from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6005, base_ring=CyclotomicField(80))
M = H._module
chi = DirichletCharacter(H, M([60,69]))
chi.galois_orbit()
[g,chi] = znchar(Mod(83,6005))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(6005\) | |
| Conductor: | \(6005\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(80\) | 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: | $\Q(\zeta_{80})$ |
| Fixed field: | Number field defined by a degree 80 polynomial |
First 31 of 32 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{6005}(83,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(1\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{69}{80}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{21}{80}\right)\) |
| \(\chi_{6005}(308,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(1\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{21}{80}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{69}{80}\right)\) |
| \(\chi_{6005}(592,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(1\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{3}{80}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{67}{80}\right)\) |
| \(\chi_{6005}(893,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(1\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{61}{80}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{29}{80}\right)\) |
| \(\chi_{6005}(1017,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(1\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{23}{80}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{7}{80}\right)\) |
| \(\chi_{6005}(1118,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(1\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{29}{80}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{61}{80}\right)\) |
| \(\chi_{6005}(1157,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(1\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{31}{80}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{79}{80}\right)\) |
| \(\chi_{6005}(1288,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(1\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{13}{80}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{77}{80}\right)\) |
| \(\chi_{6005}(1312,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(1\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{47}{80}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{63}{80}\right)\) |
| \(\chi_{6005}(1447,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(1\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{11}{80}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{59}{80}\right)\) |
| \(\chi_{6005}(1767,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(1\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{67}{80}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{3}{80}\right)\) |
| \(\chi_{6005}(1782,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(1\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{19}{80}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{51}{80}\right)\) |
| \(\chi_{6005}(1938,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(1\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{9}{80}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{41}{80}\right)\) |
| \(\chi_{6005}(1978,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(1\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{77}{80}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{13}{80}\right)\) |
| \(\chi_{6005}(2043,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(1\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{57}{80}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{73}{80}\right)\) |
| \(\chi_{6005}(2047,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(1\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{79}{80}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{31}{80}\right)\) |
| \(\chi_{6005}(2757,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(1\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{39}{80}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{71}{80}\right)\) |
| \(\chi_{6005}(2923,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(1\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{41}{80}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{9}{80}\right)\) |
| \(\chi_{6005}(2943,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(1\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{33}{80}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{17}{80}\right)\) |
| \(\chi_{6005}(3022,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(1\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{59}{80}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{11}{80}\right)\) |
| \(\chi_{6005}(3037,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(1\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{27}{80}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{43}{80}\right)\) |
| \(\chi_{6005}(3357,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(1\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{51}{80}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{19}{80}\right)\) |
| \(\chi_{6005}(3492,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(1\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{7}{80}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{23}{80}\right)\) |
| \(\chi_{6005}(3647,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(1\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{71}{80}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{39}{80}\right)\) |
| \(\chi_{6005}(3787,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(1\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{63}{80}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{47}{80}\right)\) |
| \(\chi_{6005}(4212,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(1\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{43}{80}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{27}{80}\right)\) |
| \(\chi_{6005}(4263,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(1\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{73}{80}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{57}{80}\right)\) |
| \(\chi_{6005}(4283,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(1\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{1}{80}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{49}{80}\right)\) |
| \(\chi_{6005}(5163,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(1\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{17}{80}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{33}{80}\right)\) |
| \(\chi_{6005}(5228,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(1\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{37}{80}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{53}{80}\right)\) |
| \(\chi_{6005}(5268,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(1\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{49}{80}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{1}{80}\right)\) |