from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(916, base_ring=CyclotomicField(76))
M = H._module
chi = DirichletCharacter(H, M([0,39]))
chi.galois_orbit()
[g,chi] = znchar(Mod(13,916))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(916\) | |
| Conductor: | \(229\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(76\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 229.j | 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: | $\Q(\zeta_{76})$ |
| Fixed field: | Number field defined by a degree 76 polynomial |
First 31 of 36 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{916}(13,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{11}{38}\right)\) | \(e\left(\frac{69}{76}\right)\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{5}{38}\right)\) | \(e\left(\frac{3}{76}\right)\) | \(e\left(\frac{1}{38}\right)\) | \(e\left(\frac{15}{19}\right)\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{49}{76}\right)\) |
| \(\chi_{916}(21,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{19}\right)\) | \(e\left(\frac{15}{38}\right)\) | \(e\left(\frac{63}{76}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{31}{38}\right)\) | \(e\left(\frac{49}{76}\right)\) | \(e\left(\frac{29}{38}\right)\) | \(e\left(\frac{17}{19}\right)\) | \(e\left(\frac{2}{19}\right)\) | \(e\left(\frac{15}{76}\right)\) |
| \(\chi_{916}(93,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{33}{38}\right)\) | \(e\left(\frac{17}{76}\right)\) | \(e\left(\frac{8}{19}\right)\) | \(e\left(\frac{15}{38}\right)\) | \(e\left(\frac{47}{76}\right)\) | \(e\left(\frac{3}{38}\right)\) | \(e\left(\frac{7}{19}\right)\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{33}{76}\right)\) |
| \(\chi_{916}(101,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{2}{19}\right)\) | \(e\left(\frac{7}{38}\right)\) | \(e\left(\frac{37}{76}\right)\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{17}{38}\right)\) | \(e\left(\frac{71}{76}\right)\) | \(e\left(\frac{11}{38}\right)\) | \(e\left(\frac{13}{19}\right)\) | \(e\left(\frac{6}{19}\right)\) | \(e\left(\frac{45}{76}\right)\) |
| \(\chi_{916}(109,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{23}{38}\right)\) | \(e\left(\frac{51}{76}\right)\) | \(e\left(\frac{5}{19}\right)\) | \(e\left(\frac{7}{38}\right)\) | \(e\left(\frac{65}{76}\right)\) | \(e\left(\frac{9}{38}\right)\) | \(e\left(\frac{2}{19}\right)\) | \(e\left(\frac{17}{19}\right)\) | \(e\left(\frac{23}{76}\right)\) |
| \(\chi_{916}(141,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{19}\right)\) | \(e\left(\frac{27}{38}\right)\) | \(e\left(\frac{7}{76}\right)\) | \(e\left(\frac{10}{19}\right)\) | \(e\left(\frac{33}{38}\right)\) | \(e\left(\frac{73}{76}\right)\) | \(e\left(\frac{37}{38}\right)\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{15}{19}\right)\) | \(e\left(\frac{27}{76}\right)\) |
| \(\chi_{916}(145,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{19}\right)\) | \(e\left(\frac{17}{38}\right)\) | \(e\left(\frac{3}{76}\right)\) | \(e\left(\frac{7}{19}\right)\) | \(e\left(\frac{25}{38}\right)\) | \(e\left(\frac{53}{76}\right)\) | \(e\left(\frac{5}{38}\right)\) | \(e\left(\frac{18}{19}\right)\) | \(e\left(\frac{1}{19}\right)\) | \(e\left(\frac{55}{76}\right)\) |
| \(\chi_{916}(177,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{19}\right)\) | \(e\left(\frac{13}{38}\right)\) | \(e\left(\frac{9}{76}\right)\) | \(e\left(\frac{2}{19}\right)\) | \(e\left(\frac{37}{38}\right)\) | \(e\left(\frac{7}{76}\right)\) | \(e\left(\frac{15}{38}\right)\) | \(e\left(\frac{16}{19}\right)\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{13}{76}\right)\) |
| \(\chi_{916}(197,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{15}{19}\right)\) | \(e\left(\frac{5}{38}\right)\) | \(e\left(\frac{59}{76}\right)\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{23}{38}\right)\) | \(e\left(\frac{29}{76}\right)\) | \(e\left(\frac{35}{38}\right)\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{7}{19}\right)\) | \(e\left(\frac{43}{76}\right)\) |
| \(\chi_{916}(221,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{3}{38}\right)\) | \(e\left(\frac{5}{76}\right)\) | \(e\left(\frac{18}{19}\right)\) | \(e\left(\frac{29}{38}\right)\) | \(e\left(\frac{63}{76}\right)\) | \(e\left(\frac{21}{38}\right)\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{8}{19}\right)\) | \(e\left(\frac{41}{76}\right)\) |
| \(\chi_{916}(237,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{3}{38}\right)\) | \(e\left(\frac{43}{76}\right)\) | \(e\left(\frac{18}{19}\right)\) | \(e\left(\frac{29}{38}\right)\) | \(e\left(\frac{25}{76}\right)\) | \(e\left(\frac{21}{38}\right)\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{8}{19}\right)\) | \(e\left(\frac{3}{76}\right)\) |
| \(\chi_{916}(261,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{15}{19}\right)\) | \(e\left(\frac{5}{38}\right)\) | \(e\left(\frac{21}{76}\right)\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{23}{38}\right)\) | \(e\left(\frac{67}{76}\right)\) | \(e\left(\frac{35}{38}\right)\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{7}{19}\right)\) | \(e\left(\frac{5}{76}\right)\) |
| \(\chi_{916}(281,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{19}\right)\) | \(e\left(\frac{13}{38}\right)\) | \(e\left(\frac{47}{76}\right)\) | \(e\left(\frac{2}{19}\right)\) | \(e\left(\frac{37}{38}\right)\) | \(e\left(\frac{45}{76}\right)\) | \(e\left(\frac{15}{38}\right)\) | \(e\left(\frac{16}{19}\right)\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{51}{76}\right)\) |
| \(\chi_{916}(313,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{19}\right)\) | \(e\left(\frac{17}{38}\right)\) | \(e\left(\frac{41}{76}\right)\) | \(e\left(\frac{7}{19}\right)\) | \(e\left(\frac{25}{38}\right)\) | \(e\left(\frac{15}{76}\right)\) | \(e\left(\frac{5}{38}\right)\) | \(e\left(\frac{18}{19}\right)\) | \(e\left(\frac{1}{19}\right)\) | \(e\left(\frac{17}{76}\right)\) |
| \(\chi_{916}(317,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{19}\right)\) | \(e\left(\frac{27}{38}\right)\) | \(e\left(\frac{45}{76}\right)\) | \(e\left(\frac{10}{19}\right)\) | \(e\left(\frac{33}{38}\right)\) | \(e\left(\frac{35}{76}\right)\) | \(e\left(\frac{37}{38}\right)\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{15}{19}\right)\) | \(e\left(\frac{65}{76}\right)\) |
| \(\chi_{916}(349,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{23}{38}\right)\) | \(e\left(\frac{13}{76}\right)\) | \(e\left(\frac{5}{19}\right)\) | \(e\left(\frac{7}{38}\right)\) | \(e\left(\frac{27}{76}\right)\) | \(e\left(\frac{9}{38}\right)\) | \(e\left(\frac{2}{19}\right)\) | \(e\left(\frac{17}{19}\right)\) | \(e\left(\frac{61}{76}\right)\) |
| \(\chi_{916}(357,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{2}{19}\right)\) | \(e\left(\frac{7}{38}\right)\) | \(e\left(\frac{75}{76}\right)\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{17}{38}\right)\) | \(e\left(\frac{33}{76}\right)\) | \(e\left(\frac{11}{38}\right)\) | \(e\left(\frac{13}{19}\right)\) | \(e\left(\frac{6}{19}\right)\) | \(e\left(\frac{7}{76}\right)\) |
| \(\chi_{916}(365,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{33}{38}\right)\) | \(e\left(\frac{55}{76}\right)\) | \(e\left(\frac{8}{19}\right)\) | \(e\left(\frac{15}{38}\right)\) | \(e\left(\frac{9}{76}\right)\) | \(e\left(\frac{3}{38}\right)\) | \(e\left(\frac{7}{19}\right)\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{71}{76}\right)\) |
| \(\chi_{916}(437,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{19}\right)\) | \(e\left(\frac{15}{38}\right)\) | \(e\left(\frac{25}{76}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{31}{38}\right)\) | \(e\left(\frac{11}{76}\right)\) | \(e\left(\frac{29}{38}\right)\) | \(e\left(\frac{17}{19}\right)\) | \(e\left(\frac{2}{19}\right)\) | \(e\left(\frac{53}{76}\right)\) |
| \(\chi_{916}(445,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{11}{38}\right)\) | \(e\left(\frac{31}{76}\right)\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{5}{38}\right)\) | \(e\left(\frac{41}{76}\right)\) | \(e\left(\frac{1}{38}\right)\) | \(e\left(\frac{15}{19}\right)\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{11}{76}\right)\) |
| \(\chi_{916}(573,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{16}{19}\right)\) | \(e\left(\frac{37}{38}\right)\) | \(e\left(\frac{11}{76}\right)\) | \(e\left(\frac{13}{19}\right)\) | \(e\left(\frac{3}{38}\right)\) | \(e\left(\frac{17}{76}\right)\) | \(e\left(\frac{31}{38}\right)\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{10}{19}\right)\) | \(e\left(\frac{75}{76}\right)\) |
| \(\chi_{916}(581,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{29}{38}\right)\) | \(e\left(\frac{23}{76}\right)\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{27}{38}\right)\) | \(e\left(\frac{1}{76}\right)\) | \(e\left(\frac{13}{38}\right)\) | \(e\left(\frac{5}{19}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{67}{76}\right)\) |
| \(\chi_{916}(601,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{10}{19}\right)\) | \(e\left(\frac{35}{38}\right)\) | \(e\left(\frac{71}{76}\right)\) | \(e\left(\frac{1}{19}\right)\) | \(e\left(\frac{9}{38}\right)\) | \(e\left(\frac{13}{76}\right)\) | \(e\left(\frac{17}{38}\right)\) | \(e\left(\frac{8}{19}\right)\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{35}{76}\right)\) |
| \(\chi_{916}(633,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{8}{19}\right)\) | \(e\left(\frac{9}{38}\right)\) | \(e\left(\frac{15}{76}\right)\) | \(e\left(\frac{16}{19}\right)\) | \(e\left(\frac{11}{38}\right)\) | \(e\left(\frac{37}{76}\right)\) | \(e\left(\frac{25}{38}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{5}{19}\right)\) | \(e\left(\frac{47}{76}\right)\) |
| \(\chi_{916}(653,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{19}\right)\) | \(e\left(\frac{31}{38}\right)\) | \(e\left(\frac{39}{76}\right)\) | \(e\left(\frac{15}{19}\right)\) | \(e\left(\frac{21}{38}\right)\) | \(e\left(\frac{5}{76}\right)\) | \(e\left(\frac{27}{38}\right)\) | \(e\left(\frac{6}{19}\right)\) | \(e\left(\frac{13}{19}\right)\) | \(e\left(\frac{31}{76}\right)\) |
| \(\chi_{916}(657,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{6}{19}\right)\) | \(e\left(\frac{21}{38}\right)\) | \(e\left(\frac{73}{76}\right)\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{13}{38}\right)\) | \(e\left(\frac{23}{76}\right)\) | \(e\left(\frac{33}{38}\right)\) | \(e\left(\frac{1}{19}\right)\) | \(e\left(\frac{18}{19}\right)\) | \(e\left(\frac{21}{76}\right)\) |
| \(\chi_{916}(665,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{18}{19}\right)\) | \(e\left(\frac{25}{38}\right)\) | \(e\left(\frac{29}{76}\right)\) | \(e\left(\frac{17}{19}\right)\) | \(e\left(\frac{1}{38}\right)\) | \(e\left(\frac{31}{76}\right)\) | \(e\left(\frac{23}{38}\right)\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{16}{19}\right)\) | \(e\left(\frac{25}{76}\right)\) |
| \(\chi_{916}(685,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{1}{38}\right)\) | \(e\left(\frac{27}{76}\right)\) | \(e\left(\frac{6}{19}\right)\) | \(e\left(\frac{35}{38}\right)\) | \(e\left(\frac{21}{76}\right)\) | \(e\left(\frac{7}{38}\right)\) | \(e\left(\frac{10}{19}\right)\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{39}{76}\right)\) |
| \(\chi_{916}(689,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{1}{38}\right)\) | \(e\left(\frac{65}{76}\right)\) | \(e\left(\frac{6}{19}\right)\) | \(e\left(\frac{35}{38}\right)\) | \(e\left(\frac{59}{76}\right)\) | \(e\left(\frac{7}{38}\right)\) | \(e\left(\frac{10}{19}\right)\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{1}{76}\right)\) |
| \(\chi_{916}(709,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{18}{19}\right)\) | \(e\left(\frac{25}{38}\right)\) | \(e\left(\frac{67}{76}\right)\) | \(e\left(\frac{17}{19}\right)\) | \(e\left(\frac{1}{38}\right)\) | \(e\left(\frac{69}{76}\right)\) | \(e\left(\frac{23}{38}\right)\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{16}{19}\right)\) | \(e\left(\frac{63}{76}\right)\) |
| \(\chi_{916}(717,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{6}{19}\right)\) | \(e\left(\frac{21}{38}\right)\) | \(e\left(\frac{35}{76}\right)\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{13}{38}\right)\) | \(e\left(\frac{61}{76}\right)\) | \(e\left(\frac{33}{38}\right)\) | \(e\left(\frac{1}{19}\right)\) | \(e\left(\frac{18}{19}\right)\) | \(e\left(\frac{59}{76}\right)\) |