from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4010, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([20,23]))
chi.galois_orbit()
[g,chi] = znchar(Mod(239,4010))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(4010\) | |
| Conductor: | \(2005\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(40\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 2005.be | 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 | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{4010}(239,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{9}{40}\right)\) |
| \(\chi_{4010}(299,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{3}{40}\right)\) |
| \(\chi_{4010}(369,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{39}{40}\right)\) |
| \(\chi_{4010}(589,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{21}{40}\right)\) |
| \(\chi_{4010}(1089,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{17}{40}\right)\) |
| \(\chi_{4010}(1329,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{7}{40}\right)\) |
| \(\chi_{4010}(1569,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{33}{40}\right)\) |
| \(\chi_{4010}(1639,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{13}{40}\right)\) |
| \(\chi_{4010}(1879,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{27}{40}\right)\) |
| \(\chi_{4010}(2119,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{37}{40}\right)\) |
| \(\chi_{4010}(2619,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{1}{40}\right)\) |
| \(\chi_{4010}(2839,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{19}{40}\right)\) |
| \(\chi_{4010}(2909,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{23}{40}\right)\) |
| \(\chi_{4010}(2969,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{29}{40}\right)\) |
| \(\chi_{4010}(3359,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{11}{40}\right)\) |
| \(\chi_{4010}(3859,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{31}{40}\right)\) |