from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8470, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([33,0,10]))
chi.galois_orbit()
[g,chi] = znchar(Mod(43,8470))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(8470\) | |
| Conductor: | \(605\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 605.r | 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: | \(\Q(\zeta_{44})\) |
| Fixed field: | 44.44.2885428559557085084648615903962269104974580506944665166312236845353556846511909399754484184086322784423828125.1 |
Characters in Galois orbit
| Character | \(-1\) | \(1\) | \(3\) | \(9\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) | \(37\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{8470}(43,\cdot)\) | \(1\) | \(1\) | \(i\) | \(-1\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{39}{44}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{7}{44}\right)\) | \(-i\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{13}{44}\right)\) |
| \(\chi_{8470}(197,\cdot)\) | \(1\) | \(1\) | \(-i\) | \(-1\) | \(e\left(\frac{35}{44}\right)\) | \(e\left(\frac{5}{44}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{37}{44}\right)\) | \(i\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{31}{44}\right)\) |
| \(\chi_{8470}(813,\cdot)\) | \(1\) | \(1\) | \(i\) | \(-1\) | \(e\left(\frac{29}{44}\right)\) | \(e\left(\frac{23}{44}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{3}{44}\right)\) | \(-i\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{37}{44}\right)\) |
| \(\chi_{8470}(1583,\cdot)\) | \(1\) | \(1\) | \(i\) | \(-1\) | \(e\left(\frac{5}{44}\right)\) | \(e\left(\frac{7}{44}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{43}{44}\right)\) | \(-i\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{17}{44}\right)\) |
| \(\chi_{8470}(1737,\cdot)\) | \(1\) | \(1\) | \(-i\) | \(-1\) | \(e\left(\frac{31}{44}\right)\) | \(e\left(\frac{17}{44}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{29}{44}\right)\) | \(i\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{35}{44}\right)\) |
| \(\chi_{8470}(2353,\cdot)\) | \(1\) | \(1\) | \(i\) | \(-1\) | \(e\left(\frac{25}{44}\right)\) | \(e\left(\frac{35}{44}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{39}{44}\right)\) | \(-i\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{41}{44}\right)\) |
| \(\chi_{8470}(2507,\cdot)\) | \(1\) | \(1\) | \(-i\) | \(-1\) | \(e\left(\frac{7}{44}\right)\) | \(e\left(\frac{1}{44}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{25}{44}\right)\) | \(i\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{15}{44}\right)\) |
| \(\chi_{8470}(3123,\cdot)\) | \(1\) | \(1\) | \(i\) | \(-1\) | \(e\left(\frac{1}{44}\right)\) | \(e\left(\frac{19}{44}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{35}{44}\right)\) | \(-i\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{21}{44}\right)\) |
| \(\chi_{8470}(3277,\cdot)\) | \(1\) | \(1\) | \(-i\) | \(-1\) | \(e\left(\frac{27}{44}\right)\) | \(e\left(\frac{29}{44}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{21}{44}\right)\) | \(i\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{39}{44}\right)\) |
| \(\chi_{8470}(3893,\cdot)\) | \(1\) | \(1\) | \(i\) | \(-1\) | \(e\left(\frac{21}{44}\right)\) | \(e\left(\frac{3}{44}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{31}{44}\right)\) | \(-i\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{1}{44}\right)\) |
| \(\chi_{8470}(4047,\cdot)\) | \(1\) | \(1\) | \(-i\) | \(-1\) | \(e\left(\frac{3}{44}\right)\) | \(e\left(\frac{13}{44}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{17}{44}\right)\) | \(i\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{19}{44}\right)\) |
| \(\chi_{8470}(4663,\cdot)\) | \(1\) | \(1\) | \(i\) | \(-1\) | \(e\left(\frac{41}{44}\right)\) | \(e\left(\frac{31}{44}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{27}{44}\right)\) | \(-i\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{25}{44}\right)\) |
| \(\chi_{8470}(4817,\cdot)\) | \(1\) | \(1\) | \(-i\) | \(-1\) | \(e\left(\frac{23}{44}\right)\) | \(e\left(\frac{41}{44}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{13}{44}\right)\) | \(i\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{43}{44}\right)\) |
| \(\chi_{8470}(5433,\cdot)\) | \(1\) | \(1\) | \(i\) | \(-1\) | \(e\left(\frac{17}{44}\right)\) | \(e\left(\frac{15}{44}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{23}{44}\right)\) | \(-i\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{5}{44}\right)\) |
| \(\chi_{8470}(5587,\cdot)\) | \(1\) | \(1\) | \(-i\) | \(-1\) | \(e\left(\frac{43}{44}\right)\) | \(e\left(\frac{25}{44}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(i\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{23}{44}\right)\) |
| \(\chi_{8470}(6203,\cdot)\) | \(1\) | \(1\) | \(i\) | \(-1\) | \(e\left(\frac{37}{44}\right)\) | \(e\left(\frac{43}{44}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{19}{44}\right)\) | \(-i\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{29}{44}\right)\) |
| \(\chi_{8470}(6357,\cdot)\) | \(1\) | \(1\) | \(-i\) | \(-1\) | \(e\left(\frac{19}{44}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{5}{44}\right)\) | \(i\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{3}{44}\right)\) |
| \(\chi_{8470}(6973,\cdot)\) | \(1\) | \(1\) | \(i\) | \(-1\) | \(e\left(\frac{13}{44}\right)\) | \(e\left(\frac{27}{44}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{15}{44}\right)\) | \(-i\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{9}{44}\right)\) |
| \(\chi_{8470}(7127,\cdot)\) | \(1\) | \(1\) | \(-i\) | \(-1\) | \(e\left(\frac{39}{44}\right)\) | \(e\left(\frac{37}{44}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{1}{44}\right)\) | \(i\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{27}{44}\right)\) |
| \(\chi_{8470}(7897,\cdot)\) | \(1\) | \(1\) | \(-i\) | \(-1\) | \(e\left(\frac{15}{44}\right)\) | \(e\left(\frac{21}{44}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{41}{44}\right)\) | \(i\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{7}{44}\right)\) |