from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4675, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([32,16,15]))
chi.galois_orbit()
[g,chi] = znchar(Mod(236,4675))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(4675\) | |
| Conductor: | \(4675\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(40\) | 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: | \(\Q(\zeta_{40})\) |
| Fixed field: | Number field defined by a degree 40 polynomial |
Characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(12\) | \(13\) | \(14\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{4675}(236,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{3}{8}\right)\) |
| \(\chi_{4675}(416,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{5}{8}\right)\) |
| \(\chi_{4675}(631,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{7}{8}\right)\) |
| \(\chi_{4675}(971,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{7}{8}\right)\) |
| \(\chi_{4675}(1181,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{5}{8}\right)\) |
| \(\chi_{4675}(1521,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{5}{8}\right)\) |
| \(\chi_{4675}(2066,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{1}{8}\right)\) |
| \(\chi_{4675}(2161,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{7}{8}\right)\) |
| \(\chi_{4675}(2616,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{3}{8}\right)\) |
| \(\chi_{4675}(2711,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{5}{8}\right)\) |
| \(\chi_{4675}(2831,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{1}{8}\right)\) |
| \(\chi_{4675}(3171,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{1}{8}\right)\) |
| \(\chi_{4675}(3381,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{3}{8}\right)\) |
| \(\chi_{4675}(3721,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{3}{8}\right)\) |
| \(\chi_{4675}(4361,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{1}{8}\right)\) |
| \(\chi_{4675}(4541,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{7}{8}\right)\) |