from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(605, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([11,36]))
chi.galois_orbit()
[g,chi] = znchar(Mod(12,605))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(605\) | |
| Conductor: | \(605\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | 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: | \(\Q(\zeta_{44})\) |
| Fixed field: | 44.0.23846517021132934583872858710431976074170086834253431126547411945070717739767846278962679207324981689453125.1 |
Characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(12\) | \(13\) | \(14\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{605}(12,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{44}\right)\) | \(-i\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{43}{44}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(-1\) | \(e\left(\frac{39}{44}\right)\) | \(e\left(\frac{17}{44}\right)\) | \(e\left(\frac{1}{22}\right)\) |
| \(\chi_{605}(23,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{44}\right)\) | \(i\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{7}{44}\right)\) | \(-1\) | \(e\left(\frac{1}{44}\right)\) | \(e\left(\frac{23}{44}\right)\) | \(e\left(\frac{13}{22}\right)\) |
| \(\chi_{605}(67,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{44}\right)\) | \(-i\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{27}{44}\right)\) | \(e\left(\frac{21}{44}\right)\) | \(-1\) | \(e\left(\frac{3}{44}\right)\) | \(e\left(\frac{25}{44}\right)\) | \(e\left(\frac{17}{22}\right)\) |
| \(\chi_{605}(78,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{21}{44}\right)\) | \(i\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{37}{44}\right)\) | \(e\left(\frac{19}{44}\right)\) | \(-1\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{31}{44}\right)\) | \(e\left(\frac{7}{22}\right)\) |
| \(\chi_{605}(133,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{25}{44}\right)\) | \(i\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{21}{44}\right)\) | \(e\left(\frac{31}{44}\right)\) | \(-1\) | \(e\left(\frac{17}{44}\right)\) | \(e\left(\frac{39}{44}\right)\) | \(e\left(\frac{1}{22}\right)\) |
| \(\chi_{605}(177,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{15}{44}\right)\) | \(-i\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{39}{44}\right)\) | \(e\left(\frac{1}{44}\right)\) | \(-1\) | \(e\left(\frac{19}{44}\right)\) | \(e\left(\frac{41}{44}\right)\) | \(e\left(\frac{5}{22}\right)\) |
| \(\chi_{605}(188,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{44}\right)\) | \(i\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{5}{44}\right)\) | \(e\left(\frac{43}{44}\right)\) | \(-1\) | \(e\left(\frac{25}{44}\right)\) | \(e\left(\frac{3}{44}\right)\) | \(e\left(\frac{17}{22}\right)\) |
| \(\chi_{605}(232,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{44}\right)\) | \(-i\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{23}{44}\right)\) | \(e\left(\frac{13}{44}\right)\) | \(-1\) | \(e\left(\frac{27}{44}\right)\) | \(e\left(\frac{5}{44}\right)\) | \(e\left(\frac{21}{22}\right)\) |
| \(\chi_{605}(287,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{44}\right)\) | \(-i\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{7}{44}\right)\) | \(e\left(\frac{25}{44}\right)\) | \(-1\) | \(e\left(\frac{35}{44}\right)\) | \(e\left(\frac{13}{44}\right)\) | \(e\left(\frac{15}{22}\right)\) |
| \(\chi_{605}(298,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{37}{44}\right)\) | \(i\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{17}{44}\right)\) | \(e\left(\frac{23}{44}\right)\) | \(-1\) | \(e\left(\frac{41}{44}\right)\) | \(e\left(\frac{19}{44}\right)\) | \(e\left(\frac{5}{22}\right)\) |
| \(\chi_{605}(342,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{27}{44}\right)\) | \(-i\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{35}{44}\right)\) | \(e\left(\frac{37}{44}\right)\) | \(-1\) | \(e\left(\frac{43}{44}\right)\) | \(e\left(\frac{21}{44}\right)\) | \(e\left(\frac{9}{22}\right)\) |
| \(\chi_{605}(353,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{41}{44}\right)\) | \(i\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{1}{44}\right)\) | \(e\left(\frac{35}{44}\right)\) | \(-1\) | \(e\left(\frac{5}{44}\right)\) | \(e\left(\frac{27}{44}\right)\) | \(e\left(\frac{21}{22}\right)\) |
| \(\chi_{605}(397,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{44}\right)\) | \(-i\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{19}{44}\right)\) | \(e\left(\frac{5}{44}\right)\) | \(-1\) | \(e\left(\frac{7}{44}\right)\) | \(e\left(\frac{29}{44}\right)\) | \(e\left(\frac{3}{22}\right)\) |
| \(\chi_{605}(408,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{44}\right)\) | \(i\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{29}{44}\right)\) | \(e\left(\frac{3}{44}\right)\) | \(-1\) | \(e\left(\frac{13}{44}\right)\) | \(e\left(\frac{35}{44}\right)\) | \(e\left(\frac{15}{22}\right)\) |
| \(\chi_{605}(452,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{35}{44}\right)\) | \(-i\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{3}{44}\right)\) | \(e\left(\frac{17}{44}\right)\) | \(-1\) | \(e\left(\frac{15}{44}\right)\) | \(e\left(\frac{37}{44}\right)\) | \(e\left(\frac{19}{22}\right)\) |
| \(\chi_{605}(463,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{44}\right)\) | \(i\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{13}{44}\right)\) | \(e\left(\frac{15}{44}\right)\) | \(-1\) | \(e\left(\frac{21}{44}\right)\) | \(e\left(\frac{43}{44}\right)\) | \(e\left(\frac{9}{22}\right)\) |
| \(\chi_{605}(507,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{39}{44}\right)\) | \(-i\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{31}{44}\right)\) | \(e\left(\frac{29}{44}\right)\) | \(-1\) | \(e\left(\frac{23}{44}\right)\) | \(e\left(\frac{1}{44}\right)\) | \(e\left(\frac{13}{22}\right)\) |
| \(\chi_{605}(518,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{9}{44}\right)\) | \(i\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{41}{44}\right)\) | \(e\left(\frac{27}{44}\right)\) | \(-1\) | \(e\left(\frac{29}{44}\right)\) | \(e\left(\frac{7}{44}\right)\) | \(e\left(\frac{3}{22}\right)\) |
| \(\chi_{605}(562,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{44}\right)\) | \(-i\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{15}{44}\right)\) | \(e\left(\frac{41}{44}\right)\) | \(-1\) | \(e\left(\frac{31}{44}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{7}{22}\right)\) |
| \(\chi_{605}(573,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{44}\right)\) | \(i\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{25}{44}\right)\) | \(e\left(\frac{39}{44}\right)\) | \(-1\) | \(e\left(\frac{37}{44}\right)\) | \(e\left(\frac{15}{44}\right)\) | \(e\left(\frac{19}{22}\right)\) |