from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1805, base_ring=CyclotomicField(76))
M = H._module
chi = DirichletCharacter(H, M([57,62]))
chi.galois_orbit()
[g,chi] = znchar(Mod(18,1805))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(1805\) | |
| Conductor: | \(1805\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(76\) | 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: | $\Q(\zeta_{76})$ |
| Fixed field: | Number field defined by a degree 76 polynomial |
First 31 of 36 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{1805}(18,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{76}\right)\) | \(e\left(\frac{49}{76}\right)\) | \(e\left(\frac{5}{38}\right)\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{9}{76}\right)\) | \(e\left(\frac{53}{76}\right)\) | \(e\left(\frac{11}{38}\right)\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{59}{76}\right)\) | \(e\left(\frac{49}{76}\right)\) |
| \(\chi_{1805}(37,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{76}\right)\) | \(e\left(\frac{3}{76}\right)\) | \(e\left(\frac{29}{38}\right)\) | \(e\left(\frac{8}{19}\right)\) | \(e\left(\frac{75}{76}\right)\) | \(e\left(\frac{11}{76}\right)\) | \(e\left(\frac{3}{38}\right)\) | \(e\left(\frac{8}{19}\right)\) | \(e\left(\frac{61}{76}\right)\) | \(e\left(\frac{3}{76}\right)\) |
| \(\chi_{1805}(113,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{76}\right)\) | \(e\left(\frac{9}{76}\right)\) | \(e\left(\frac{11}{38}\right)\) | \(e\left(\frac{5}{19}\right)\) | \(e\left(\frac{73}{76}\right)\) | \(e\left(\frac{33}{76}\right)\) | \(e\left(\frac{9}{38}\right)\) | \(e\left(\frac{5}{19}\right)\) | \(e\left(\frac{31}{76}\right)\) | \(e\left(\frac{9}{76}\right)\) |
| \(\chi_{1805}(132,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{73}{76}\right)\) | \(e\left(\frac{39}{76}\right)\) | \(e\left(\frac{35}{38}\right)\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{63}{76}\right)\) | \(e\left(\frac{67}{76}\right)\) | \(e\left(\frac{1}{38}\right)\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{33}{76}\right)\) | \(e\left(\frac{39}{76}\right)\) |
| \(\chi_{1805}(208,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{55}{76}\right)\) | \(e\left(\frac{45}{76}\right)\) | \(e\left(\frac{17}{38}\right)\) | \(e\left(\frac{6}{19}\right)\) | \(e\left(\frac{61}{76}\right)\) | \(e\left(\frac{13}{76}\right)\) | \(e\left(\frac{7}{38}\right)\) | \(e\left(\frac{6}{19}\right)\) | \(e\left(\frac{3}{76}\right)\) | \(e\left(\frac{45}{76}\right)\) |
| \(\chi_{1805}(227,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{76}\right)\) | \(e\left(\frac{75}{76}\right)\) | \(e\left(\frac{3}{38}\right)\) | \(e\left(\frac{10}{19}\right)\) | \(e\left(\frac{51}{76}\right)\) | \(e\left(\frac{47}{76}\right)\) | \(e\left(\frac{37}{38}\right)\) | \(e\left(\frac{10}{19}\right)\) | \(e\left(\frac{5}{76}\right)\) | \(e\left(\frac{75}{76}\right)\) |
| \(\chi_{1805}(303,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{76}\right)\) | \(e\left(\frac{5}{76}\right)\) | \(e\left(\frac{23}{38}\right)\) | \(e\left(\frac{7}{19}\right)\) | \(e\left(\frac{49}{76}\right)\) | \(e\left(\frac{69}{76}\right)\) | \(e\left(\frac{5}{38}\right)\) | \(e\left(\frac{7}{19}\right)\) | \(e\left(\frac{51}{76}\right)\) | \(e\left(\frac{5}{76}\right)\) |
| \(\chi_{1805}(322,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{76}\right)\) | \(e\left(\frac{35}{76}\right)\) | \(e\left(\frac{9}{38}\right)\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{39}{76}\right)\) | \(e\left(\frac{27}{76}\right)\) | \(e\left(\frac{35}{38}\right)\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{53}{76}\right)\) | \(e\left(\frac{35}{76}\right)\) |
| \(\chi_{1805}(398,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{67}{76}\right)\) | \(e\left(\frac{41}{76}\right)\) | \(e\left(\frac{29}{38}\right)\) | \(e\left(\frac{8}{19}\right)\) | \(e\left(\frac{37}{76}\right)\) | \(e\left(\frac{49}{76}\right)\) | \(e\left(\frac{3}{38}\right)\) | \(e\left(\frac{8}{19}\right)\) | \(e\left(\frac{23}{76}\right)\) | \(e\left(\frac{41}{76}\right)\) |
| \(\chi_{1805}(417,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{53}{76}\right)\) | \(e\left(\frac{71}{76}\right)\) | \(e\left(\frac{15}{38}\right)\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{27}{76}\right)\) | \(e\left(\frac{7}{76}\right)\) | \(e\left(\frac{33}{38}\right)\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{25}{76}\right)\) | \(e\left(\frac{71}{76}\right)\) |
| \(\chi_{1805}(493,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{35}{76}\right)\) | \(e\left(\frac{1}{76}\right)\) | \(e\left(\frac{35}{38}\right)\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{25}{76}\right)\) | \(e\left(\frac{29}{76}\right)\) | \(e\left(\frac{1}{38}\right)\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{71}{76}\right)\) | \(e\left(\frac{1}{76}\right)\) |
| \(\chi_{1805}(512,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{21}{76}\right)\) | \(e\left(\frac{31}{76}\right)\) | \(e\left(\frac{21}{38}\right)\) | \(e\left(\frac{13}{19}\right)\) | \(e\left(\frac{15}{76}\right)\) | \(e\left(\frac{63}{76}\right)\) | \(e\left(\frac{31}{38}\right)\) | \(e\left(\frac{13}{19}\right)\) | \(e\left(\frac{73}{76}\right)\) | \(e\left(\frac{31}{76}\right)\) |
| \(\chi_{1805}(588,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{76}\right)\) | \(e\left(\frac{37}{76}\right)\) | \(e\left(\frac{3}{38}\right)\) | \(e\left(\frac{10}{19}\right)\) | \(e\left(\frac{13}{76}\right)\) | \(e\left(\frac{9}{76}\right)\) | \(e\left(\frac{37}{38}\right)\) | \(e\left(\frac{10}{19}\right)\) | \(e\left(\frac{43}{76}\right)\) | \(e\left(\frac{37}{76}\right)\) |
| \(\chi_{1805}(607,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{65}{76}\right)\) | \(e\left(\frac{67}{76}\right)\) | \(e\left(\frac{27}{38}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{3}{76}\right)\) | \(e\left(\frac{43}{76}\right)\) | \(e\left(\frac{29}{38}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{45}{76}\right)\) | \(e\left(\frac{67}{76}\right)\) |
| \(\chi_{1805}(683,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{47}{76}\right)\) | \(e\left(\frac{73}{76}\right)\) | \(e\left(\frac{9}{38}\right)\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{1}{76}\right)\) | \(e\left(\frac{65}{76}\right)\) | \(e\left(\frac{35}{38}\right)\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{15}{76}\right)\) | \(e\left(\frac{73}{76}\right)\) |
| \(\chi_{1805}(702,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{33}{76}\right)\) | \(e\left(\frac{27}{76}\right)\) | \(e\left(\frac{33}{38}\right)\) | \(e\left(\frac{15}{19}\right)\) | \(e\left(\frac{67}{76}\right)\) | \(e\left(\frac{23}{76}\right)\) | \(e\left(\frac{27}{38}\right)\) | \(e\left(\frac{15}{19}\right)\) | \(e\left(\frac{17}{76}\right)\) | \(e\left(\frac{27}{76}\right)\) |
| \(\chi_{1805}(778,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{15}{76}\right)\) | \(e\left(\frac{33}{76}\right)\) | \(e\left(\frac{15}{38}\right)\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{65}{76}\right)\) | \(e\left(\frac{45}{76}\right)\) | \(e\left(\frac{33}{38}\right)\) | \(e\left(\frac{12}{19}\right)\) | \(e\left(\frac{63}{76}\right)\) | \(e\left(\frac{33}{76}\right)\) |
| \(\chi_{1805}(797,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{76}\right)\) | \(e\left(\frac{63}{76}\right)\) | \(e\left(\frac{1}{38}\right)\) | \(e\left(\frac{16}{19}\right)\) | \(e\left(\frac{55}{76}\right)\) | \(e\left(\frac{3}{76}\right)\) | \(e\left(\frac{25}{38}\right)\) | \(e\left(\frac{16}{19}\right)\) | \(e\left(\frac{65}{76}\right)\) | \(e\left(\frac{63}{76}\right)\) |
| \(\chi_{1805}(873,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{59}{76}\right)\) | \(e\left(\frac{69}{76}\right)\) | \(e\left(\frac{21}{38}\right)\) | \(e\left(\frac{13}{19}\right)\) | \(e\left(\frac{53}{76}\right)\) | \(e\left(\frac{25}{76}\right)\) | \(e\left(\frac{31}{38}\right)\) | \(e\left(\frac{13}{19}\right)\) | \(e\left(\frac{35}{76}\right)\) | \(e\left(\frac{69}{76}\right)\) |
| \(\chi_{1805}(892,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{45}{76}\right)\) | \(e\left(\frac{23}{76}\right)\) | \(e\left(\frac{7}{38}\right)\) | \(e\left(\frac{17}{19}\right)\) | \(e\left(\frac{43}{76}\right)\) | \(e\left(\frac{59}{76}\right)\) | \(e\left(\frac{23}{38}\right)\) | \(e\left(\frac{17}{19}\right)\) | \(e\left(\frac{37}{76}\right)\) | \(e\left(\frac{23}{76}\right)\) |
| \(\chi_{1805}(968,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{27}{76}\right)\) | \(e\left(\frac{29}{76}\right)\) | \(e\left(\frac{27}{38}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{41}{76}\right)\) | \(e\left(\frac{5}{76}\right)\) | \(e\left(\frac{29}{38}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{7}{76}\right)\) | \(e\left(\frac{29}{76}\right)\) |
| \(\chi_{1805}(987,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{76}\right)\) | \(e\left(\frac{59}{76}\right)\) | \(e\left(\frac{13}{38}\right)\) | \(e\left(\frac{18}{19}\right)\) | \(e\left(\frac{31}{76}\right)\) | \(e\left(\frac{39}{76}\right)\) | \(e\left(\frac{21}{38}\right)\) | \(e\left(\frac{18}{19}\right)\) | \(e\left(\frac{9}{76}\right)\) | \(e\left(\frac{59}{76}\right)\) |
| \(\chi_{1805}(1063,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{71}{76}\right)\) | \(e\left(\frac{65}{76}\right)\) | \(e\left(\frac{33}{38}\right)\) | \(e\left(\frac{15}{19}\right)\) | \(e\left(\frac{29}{76}\right)\) | \(e\left(\frac{61}{76}\right)\) | \(e\left(\frac{27}{38}\right)\) | \(e\left(\frac{15}{19}\right)\) | \(e\left(\frac{55}{76}\right)\) | \(e\left(\frac{65}{76}\right)\) |
| \(\chi_{1805}(1158,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{39}{76}\right)\) | \(e\left(\frac{25}{76}\right)\) | \(e\left(\frac{1}{38}\right)\) | \(e\left(\frac{16}{19}\right)\) | \(e\left(\frac{17}{76}\right)\) | \(e\left(\frac{41}{76}\right)\) | \(e\left(\frac{25}{38}\right)\) | \(e\left(\frac{16}{19}\right)\) | \(e\left(\frac{27}{76}\right)\) | \(e\left(\frac{25}{76}\right)\) |
| \(\chi_{1805}(1177,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{76}\right)\) | \(e\left(\frac{55}{76}\right)\) | \(e\left(\frac{25}{38}\right)\) | \(e\left(\frac{1}{19}\right)\) | \(e\left(\frac{7}{76}\right)\) | \(e\left(\frac{75}{76}\right)\) | \(e\left(\frac{17}{38}\right)\) | \(e\left(\frac{1}{19}\right)\) | \(e\left(\frac{29}{76}\right)\) | \(e\left(\frac{55}{76}\right)\) |
| \(\chi_{1805}(1253,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{76}\right)\) | \(e\left(\frac{61}{76}\right)\) | \(e\left(\frac{7}{38}\right)\) | \(e\left(\frac{17}{19}\right)\) | \(e\left(\frac{5}{76}\right)\) | \(e\left(\frac{21}{76}\right)\) | \(e\left(\frac{23}{38}\right)\) | \(e\left(\frac{17}{19}\right)\) | \(e\left(\frac{75}{76}\right)\) | \(e\left(\frac{61}{76}\right)\) |
| \(\chi_{1805}(1272,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{69}{76}\right)\) | \(e\left(\frac{15}{76}\right)\) | \(e\left(\frac{31}{38}\right)\) | \(e\left(\frac{2}{19}\right)\) | \(e\left(\frac{71}{76}\right)\) | \(e\left(\frac{55}{76}\right)\) | \(e\left(\frac{15}{38}\right)\) | \(e\left(\frac{2}{19}\right)\) | \(e\left(\frac{1}{76}\right)\) | \(e\left(\frac{15}{76}\right)\) |
| \(\chi_{1805}(1348,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{51}{76}\right)\) | \(e\left(\frac{21}{76}\right)\) | \(e\left(\frac{13}{38}\right)\) | \(e\left(\frac{18}{19}\right)\) | \(e\left(\frac{69}{76}\right)\) | \(e\left(\frac{1}{76}\right)\) | \(e\left(\frac{21}{38}\right)\) | \(e\left(\frac{18}{19}\right)\) | \(e\left(\frac{47}{76}\right)\) | \(e\left(\frac{21}{76}\right)\) |
| \(\chi_{1805}(1367,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{76}\right)\) | \(e\left(\frac{51}{76}\right)\) | \(e\left(\frac{37}{38}\right)\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{59}{76}\right)\) | \(e\left(\frac{35}{76}\right)\) | \(e\left(\frac{13}{38}\right)\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{49}{76}\right)\) | \(e\left(\frac{51}{76}\right)\) |
| \(\chi_{1805}(1462,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{76}\right)\) | \(e\left(\frac{11}{76}\right)\) | \(e\left(\frac{5}{38}\right)\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{47}{76}\right)\) | \(e\left(\frac{15}{76}\right)\) | \(e\left(\frac{11}{38}\right)\) | \(e\left(\frac{4}{19}\right)\) | \(e\left(\frac{21}{76}\right)\) | \(e\left(\frac{11}{76}\right)\) |
| \(\chi_{1805}(1538,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{63}{76}\right)\) | \(e\left(\frac{17}{76}\right)\) | \(e\left(\frac{25}{38}\right)\) | \(e\left(\frac{1}{19}\right)\) | \(e\left(\frac{45}{76}\right)\) | \(e\left(\frac{37}{76}\right)\) | \(e\left(\frac{17}{38}\right)\) | \(e\left(\frac{1}{19}\right)\) | \(e\left(\frac{67}{76}\right)\) | \(e\left(\frac{17}{76}\right)\) |