from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(376, base_ring=CyclotomicField(46))
M = H._module
chi = DirichletCharacter(H, M([0,0,40]))
chi.galois_orbit()
[g,chi] = znchar(Mod(9,376))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(376\) | |
| Conductor: | \(47\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(23\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 47.c | 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: | \(\Q(\zeta_{23})\) |
| Fixed field: | Number field defined by a degree 23 polynomial |
Characters in Galois orbit
| Character | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{376}(9,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{23}\right)\) | \(e\left(\frac{20}{23}\right)\) | \(e\left(\frac{19}{23}\right)\) | \(e\left(\frac{18}{23}\right)\) | \(e\left(\frac{2}{23}\right)\) | \(e\left(\frac{13}{23}\right)\) | \(e\left(\frac{6}{23}\right)\) | \(e\left(\frac{21}{23}\right)\) | \(e\left(\frac{3}{23}\right)\) | \(e\left(\frac{5}{23}\right)\) |
| \(\chi_{376}(17,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{22}{23}\right)\) | \(e\left(\frac{8}{23}\right)\) | \(e\left(\frac{3}{23}\right)\) | \(e\left(\frac{21}{23}\right)\) | \(e\left(\frac{10}{23}\right)\) | \(e\left(\frac{19}{23}\right)\) | \(e\left(\frac{7}{23}\right)\) | \(e\left(\frac{13}{23}\right)\) | \(e\left(\frac{15}{23}\right)\) | \(e\left(\frac{2}{23}\right)\) |
| \(\chi_{376}(25,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{20}{23}\right)\) | \(e\left(\frac{1}{23}\right)\) | \(e\left(\frac{9}{23}\right)\) | \(e\left(\frac{17}{23}\right)\) | \(e\left(\frac{7}{23}\right)\) | \(e\left(\frac{11}{23}\right)\) | \(e\left(\frac{21}{23}\right)\) | \(e\left(\frac{16}{23}\right)\) | \(e\left(\frac{22}{23}\right)\) | \(e\left(\frac{6}{23}\right)\) |
| \(\chi_{376}(49,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{23}\right)\) | \(e\left(\frac{9}{23}\right)\) | \(e\left(\frac{12}{23}\right)\) | \(e\left(\frac{15}{23}\right)\) | \(e\left(\frac{17}{23}\right)\) | \(e\left(\frac{7}{23}\right)\) | \(e\left(\frac{5}{23}\right)\) | \(e\left(\frac{6}{23}\right)\) | \(e\left(\frac{14}{23}\right)\) | \(e\left(\frac{8}{23}\right)\) |
| \(\chi_{376}(65,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{23}\right)\) | \(e\left(\frac{6}{23}\right)\) | \(e\left(\frac{8}{23}\right)\) | \(e\left(\frac{10}{23}\right)\) | \(e\left(\frac{19}{23}\right)\) | \(e\left(\frac{20}{23}\right)\) | \(e\left(\frac{11}{23}\right)\) | \(e\left(\frac{4}{23}\right)\) | \(e\left(\frac{17}{23}\right)\) | \(e\left(\frac{13}{23}\right)\) |
| \(\chi_{376}(81,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{18}{23}\right)\) | \(e\left(\frac{17}{23}\right)\) | \(e\left(\frac{15}{23}\right)\) | \(e\left(\frac{13}{23}\right)\) | \(e\left(\frac{4}{23}\right)\) | \(e\left(\frac{3}{23}\right)\) | \(e\left(\frac{12}{23}\right)\) | \(e\left(\frac{19}{23}\right)\) | \(e\left(\frac{6}{23}\right)\) | \(e\left(\frac{10}{23}\right)\) |
| \(\chi_{376}(89,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{10}{23}\right)\) | \(e\left(\frac{12}{23}\right)\) | \(e\left(\frac{16}{23}\right)\) | \(e\left(\frac{20}{23}\right)\) | \(e\left(\frac{15}{23}\right)\) | \(e\left(\frac{17}{23}\right)\) | \(e\left(\frac{22}{23}\right)\) | \(e\left(\frac{8}{23}\right)\) | \(e\left(\frac{11}{23}\right)\) | \(e\left(\frac{3}{23}\right)\) |
| \(\chi_{376}(97,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{16}{23}\right)\) | \(e\left(\frac{10}{23}\right)\) | \(e\left(\frac{21}{23}\right)\) | \(e\left(\frac{9}{23}\right)\) | \(e\left(\frac{1}{23}\right)\) | \(e\left(\frac{18}{23}\right)\) | \(e\left(\frac{3}{23}\right)\) | \(e\left(\frac{22}{23}\right)\) | \(e\left(\frac{13}{23}\right)\) | \(e\left(\frac{14}{23}\right)\) |
| \(\chi_{376}(121,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{23}\right)\) | \(e\left(\frac{7}{23}\right)\) | \(e\left(\frac{17}{23}\right)\) | \(e\left(\frac{4}{23}\right)\) | \(e\left(\frac{3}{23}\right)\) | \(e\left(\frac{8}{23}\right)\) | \(e\left(\frac{9}{23}\right)\) | \(e\left(\frac{20}{23}\right)\) | \(e\left(\frac{16}{23}\right)\) | \(e\left(\frac{19}{23}\right)\) |
| \(\chi_{376}(145,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{15}{23}\right)\) | \(e\left(\frac{18}{23}\right)\) | \(e\left(\frac{1}{23}\right)\) | \(e\left(\frac{7}{23}\right)\) | \(e\left(\frac{11}{23}\right)\) | \(e\left(\frac{14}{23}\right)\) | \(e\left(\frac{10}{23}\right)\) | \(e\left(\frac{12}{23}\right)\) | \(e\left(\frac{5}{23}\right)\) | \(e\left(\frac{16}{23}\right)\) |
| \(\chi_{376}(153,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{23}\right)\) | \(e\left(\frac{5}{23}\right)\) | \(e\left(\frac{22}{23}\right)\) | \(e\left(\frac{16}{23}\right)\) | \(e\left(\frac{12}{23}\right)\) | \(e\left(\frac{9}{23}\right)\) | \(e\left(\frac{13}{23}\right)\) | \(e\left(\frac{11}{23}\right)\) | \(e\left(\frac{18}{23}\right)\) | \(e\left(\frac{7}{23}\right)\) |
| \(\chi_{376}(169,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{23}\right)\) | \(e\left(\frac{11}{23}\right)\) | \(e\left(\frac{7}{23}\right)\) | \(e\left(\frac{3}{23}\right)\) | \(e\left(\frac{8}{23}\right)\) | \(e\left(\frac{6}{23}\right)\) | \(e\left(\frac{1}{23}\right)\) | \(e\left(\frac{15}{23}\right)\) | \(e\left(\frac{12}{23}\right)\) | \(e\left(\frac{20}{23}\right)\) |
| \(\chi_{376}(177,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{23}\right)\) | \(e\left(\frac{15}{23}\right)\) | \(e\left(\frac{20}{23}\right)\) | \(e\left(\frac{2}{23}\right)\) | \(e\left(\frac{13}{23}\right)\) | \(e\left(\frac{4}{23}\right)\) | \(e\left(\frac{16}{23}\right)\) | \(e\left(\frac{10}{23}\right)\) | \(e\left(\frac{8}{23}\right)\) | \(e\left(\frac{21}{23}\right)\) |
| \(\chi_{376}(209,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{14}{23}\right)\) | \(e\left(\frac{3}{23}\right)\) | \(e\left(\frac{4}{23}\right)\) | \(e\left(\frac{5}{23}\right)\) | \(e\left(\frac{21}{23}\right)\) | \(e\left(\frac{10}{23}\right)\) | \(e\left(\frac{17}{23}\right)\) | \(e\left(\frac{2}{23}\right)\) | \(e\left(\frac{20}{23}\right)\) | \(e\left(\frac{18}{23}\right)\) |
| \(\chi_{376}(225,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{6}{23}\right)\) | \(e\left(\frac{21}{23}\right)\) | \(e\left(\frac{5}{23}\right)\) | \(e\left(\frac{12}{23}\right)\) | \(e\left(\frac{9}{23}\right)\) | \(e\left(\frac{1}{23}\right)\) | \(e\left(\frac{4}{23}\right)\) | \(e\left(\frac{14}{23}\right)\) | \(e\left(\frac{2}{23}\right)\) | \(e\left(\frac{11}{23}\right)\) |
| \(\chi_{376}(241,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{12}{23}\right)\) | \(e\left(\frac{19}{23}\right)\) | \(e\left(\frac{10}{23}\right)\) | \(e\left(\frac{1}{23}\right)\) | \(e\left(\frac{18}{23}\right)\) | \(e\left(\frac{2}{23}\right)\) | \(e\left(\frac{8}{23}\right)\) | \(e\left(\frac{5}{23}\right)\) | \(e\left(\frac{4}{23}\right)\) | \(e\left(\frac{22}{23}\right)\) |
| \(\chi_{376}(249,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{23}\right)\) | \(e\left(\frac{2}{23}\right)\) | \(e\left(\frac{18}{23}\right)\) | \(e\left(\frac{11}{23}\right)\) | \(e\left(\frac{14}{23}\right)\) | \(e\left(\frac{22}{23}\right)\) | \(e\left(\frac{19}{23}\right)\) | \(e\left(\frac{9}{23}\right)\) | \(e\left(\frac{21}{23}\right)\) | \(e\left(\frac{12}{23}\right)\) |
| \(\chi_{376}(289,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{21}{23}\right)\) | \(e\left(\frac{16}{23}\right)\) | \(e\left(\frac{6}{23}\right)\) | \(e\left(\frac{19}{23}\right)\) | \(e\left(\frac{20}{23}\right)\) | \(e\left(\frac{15}{23}\right)\) | \(e\left(\frac{14}{23}\right)\) | \(e\left(\frac{3}{23}\right)\) | \(e\left(\frac{7}{23}\right)\) | \(e\left(\frac{4}{23}\right)\) |
| \(\chi_{376}(337,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{23}\right)\) | \(e\left(\frac{4}{23}\right)\) | \(e\left(\frac{13}{23}\right)\) | \(e\left(\frac{22}{23}\right)\) | \(e\left(\frac{5}{23}\right)\) | \(e\left(\frac{21}{23}\right)\) | \(e\left(\frac{15}{23}\right)\) | \(e\left(\frac{18}{23}\right)\) | \(e\left(\frac{19}{23}\right)\) | \(e\left(\frac{1}{23}\right)\) |
| \(\chi_{376}(345,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{23}\right)\) | \(e\left(\frac{13}{23}\right)\) | \(e\left(\frac{2}{23}\right)\) | \(e\left(\frac{14}{23}\right)\) | \(e\left(\frac{22}{23}\right)\) | \(e\left(\frac{5}{23}\right)\) | \(e\left(\frac{20}{23}\right)\) | \(e\left(\frac{1}{23}\right)\) | \(e\left(\frac{10}{23}\right)\) | \(e\left(\frac{9}{23}\right)\) |
| \(\chi_{376}(353,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{23}\right)\) | \(e\left(\frac{14}{23}\right)\) | \(e\left(\frac{11}{23}\right)\) | \(e\left(\frac{8}{23}\right)\) | \(e\left(\frac{6}{23}\right)\) | \(e\left(\frac{16}{23}\right)\) | \(e\left(\frac{18}{23}\right)\) | \(e\left(\frac{17}{23}\right)\) | \(e\left(\frac{9}{23}\right)\) | \(e\left(\frac{15}{23}\right)\) |
| \(\chi_{376}(361,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{23}\right)\) | \(e\left(\frac{22}{23}\right)\) | \(e\left(\frac{14}{23}\right)\) | \(e\left(\frac{6}{23}\right)\) | \(e\left(\frac{16}{23}\right)\) | \(e\left(\frac{12}{23}\right)\) | \(e\left(\frac{2}{23}\right)\) | \(e\left(\frac{7}{23}\right)\) | \(e\left(\frac{1}{23}\right)\) | \(e\left(\frac{17}{23}\right)\) |