \(e\left(\frac{63}{68}\right)\) | \(e\left(\frac{39}{68}\right)\) |
| \(\chi_{289}(21,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{33}{34}\right)\) | \(e\left(\frac{39}{68}\right)\) | \(e\left(\frac{16}{17}\right)\) | \(e\left(\frac{23}{68}\right)\) | \(e\left(\frac{37}{68}\right)\) | \(e\left(\frac{61}{68}\right)\) | \(e\left(\frac{31}{34}\right)\) | \(e\left(\frac{5}{34}\right)\) | \(e\left(\frac{21}{68}\right)\) | \(e\left(\frac{13}{68}\right)\) |
| \(\chi_{289}(30,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{34}\right)\) | \(e\left(\frac{37}{68}\right)\) | \(e\left(\frac{13}{17}\right)\) | \(e\left(\frac{41}{68}\right)\) | \(e\left(\frac{63}{68}\right)\) | \(e\left(\frac{23}{68}\right)\) | \(e\left(\frac{5}{34}\right)\) | \(e\left(\frac{3}{34}\right)\) | \(e\left(\frac{67}{68}\right)\) | \(e\left(\frac{35}{68}\right)\) |
| \(\chi_{289}(47,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{34}\right)\) | \(e\left(\frac{25}{68}\right)\) | \(e\left(\frac{12}{17}\right)\) | \(e\left(\frac{13}{68}\right)\) | \(e\left(\frac{15}{68}\right)\) | \(e\left(\frac{67}{68}\right)\) | \(e\left(\frac{19}{34}\right)\) | \(e\left(\frac{25}{34}\right)\) | \(e\left(\frac{3}{68}\right)\) | \(e\left(\frac{31}{68}\right)\) |
| \(\chi_{289}(55,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{34}\right)\) | \(e\left(\frac{63}{68}\right)\) | \(e\left(\frac{1}{17}\right)\) | \(e\left(\frac{11}{68}\right)\) | \(e\left(\frac{65}{68}\right)\) | \(e\left(\frac{41}{68}\right)\) | \(e\left(\frac{3}{34}\right)\) | \(e\left(\frac{29}{34}\right)\) | \(e\left(\frac{13}{68}\right)\) | \(e\left(\frac{21}{68}\right)\) |
| \(\chi_{289}(64,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{34}\right)\) | \(e\left(\frac{13}{68}\right)\) | \(e\left(\frac{11}{17}\right)\) | \(e\left(\frac{53}{68}\right)\) | \(e\left(\frac{35}{68}\right)\) | \(e\left(\frac{43}{68}\right)\) | \(e\left(\frac{33}{34}\right)\) | \(e\left(\frac{13}{34}\right)\) | \(e\left(\frac{7}{68}\right)\) | \(e\left(\frac{27}{68}\right)\) |
| \(\chi_{289}(72,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{34}\right)\) | \(e\left(\frac{7}{68}\right)\) | \(e\left(\frac{2}{17}\right)\) | \(e\left(\frac{39}{68}\right)\) | \(e\left(\frac{45}{68}\right)\) | \(e\left(\frac{65}{68}\right)\) | \(e\left(\frac{23}{34}\right)\) | \(e\left(\frac{7}{34}\right)\) | \(e\left(\frac{9}{68}\right)\) | \(e\left(\frac{25}{68}\right)\) |
| \(\chi_{289}(81,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{27}{34}\right)\) | \(e\left(\frac{1}{68}\right)\) | \(e\left(\frac{10}{17}\right)\) | \(e\left(\frac{25}{68}\right)\) | \(e\left(\frac{55}{68}\right)\) | \(e\left(\frac{19}{68}\right)\) | \(e\left(\frac{13}{34}\right)\) | \(e\left(\frac{1}{34}\right)\) | \(e\left(\frac{11}{68}\right)\) | \(e\left(\frac{23}{68}\right)\) |
| \(\chi_{289}(89,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{34}\right)\) | \(e\left(\frac{19}{68}\right)\) | \(e\left(\frac{3}{17}\right)\) | \(e\left(\frac{67}{68}\right)\) | \(e\left(\frac{25}{68}\right)\) | \(e\left(\frac{21}{68}\right)\) | \(e\left(\frac{9}{34}\right)\) | \(e\left(\frac{19}{34}\right)\) | \(e\left(\frac{5}{68}\right)\) | \(e\left(\frac{29}{68}\right)\) |
| \(\chi_{289}(98,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{34}\right)\) | \(e\left(\frac{57}{68}\right)\) | \(e\left(\frac{9}{17}\right)\) | \(e\left(\frac{65}{68}\right)\) | \(e\left(\frac{7}{68}\right)\) | \(e\left(\frac{63}{68}\right)\) | \(e\left(\frac{27}{34}\right)\) | \(e\left(\frac{23}{34}\right)\) | \(e\left(\frac{15}{68}\right)\) | \(e\left(\frac{19}{68}\right)\) |
| \(\chi_{289}(106,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{21}{34}\right)\) | \(e\left(\frac{31}{68}\right)\) | \(e\left(\frac{4}{17}\right)\) | \(e\left(\frac{27}{68}\right)\) | \(e\left(\frac{5}{68}\right)\) | \(e\left(\frac{45}{68}\right)\) | \(e\left(\frac{29}{34}\right)\) | \(e\left(\frac{31}{34}\right)\) | \(e\left(\frac{1}{68}\right)\) | \(e\left(\frac{33}{68}\right)\) |
| \(\chi_{289}(115,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{34}\right)\) | \(e\left(\frac{45}{68}\right)\) | \(e\left(\frac{8}{17}\right)\) | \(e\left(\frac{37}{68}\right)\) | \(e\left(\frac{27}{68}\right)\) | \(e\left(\frac{39}{68}\right)\) | \(e\left(\frac{7}{34}\right)\) | \(e\left(\frac{11}{34}\right)\) | \(e\left(\frac{19}{68}\right)\) | \(e\left(\frac{15}{68}\right)\) |
| \(\chi_{289}(123,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{34}\right)\) | \(e\left(\frac{43}{68}\right)\) | \(e\left(\frac{5}{17}\right)\) | \(e\left(\frac{55}{68}\right)\) | \(e\left(\frac{53}{68}\right)\) | \(e\left(\frac{1}{68}\right)\) | \(e\left(\frac{15}{34}\right)\) | \(e\left(\frac{9}{34}\right)\) | \(e\left(\frac{65}{68}\right)\) | \(e\left(\frac{37}{68}\right)\) |
| \(\chi_{289}(132,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{34}\right)\) | \(e\left(\frac{33}{68}\right)\) | \(e\left(\frac{7}{17}\right)\) | \(e\left(\frac{9}{68}\right)\) | \(e\left(\frac{47}{68}\right)\) | \(e\left(\frac{15}{68}\right)\) | \(e\left(\frac{21}{34}\right)\) | \(e\left(\frac{33}{34}\right)\) | \(e\left(\frac{23}{68}\right)\) | \(e\left(\frac{11}{68}\right)\) |
| \(\chi_{289}(140,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{34}\right)\) | \(e\left(\frac{55}{68}\right)\) | \(e\left(\frac{6}{17}\right)\) | \(e\left(\frac{15}{68}\right)\) | \(e\left(\frac{33}{68}\right)\) | \(e\left(\frac{25}{68}\right)\) | \(e\left(\frac{1}{34}\right)\) | \(e\left(\frac{21}{34}\right)\) | \(e\left(\frac{61}{68}\right)\) | \(e\left(\frac{41}{68}\right)\) |
| \(\chi_{289}(149,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{34}\right)\) | \(e\left(\frac{21}{68}\right)\) | \(e\left(\frac{6}{17}\right)\) | \(e\left(\frac{49}{68}\right)\) | \(e\left(\frac{67}{68}\right)\) | \(e\left(\frac{59}{68}\right)\) | \(e\left(\frac{1}{34}\right)\) | \(e\left(\frac{21}{34}\right)\) | \(e\left(\frac{27}{68}\right)\) | \(e\left(\frac{7}{68}\right)\) |
| \(\chi_{289}(157,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{34}\right)\) | \(e\left(\frac{67}{68}\right)\) | \(e\left(\frac{7}{17}\right)\) | \(e\left(\frac{43}{68}\right)\) | \(e\left(\frac{13}{68}\right)\) | \(e\left(\frac{49}{68}\right)\) | \(e\left(\frac{21}{34}\right)\) | \(e\left(\frac{33}{34}\right)\) | \(e\left(\frac{57}{68}\right)\) | \(e\left(\frac{45}{68}\right)\) |
| \(\chi_{289}(166,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{34}\right)\) | \(e\left(\frac{9}{68}\right)\) | \(e\left(\frac{5}{17}\right)\) | \(e\left(\frac{21}{68}\right)\) | \(e\left(\frac{19}{68}\right)\) | \(e\left(\frac{35}{68}\right)\) | \(e\left(\frac{15}{34}\right)\) | \(e\left(\frac{9}{34}\right)\) | \(e\left(\frac{31}{68}\right)\) | \(e\left(\frac{3}{68}\right)\) |
| \(\chi_{289}(174,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{34}\right)\) | \(e\left(\frac{11}{68}\right)\) | \(e\left(\frac{8}{17}\right)\) | \(e\left(\frac{3}{68}\right)\) | \(e\left(\frac{61}{68}\right)\) | \(e\left(\frac{5}{68}\right)\) | \(e\left(\frac{7}{34}\right)\) | \(e\left(\frac{11}{34}\right)\) | \(e\left(\frac{53}{68}\right)\) | \(e\left(\frac{49}{68}\right)\) |
| \(\chi_{289}(183,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{21}{34}\right)\) | \(e\left(\frac{65}{68}\right)\) | \(e\left(\frac{4}{17}\right)\) | \(e\left(\frac{61}{68}\right)\) | \(e\left(\frac{39}{68}\right)\) | \(e\left(\frac{11}{68}\right)\) | \(e\left(\frac{29}{34}\right)\) | \(e\left(\frac{31}{34}\right)\) | \(e\left(\frac{35}{68}\right)\) | \(e\left(\frac{67}{68}\right)\) |
| \(\chi_{289}(191,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{34}\right)\) | \(e\left(\frac{23}{68}\right)\) | \(e\left(\frac{9}{17}\right)\) | \(e\left(\frac{31}{68}\right)\) | \(e\left(\frac{41}{68}\right)\) | \(e\left(\frac{29}{68}\right)\) | \(e\left(\frac{27}{34}\right)\) | \(e\left(\frac{23}{34}\right)\) | \(e\left(\frac{49}{68}\right)\) | \(e\left(\frac{53}{68}\right)\) |
| \(\chi_{289}(200,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{34}\right)\) | \(e\left(\frac{53}{68}\right)\) | \(e\left(\frac{3}{17}\right)\) | \(e\left(\frac{33}{68}\right)\) | \(e\left(\frac{59}{68}\right)\) | \(e\left(\frac{55}{68}\right)\) | \(e\left(\frac{9}{34}\right)\) | \(e\left(\frac{19}{34}\right)\) | \(e\left(\frac{39}{68}\right)\) | \(e\left(\frac{63}{68}\right)\) |
| \(\chi_{289}(208,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{27}{34}\right)\) | \(e\left(\frac{35}{68}\right)\) | \(e\left(\frac{10}{17}\right)\) | \(e\left(\frac{59}{68}\right)\) | \(e\left(\frac{21}{68}\right)\) | \(e\left(\frac{53}{68}\right)\) | \(e\left(\frac{13}{34}\right)\) | \(e\left(\frac{1}{34}\right)\) | \(e\left(\frac{45}{68}\right)\) | \(e\left(\frac{57}{68}\right)\) |
| \(\chi_{289}(217,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{34}\right)\) | \(e\left(\frac{41}{68}\right)\) | \(e\left(\frac{2}{17}\right)\) | \(e\left(\frac{5}{68}\right)\) | \(e\left(\frac{11}{68}\right)\) | \(e\left(\frac{31}{68}\right)\) | \(e\left(\frac{23}{34}\right)\) | \(e\left(\frac{7}{34}\right)\) | \(e\left(\frac{43}{68}\right)\) | \(e\left(\frac{59}{68}\right)\) |
| \(\chi_{289}(225,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{34}\right)\) | \(e\left(\frac{47}{68}\right)\) | \(e\left(\frac{11}{17}\right)\) | \(e\left(\frac{19}{68}\right)\) | \(e\left(\frac{1}{68}\right)\) | \(e\left(\frac{9}{68}\right)\) | \(e\left(\frac{33}{34}\right)\) | \(e\left(\frac{13}{34}\right)\) | \(e\left(\frac{41}{68}\right)\) | \(e\left(\frac{61}{68}\right)\) |
| \(\chi_{289}(234,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{34}\right)\) | \(e\left(\frac{29}{68}\right)\) | \(e\left(\frac{1}{17}\right)\) | \(e\left(\frac{45}{68}\right)\) | \(e\left(\frac{31}{68}\right)\) | \(e\left(\frac{7}{68}\right)\) | \(e\left(\frac{3}{34}\right)\) | \(e\left(\frac{29}{34}\right)\) | \(e\left(\frac{47}{68}\right)\) | \(e\left(\frac{55}{68}\right)\) |
| \(\chi_{289}(242,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{34}\right)\) | \(e\left(\frac{59}{68}\right)\) | \(e\left(\frac{12}{17}\right)\) | \(e\left(\frac{47}{68}\right)\) | \(e\left(\frac{49}{68}\right)\) | \(e\left(\frac{33}{68}\right)\) | \(e\left(\frac{19}{34}\right)\) | \(e\left(\frac{25}{34}\right)\) | \(e\left(\frac{37}{68}\right)\) | \(e\left(\frac{65}{68}\right)\) |
| \(\chi_{289}(259,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{34}\right)\) | \(e\left(\frac{3}{68}\right)\) | \(e\left(\frac{13}{17}\right)\) | \(e\left(\frac{7}{68}\right)\) | \(e\left(\frac{29}{68}\right)\) | \(e\left(\frac{57}{68}\right)\) | \(e\left(\frac{5}{34}\right)\) | \(e\left(\frac{3}{34}\right)\) | \(e\left(\frac{33}{68}\right)\) | \(e\left(\frac{1}{68}\right)\) |
| \(\chi_{289}(268,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{33}{34}\right)\) | \(e\left(\frac{5}{68}\right)\) | \(e\left(\frac{16}{17}\right)\) | \(e\left(\frac{57}{68}\right)\) | \(e\left(\frac{3}{68}\right)\) | \(e\left(\frac{27}{68}\right)\) | \(e\left(\frac{31}{34}\right)\) | \(e\left(\frac{5}{34}\right)\) | \(e\left(\frac{55}{68}\right)\) | \(e\left(\frac{47}{68}\right)\) |
| \(\chi_{289}(276,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{34}\right)\) | \(e\left(\frac{15}{68}\right)\) | \(e\left(\frac{14}{17}\right)\) | \(e\left(\frac{35}{68}\right)\) | \(e\left(\frac{9}{68}\right)\) | \(e\left(\frac{13}{68}\right)\) | \(e\left(\frac{25}{34}\right)\) | \(e\left(\frac{15}{34}\right)\) | \(e\left(\frac{29}{68}\right)\) | \(e\left(\frac{5}{68}\right)\) |