from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2805, base_ring=CyclotomicField(80))
M = H._module
chi = DirichletCharacter(H, M([0,0,8,65]))
chi.galois_orbit()
[g,chi] = znchar(Mod(46,2805))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(2805\) | |
| Conductor: | \(187\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(80\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 187.t | 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: | $\Q(\zeta_{80})$ |
| Fixed field: | Number field defined by a degree 80 polynomial |
First 31 of 32 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(13\) | \(14\) | \(16\) | \(19\) | \(23\) | \(26\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{2805}(46,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{51}{80}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{9}{80}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{33}{40}\right)\) |
| \(\chi_{2805}(61,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{29}{80}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{71}{80}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{7}{40}\right)\) |
| \(\chi_{2805}(211,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{21}{80}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{79}{80}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{23}{40}\right)\) |
| \(\chi_{2805}(226,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{59}{80}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{1}{80}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{17}{40}\right)\) |
| \(\chi_{2805}(316,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{13}{80}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{7}{80}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{39}{40}\right)\) |
| \(\chi_{2805}(436,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{57}{80}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{43}{80}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{11}{40}\right)\) |
| \(\chi_{2805}(481,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{43}{80}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{17}{80}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{9}{40}\right)\) |
| \(\chi_{2805}(541,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{71}{80}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{69}{80}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{13}{40}\right)\) |
| \(\chi_{2805}(556,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{19}{80}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{41}{80}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{17}{40}\right)\) |
| \(\chi_{2805}(601,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{17}{80}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{3}{80}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{11}{40}\right)\) |
| \(\chi_{2805}(721,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{69}{80}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{31}{80}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{7}{40}\right)\) |
| \(\chi_{2805}(811,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{3}{80}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{57}{80}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{9}{40}\right)\) |
| \(\chi_{2805}(976,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{53}{80}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{47}{80}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{39}{40}\right)\) |
| \(\chi_{2805}(1051,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{39}{80}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{21}{80}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{37}{40}\right)\) |
| \(\chi_{2805}(1201,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{41}{80}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{59}{80}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{3}{40}\right)\) |
| \(\chi_{2805}(1261,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{47}{80}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{13}{80}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{21}{40}\right)\) |
| \(\chi_{2805}(1306,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{23}{80}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{37}{80}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{29}{40}\right)\) |
| \(\chi_{2805}(1366,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{1}{80}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{19}{80}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{3}{40}\right)\) |
| \(\chi_{2805}(1591,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{77}{80}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{23}{80}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{31}{40}\right)\) |
| \(\chi_{2805}(1711,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{9}{80}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{11}{80}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{27}{40}\right)\) |
| \(\chi_{2805}(1756,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{27}{80}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{33}{80}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{1}{40}\right)\) |
| \(\chi_{2805}(1876,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{49}{80}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{51}{80}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{27}{40}\right)\) |
| \(\chi_{2805}(1966,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{73}{80}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{27}{80}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{19}{40}\right)\) |
| \(\chi_{2805}(2026,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{31}{80}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{29}{80}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{13}{40}\right)\) |
| \(\chi_{2805}(2086,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{67}{80}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{73}{80}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{1}{40}\right)\) |
| \(\chi_{2805}(2131,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{33}{80}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{67}{80}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{19}{40}\right)\) |
| \(\chi_{2805}(2251,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{37}{80}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{63}{80}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{31}{40}\right)\) |
| \(\chi_{2805}(2356,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{61}{80}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{39}{80}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{23}{40}\right)\) |
| \(\chi_{2805}(2521,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{11}{80}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{49}{80}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{33}{40}\right)\) |
| \(\chi_{2805}(2536,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{79}{80}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{61}{80}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{37}{40}\right)\) |
| \(\chi_{2805}(2581,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{7}{80}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{53}{80}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{21}{40}\right)\) |