from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(221760, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([20,25,20,0,0,16]))
chi.galois_orbit()
[g,chi] = znchar(Mod(71,221760))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(221760\) | |
| Conductor: | \(1056\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(40\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 1056.ch | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Related number fields
| Field of values: | \(\Q(\zeta_{40})\) |
| Fixed field: | Number field defined by a degree 40 polynomial |
Characters in Galois orbit
| Character | \(-1\) | \(1\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) | \(47\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{221760}(71,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(-i\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{7}{10}\right)\) |
| \(\chi_{221760}(10151,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(-i\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{3}{10}\right)\) |
| \(\chi_{221760}(25271,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(i\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{1}{10}\right)\) |
| \(\chi_{221760}(50471,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(-i\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{9}{10}\right)\) |
| \(\chi_{221760}(55511,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(i\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{7}{10}\right)\) |
| \(\chi_{221760}(65591,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(i\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{3}{10}\right)\) |
| \(\chi_{221760}(80711,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(-i\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{1}{10}\right)\) |
| \(\chi_{221760}(105911,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(i\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{9}{10}\right)\) |
| \(\chi_{221760}(110951,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(-i\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{7}{10}\right)\) |
| \(\chi_{221760}(121031,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(-i\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{3}{10}\right)\) |
| \(\chi_{221760}(136151,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(i\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{1}{10}\right)\) |
| \(\chi_{221760}(161351,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(-i\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{9}{10}\right)\) |
| \(\chi_{221760}(166391,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(i\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{7}{10}\right)\) |
| \(\chi_{221760}(176471,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(i\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{3}{10}\right)\) |
| \(\chi_{221760}(191591,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(-i\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{1}{10}\right)\) |
| \(\chi_{221760}(216791,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(i\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{9}{10}\right)\) |