from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(451, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([8,31]))
chi.galois_orbit()
[g,chi] = znchar(Mod(136,451))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(451\) | |
| Conductor: | \(451\) | 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: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Related number fields
| Field of values: | \(\Q(\zeta_{40})\) |
| Fixed field: | 40.0.1671615298546273583798912557984997133996206841810275861989343966698716508910979705071744035800281.3 |
Characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{451}(136,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{37}{40}\right)\) |
| \(\chi_{451}(157,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{33}{40}\right)\) |
| \(\chi_{451}(158,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{7}{40}\right)\) |
| \(\chi_{451}(212,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{13}{40}\right)\) |
| \(\chi_{451}(229,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{31}{40}\right)\) |
| \(\chi_{451}(258,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{9}{40}\right)\) |
| \(\chi_{451}(302,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{39}{40}\right)\) |
| \(\chi_{451}(313,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{19}{40}\right)\) |
| \(\chi_{451}(317,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{21}{40}\right)\) |
| \(\chi_{451}(334,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{27}{40}\right)\) |
| \(\chi_{451}(339,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{1}{40}\right)\) |
| \(\chi_{451}(356,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{17}{40}\right)\) |
| \(\chi_{451}(357,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{29}{40}\right)\) |
| \(\chi_{451}(388,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{3}{40}\right)\) |
| \(\chi_{451}(427,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{11}{40}\right)\) |
| \(\chi_{451}(432,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{23}{40}\right)\) |