from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5952, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([0,5,20,4]))
chi.galois_orbit()
[g,chi] = znchar(Mod(89,5952))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(5952\) | |
| Conductor: | \(2976\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(40\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 2976.ee | 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 | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(35\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{5952}(89,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(i\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{27}{40}\right)\) |
| \(\chi_{5952}(953,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(i\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{23}{40}\right)\) |
| \(\chi_{5952}(1145,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(i\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{31}{40}\right)\) |
| \(\chi_{5952}(1193,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(-i\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{29}{40}\right)\) |
| \(\chi_{5952}(1577,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(-i\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{37}{40}\right)\) |
| \(\chi_{5952}(2441,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(-i\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{33}{40}\right)\) |
| \(\chi_{5952}(2633,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(-i\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{1}{40}\right)\) |
| \(\chi_{5952}(2681,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(i\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{39}{40}\right)\) |
| \(\chi_{5952}(3065,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(i\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{7}{40}\right)\) |
| \(\chi_{5952}(3929,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(i\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{3}{40}\right)\) |
| \(\chi_{5952}(4121,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(i\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{11}{40}\right)\) |
| \(\chi_{5952}(4169,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(-i\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{9}{40}\right)\) |
| \(\chi_{5952}(4553,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(-i\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{17}{40}\right)\) |
| \(\chi_{5952}(5417,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(-i\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{13}{40}\right)\) |
| \(\chi_{5952}(5609,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(-i\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{21}{40}\right)\) |
| \(\chi_{5952}(5657,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(i\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{19}{40}\right)\) |