from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1156, base_ring=CyclotomicField(68))
M = H._module
chi = DirichletCharacter(H, M([34,25]))
chi.galois_orbit()
[g,chi] = znchar(Mod(47,1156))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(1156\) | |
| Conductor: | \(1156\) | 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: | odd | 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 | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(19\) | \(21\) | \(23\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{1156}(47,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{68}\right)\) | \(e\left(\frac{13}{68}\right)\) | \(e\left(\frac{33}{68}\right)\) | \(e\left(\frac{25}{34}\right)\) | \(e\left(\frac{65}{68}\right)\) | \(e\left(\frac{1}{17}\right)\) | \(e\left(\frac{1}{17}\right)\) | \(e\left(\frac{11}{17}\right)\) | \(e\left(\frac{6}{17}\right)\) | \(e\left(\frac{33}{68}\right)\) |
| \(\chi_{1156}(55,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{68}\right)\) | \(e\left(\frac{11}{68}\right)\) | \(e\left(\frac{7}{68}\right)\) | \(e\left(\frac{29}{34}\right)\) | \(e\left(\frac{55}{68}\right)\) | \(e\left(\frac{10}{17}\right)\) | \(e\left(\frac{10}{17}\right)\) | \(e\left(\frac{8}{17}\right)\) | \(e\left(\frac{9}{17}\right)\) | \(e\left(\frac{7}{68}\right)\) |
| \(\chi_{1156}(115,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{68}\right)\) | \(e\left(\frac{37}{68}\right)\) | \(e\left(\frac{5}{68}\right)\) | \(e\left(\frac{11}{34}\right)\) | \(e\left(\frac{49}{68}\right)\) | \(e\left(\frac{12}{17}\right)\) | \(e\left(\frac{12}{17}\right)\) | \(e\left(\frac{13}{17}\right)\) | \(e\left(\frac{4}{17}\right)\) | \(e\left(\frac{5}{68}\right)\) |
| \(\chi_{1156}(123,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{9}{68}\right)\) | \(e\left(\frac{55}{68}\right)\) | \(e\left(\frac{35}{68}\right)\) | \(e\left(\frac{9}{34}\right)\) | \(e\left(\frac{3}{68}\right)\) | \(e\left(\frac{16}{17}\right)\) | \(e\left(\frac{16}{17}\right)\) | \(e\left(\frac{6}{17}\right)\) | \(e\left(\frac{11}{17}\right)\) | \(e\left(\frac{35}{68}\right)\) |
| \(\chi_{1156}(183,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{68}\right)\) | \(e\left(\frac{61}{68}\right)\) | \(e\left(\frac{45}{68}\right)\) | \(e\left(\frac{31}{34}\right)\) | \(e\left(\frac{33}{68}\right)\) | \(e\left(\frac{6}{17}\right)\) | \(e\left(\frac{6}{17}\right)\) | \(e\left(\frac{15}{17}\right)\) | \(e\left(\frac{2}{17}\right)\) | \(e\left(\frac{45}{68}\right)\) |
| \(\chi_{1156}(191,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{57}{68}\right)\) | \(e\left(\frac{31}{68}\right)\) | \(e\left(\frac{63}{68}\right)\) | \(e\left(\frac{23}{34}\right)\) | \(e\left(\frac{19}{68}\right)\) | \(e\left(\frac{5}{17}\right)\) | \(e\left(\frac{5}{17}\right)\) | \(e\left(\frac{4}{17}\right)\) | \(e\left(\frac{13}{17}\right)\) | \(e\left(\frac{63}{68}\right)\) |
| \(\chi_{1156}(259,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{37}{68}\right)\) | \(e\left(\frac{7}{68}\right)\) | \(e\left(\frac{23}{68}\right)\) | \(e\left(\frac{3}{34}\right)\) | \(e\left(\frac{35}{68}\right)\) | \(e\left(\frac{11}{17}\right)\) | \(e\left(\frac{11}{17}\right)\) | \(e\left(\frac{2}{17}\right)\) | \(e\left(\frac{15}{17}\right)\) | \(e\left(\frac{23}{68}\right)\) |
| \(\chi_{1156}(319,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{68}\right)\) | \(e\left(\frac{41}{68}\right)\) | \(e\left(\frac{57}{68}\right)\) | \(e\left(\frac{3}{34}\right)\) | \(e\left(\frac{1}{68}\right)\) | \(e\left(\frac{11}{17}\right)\) | \(e\left(\frac{11}{17}\right)\) | \(e\left(\frac{2}{17}\right)\) | \(e\left(\frac{15}{17}\right)\) | \(e\left(\frac{57}{68}\right)\) |
| \(\chi_{1156}(387,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{68}\right)\) | \(e\left(\frac{65}{68}\right)\) | \(e\left(\frac{29}{68}\right)\) | \(e\left(\frac{23}{34}\right)\) | \(e\left(\frac{53}{68}\right)\) | \(e\left(\frac{5}{17}\right)\) | \(e\left(\frac{5}{17}\right)\) | \(e\left(\frac{4}{17}\right)\) | \(e\left(\frac{13}{17}\right)\) | \(e\left(\frac{29}{68}\right)\) |
| \(\chi_{1156}(395,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{65}{68}\right)\) | \(e\left(\frac{27}{68}\right)\) | \(e\left(\frac{11}{68}\right)\) | \(e\left(\frac{31}{34}\right)\) | \(e\left(\frac{67}{68}\right)\) | \(e\left(\frac{6}{17}\right)\) | \(e\left(\frac{6}{17}\right)\) | \(e\left(\frac{15}{17}\right)\) | \(e\left(\frac{2}{17}\right)\) | \(e\left(\frac{11}{68}\right)\) |
| \(\chi_{1156}(455,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{68}\right)\) | \(e\left(\frac{21}{68}\right)\) | \(e\left(\frac{1}{68}\right)\) | \(e\left(\frac{9}{34}\right)\) | \(e\left(\frac{37}{68}\right)\) | \(e\left(\frac{16}{17}\right)\) | \(e\left(\frac{16}{17}\right)\) | \(e\left(\frac{6}{17}\right)\) | \(e\left(\frac{11}{17}\right)\) | \(e\left(\frac{1}{68}\right)\) |
| \(\chi_{1156}(463,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{45}{68}\right)\) | \(e\left(\frac{3}{68}\right)\) | \(e\left(\frac{39}{68}\right)\) | \(e\left(\frac{11}{34}\right)\) | \(e\left(\frac{15}{68}\right)\) | \(e\left(\frac{12}{17}\right)\) | \(e\left(\frac{12}{17}\right)\) | \(e\left(\frac{13}{17}\right)\) | \(e\left(\frac{4}{17}\right)\) | \(e\left(\frac{39}{68}\right)\) |
| \(\chi_{1156}(523,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{63}{68}\right)\) | \(e\left(\frac{45}{68}\right)\) | \(e\left(\frac{41}{68}\right)\) | \(e\left(\frac{29}{34}\right)\) | \(e\left(\frac{21}{68}\right)\) | \(e\left(\frac{10}{17}\right)\) | \(e\left(\frac{10}{17}\right)\) | \(e\left(\frac{8}{17}\right)\) | \(e\left(\frac{9}{17}\right)\) | \(e\left(\frac{41}{68}\right)\) |
| \(\chi_{1156}(531,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{25}{68}\right)\) | \(e\left(\frac{47}{68}\right)\) | \(e\left(\frac{67}{68}\right)\) | \(e\left(\frac{25}{34}\right)\) | \(e\left(\frac{31}{68}\right)\) | \(e\left(\frac{1}{17}\right)\) | \(e\left(\frac{1}{17}\right)\) | \(e\left(\frac{11}{17}\right)\) | \(e\left(\frac{6}{17}\right)\) | \(e\left(\frac{67}{68}\right)\) |
| \(\chi_{1156}(591,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{15}{68}\right)\) | \(e\left(\frac{1}{68}\right)\) | \(e\left(\frac{13}{68}\right)\) | \(e\left(\frac{15}{34}\right)\) | \(e\left(\frac{5}{68}\right)\) | \(e\left(\frac{4}{17}\right)\) | \(e\left(\frac{4}{17}\right)\) | \(e\left(\frac{10}{17}\right)\) | \(e\left(\frac{7}{17}\right)\) | \(e\left(\frac{13}{68}\right)\) |
| \(\chi_{1156}(599,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{68}\right)\) | \(e\left(\frac{23}{68}\right)\) | \(e\left(\frac{27}{68}\right)\) | \(e\left(\frac{5}{34}\right)\) | \(e\left(\frac{47}{68}\right)\) | \(e\left(\frac{7}{17}\right)\) | \(e\left(\frac{7}{17}\right)\) | \(e\left(\frac{9}{17}\right)\) | \(e\left(\frac{8}{17}\right)\) | \(e\left(\frac{27}{68}\right)\) |
| \(\chi_{1156}(659,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{35}{68}\right)\) | \(e\left(\frac{25}{68}\right)\) | \(e\left(\frac{53}{68}\right)\) | \(e\left(\frac{1}{34}\right)\) | \(e\left(\frac{57}{68}\right)\) | \(e\left(\frac{15}{17}\right)\) | \(e\left(\frac{15}{17}\right)\) | \(e\left(\frac{12}{17}\right)\) | \(e\left(\frac{5}{17}\right)\) | \(e\left(\frac{53}{68}\right)\) |
| \(\chi_{1156}(667,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{53}{68}\right)\) | \(e\left(\frac{67}{68}\right)\) | \(e\left(\frac{55}{68}\right)\) | \(e\left(\frac{19}{34}\right)\) | \(e\left(\frac{63}{68}\right)\) | \(e\left(\frac{13}{17}\right)\) | \(e\left(\frac{13}{17}\right)\) | \(e\left(\frac{7}{17}\right)\) | \(e\left(\frac{10}{17}\right)\) | \(e\left(\frac{55}{68}\right)\) |
| \(\chi_{1156}(727,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{55}{68}\right)\) | \(e\left(\frac{49}{68}\right)\) | \(e\left(\frac{25}{68}\right)\) | \(e\left(\frac{21}{34}\right)\) | \(e\left(\frac{41}{68}\right)\) | \(e\left(\frac{9}{17}\right)\) | \(e\left(\frac{9}{17}\right)\) | \(e\left(\frac{14}{17}\right)\) | \(e\left(\frac{3}{17}\right)\) | \(e\left(\frac{25}{68}\right)\) |
| \(\chi_{1156}(735,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{33}{68}\right)\) | \(e\left(\frac{43}{68}\right)\) | \(e\left(\frac{15}{68}\right)\) | \(e\left(\frac{33}{34}\right)\) | \(e\left(\frac{11}{68}\right)\) | \(e\left(\frac{2}{17}\right)\) | \(e\left(\frac{2}{17}\right)\) | \(e\left(\frac{5}{17}\right)\) | \(e\left(\frac{12}{17}\right)\) | \(e\left(\frac{15}{68}\right)\) |
| \(\chi_{1156}(795,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{68}\right)\) | \(e\left(\frac{5}{68}\right)\) | \(e\left(\frac{65}{68}\right)\) | \(e\left(\frac{7}{34}\right)\) | \(e\left(\frac{25}{68}\right)\) | \(e\left(\frac{3}{17}\right)\) | \(e\left(\frac{3}{17}\right)\) | \(e\left(\frac{16}{17}\right)\) | \(e\left(\frac{1}{17}\right)\) | \(e\left(\frac{65}{68}\right)\) |
| \(\chi_{1156}(803,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{68}\right)\) | \(e\left(\frac{19}{68}\right)\) | \(e\left(\frac{43}{68}\right)\) | \(e\left(\frac{13}{34}\right)\) | \(e\left(\frac{27}{68}\right)\) | \(e\left(\frac{8}{17}\right)\) | \(e\left(\frac{8}{17}\right)\) | \(e\left(\frac{3}{17}\right)\) | \(e\left(\frac{14}{17}\right)\) | \(e\left(\frac{43}{68}\right)\) |
| \(\chi_{1156}(863,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{27}{68}\right)\) | \(e\left(\frac{29}{68}\right)\) | \(e\left(\frac{37}{68}\right)\) | \(e\left(\frac{27}{34}\right)\) | \(e\left(\frac{9}{68}\right)\) | \(e\left(\frac{14}{17}\right)\) | \(e\left(\frac{14}{17}\right)\) | \(e\left(\frac{1}{17}\right)\) | \(e\left(\frac{16}{17}\right)\) | \(e\left(\frac{37}{68}\right)\) |
| \(\chi_{1156}(871,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{61}{68}\right)\) | \(e\left(\frac{63}{68}\right)\) | \(e\left(\frac{3}{68}\right)\) | \(e\left(\frac{27}{34}\right)\) | \(e\left(\frac{43}{68}\right)\) | \(e\left(\frac{14}{17}\right)\) | \(e\left(\frac{14}{17}\right)\) | \(e\left(\frac{1}{17}\right)\) | \(e\left(\frac{16}{17}\right)\) | \(e\left(\frac{3}{68}\right)\) |
| \(\chi_{1156}(931,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{47}{68}\right)\) | \(e\left(\frac{53}{68}\right)\) | \(e\left(\frac{9}{68}\right)\) | \(e\left(\frac{13}{34}\right)\) | \(e\left(\frac{61}{68}\right)\) | \(e\left(\frac{8}{17}\right)\) | \(e\left(\frac{8}{17}\right)\) | \(e\left(\frac{3}{17}\right)\) | \(e\left(\frac{14}{17}\right)\) | \(e\left(\frac{9}{68}\right)\) |
| \(\chi_{1156}(939,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{41}{68}\right)\) | \(e\left(\frac{39}{68}\right)\) | \(e\left(\frac{31}{68}\right)\) | \(e\left(\frac{7}{34}\right)\) | \(e\left(\frac{59}{68}\right)\) | \(e\left(\frac{3}{17}\right)\) | \(e\left(\frac{3}{17}\right)\) | \(e\left(\frac{16}{17}\right)\) | \(e\left(\frac{1}{17}\right)\) | \(e\left(\frac{31}{68}\right)\) |
| \(\chi_{1156}(999,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{67}{68}\right)\) | \(e\left(\frac{9}{68}\right)\) | \(e\left(\frac{49}{68}\right)\) | \(e\left(\frac{33}{34}\right)\) | \(e\left(\frac{45}{68}\right)\) | \(e\left(\frac{2}{17}\right)\) | \(e\left(\frac{2}{17}\right)\) | \(e\left(\frac{5}{17}\right)\) | \(e\left(\frac{12}{17}\right)\) | \(e\left(\frac{49}{68}\right)\) |
| \(\chi_{1156}(1007,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{21}{68}\right)\) | \(e\left(\frac{15}{68}\right)\) | \(e\left(\frac{59}{68}\right)\) | \(e\left(\frac{21}{34}\right)\) | \(e\left(\frac{7}{68}\right)\) | \(e\left(\frac{9}{17}\right)\) | \(e\left(\frac{9}{17}\right)\) | \(e\left(\frac{14}{17}\right)\) | \(e\left(\frac{3}{17}\right)\) | \(e\left(\frac{59}{68}\right)\) |
| \(\chi_{1156}(1067,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{68}\right)\) | \(e\left(\frac{33}{68}\right)\) | \(e\left(\frac{21}{68}\right)\) | \(e\left(\frac{19}{34}\right)\) | \(e\left(\frac{29}{68}\right)\) | \(e\left(\frac{13}{17}\right)\) | \(e\left(\frac{13}{17}\right)\) | \(e\left(\frac{7}{17}\right)\) | \(e\left(\frac{10}{17}\right)\) | \(e\left(\frac{21}{68}\right)\) |
| \(\chi_{1156}(1075,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{68}\right)\) | \(e\left(\frac{59}{68}\right)\) | \(e\left(\frac{19}{68}\right)\) | \(e\left(\frac{1}{34}\right)\) | \(e\left(\frac{23}{68}\right)\) | \(e\left(\frac{15}{17}\right)\) | \(e\left(\frac{15}{17}\right)\) | \(e\left(\frac{12}{17}\right)\) | \(e\left(\frac{5}{17}\right)\) | \(e\left(\frac{19}{68}\right)\) |
| \(\chi_{1156}(1135,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{39}{68}\right)\) | \(e\left(\frac{57}{68}\right)\) | \(e\left(\frac{61}{68}\right)\) | \(e\left(\frac{5}{34}\right)\) | \(e\left(\frac{13}{68}\right)\) | \(e\left(\frac{7}{17}\right)\) | \(e\left(\frac{7}{17}\right)\) | \(e\left(\frac{9}{17}\right)\) | \(e\left(\frac{8}{17}\right)\) | \(e\left(\frac{61}{68}\right)\) |