from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(987, base_ring=CyclotomicField(46))
M = H._module
chi = DirichletCharacter(H, M([0,23,11]))
chi.galois_orbit()
[g,chi] = znchar(Mod(13,987))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(987\) | |
| Conductor: | \(329\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(46\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 329.k | 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: | 46.46.48046142766286951052967407399087703375688302701387777971505263706078295125591942766227445314201.1 |
Characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{987}(13,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{23}\right)\) | \(e\left(\frac{14}{23}\right)\) | \(e\left(\frac{17}{23}\right)\) | \(e\left(\frac{21}{23}\right)\) | \(e\left(\frac{1}{23}\right)\) | \(e\left(\frac{31}{46}\right)\) | \(e\left(\frac{3}{23}\right)\) | \(e\left(\frac{5}{23}\right)\) | \(e\left(\frac{15}{46}\right)\) | \(e\left(\frac{6}{23}\right)\) |
| \(\chi_{987}(76,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{16}{23}\right)\) | \(e\left(\frac{9}{23}\right)\) | \(e\left(\frac{6}{23}\right)\) | \(e\left(\frac{2}{23}\right)\) | \(e\left(\frac{22}{23}\right)\) | \(e\left(\frac{15}{46}\right)\) | \(e\left(\frac{20}{23}\right)\) | \(e\left(\frac{18}{23}\right)\) | \(e\left(\frac{31}{46}\right)\) | \(e\left(\frac{17}{23}\right)\) |
| \(\chi_{987}(139,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{23}\right)\) | \(e\left(\frac{2}{23}\right)\) | \(e\left(\frac{9}{23}\right)\) | \(e\left(\frac{3}{23}\right)\) | \(e\left(\frac{10}{23}\right)\) | \(e\left(\frac{11}{46}\right)\) | \(e\left(\frac{7}{23}\right)\) | \(e\left(\frac{4}{23}\right)\) | \(e\left(\frac{35}{46}\right)\) | \(e\left(\frac{14}{23}\right)\) |
| \(\chi_{987}(160,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{14}{23}\right)\) | \(e\left(\frac{5}{23}\right)\) | \(e\left(\frac{11}{23}\right)\) | \(e\left(\frac{19}{23}\right)\) | \(e\left(\frac{2}{23}\right)\) | \(e\left(\frac{39}{46}\right)\) | \(e\left(\frac{6}{23}\right)\) | \(e\left(\frac{10}{23}\right)\) | \(e\left(\frac{7}{46}\right)\) | \(e\left(\frac{12}{23}\right)\) |
| \(\chi_{987}(181,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{12}{23}\right)\) | \(e\left(\frac{1}{23}\right)\) | \(e\left(\frac{16}{23}\right)\) | \(e\left(\frac{13}{23}\right)\) | \(e\left(\frac{5}{23}\right)\) | \(e\left(\frac{17}{46}\right)\) | \(e\left(\frac{15}{23}\right)\) | \(e\left(\frac{2}{23}\right)\) | \(e\left(\frac{29}{46}\right)\) | \(e\left(\frac{7}{23}\right)\) |
| \(\chi_{987}(223,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{21}{23}\right)\) | \(e\left(\frac{19}{23}\right)\) | \(e\left(\frac{5}{23}\right)\) | \(e\left(\frac{17}{23}\right)\) | \(e\left(\frac{3}{23}\right)\) | \(e\left(\frac{1}{46}\right)\) | \(e\left(\frac{9}{23}\right)\) | \(e\left(\frac{15}{23}\right)\) | \(e\left(\frac{45}{46}\right)\) | \(e\left(\frac{18}{23}\right)\) |
| \(\chi_{987}(265,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{6}{23}\right)\) | \(e\left(\frac{12}{23}\right)\) | \(e\left(\frac{8}{23}\right)\) | \(e\left(\frac{18}{23}\right)\) | \(e\left(\frac{14}{23}\right)\) | \(e\left(\frac{43}{46}\right)\) | \(e\left(\frac{19}{23}\right)\) | \(e\left(\frac{1}{23}\right)\) | \(e\left(\frac{3}{46}\right)\) | \(e\left(\frac{15}{23}\right)\) |
| \(\chi_{987}(349,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{23}\right)\) | \(e\left(\frac{22}{23}\right)\) | \(e\left(\frac{7}{23}\right)\) | \(e\left(\frac{10}{23}\right)\) | \(e\left(\frac{18}{23}\right)\) | \(e\left(\frac{29}{46}\right)\) | \(e\left(\frac{8}{23}\right)\) | \(e\left(\frac{21}{23}\right)\) | \(e\left(\frac{17}{46}\right)\) | \(e\left(\frac{16}{23}\right)\) |
| \(\chi_{987}(370,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{20}{23}\right)\) | \(e\left(\frac{17}{23}\right)\) | \(e\left(\frac{19}{23}\right)\) | \(e\left(\frac{14}{23}\right)\) | \(e\left(\frac{16}{23}\right)\) | \(e\left(\frac{13}{46}\right)\) | \(e\left(\frac{2}{23}\right)\) | \(e\left(\frac{11}{23}\right)\) | \(e\left(\frac{33}{46}\right)\) | \(e\left(\frac{4}{23}\right)\) |
| \(\chi_{987}(391,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{23}\right)\) | \(e\left(\frac{10}{23}\right)\) | \(e\left(\frac{22}{23}\right)\) | \(e\left(\frac{15}{23}\right)\) | \(e\left(\frac{4}{23}\right)\) | \(e\left(\frac{9}{46}\right)\) | \(e\left(\frac{12}{23}\right)\) | \(e\left(\frac{20}{23}\right)\) | \(e\left(\frac{37}{46}\right)\) | \(e\left(\frac{1}{23}\right)\) |
| \(\chi_{987}(433,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{10}{23}\right)\) | \(e\left(\frac{20}{23}\right)\) | \(e\left(\frac{21}{23}\right)\) | \(e\left(\frac{7}{23}\right)\) | \(e\left(\frac{8}{23}\right)\) | \(e\left(\frac{41}{46}\right)\) | \(e\left(\frac{1}{23}\right)\) | \(e\left(\frac{17}{23}\right)\) | \(e\left(\frac{5}{46}\right)\) | \(e\left(\frac{2}{23}\right)\) |
| \(\chi_{987}(454,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{23}\right)\) | \(e\left(\frac{8}{23}\right)\) | \(e\left(\frac{13}{23}\right)\) | \(e\left(\frac{12}{23}\right)\) | \(e\left(\frac{17}{23}\right)\) | \(e\left(\frac{21}{46}\right)\) | \(e\left(\frac{5}{23}\right)\) | \(e\left(\frac{16}{23}\right)\) | \(e\left(\frac{25}{46}\right)\) | \(e\left(\frac{10}{23}\right)\) |
| \(\chi_{987}(475,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{23}\right)\) | \(e\left(\frac{18}{23}\right)\) | \(e\left(\frac{12}{23}\right)\) | \(e\left(\frac{4}{23}\right)\) | \(e\left(\frac{21}{23}\right)\) | \(e\left(\frac{7}{46}\right)\) | \(e\left(\frac{17}{23}\right)\) | \(e\left(\frac{13}{23}\right)\) | \(e\left(\frac{39}{46}\right)\) | \(e\left(\frac{11}{23}\right)\) |
| \(\chi_{987}(496,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{23}\right)\) | \(e\left(\frac{16}{23}\right)\) | \(e\left(\frac{3}{23}\right)\) | \(e\left(\frac{1}{23}\right)\) | \(e\left(\frac{11}{23}\right)\) | \(e\left(\frac{19}{46}\right)\) | \(e\left(\frac{10}{23}\right)\) | \(e\left(\frac{9}{23}\right)\) | \(e\left(\frac{27}{46}\right)\) | \(e\left(\frac{20}{23}\right)\) |
| \(\chi_{987}(622,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{23}\right)\) | \(e\left(\frac{11}{23}\right)\) | \(e\left(\frac{15}{23}\right)\) | \(e\left(\frac{5}{23}\right)\) | \(e\left(\frac{9}{23}\right)\) | \(e\left(\frac{3}{46}\right)\) | \(e\left(\frac{4}{23}\right)\) | \(e\left(\frac{22}{23}\right)\) | \(e\left(\frac{43}{46}\right)\) | \(e\left(\frac{8}{23}\right)\) |
| \(\chi_{987}(727,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{18}{23}\right)\) | \(e\left(\frac{13}{23}\right)\) | \(e\left(\frac{1}{23}\right)\) | \(e\left(\frac{8}{23}\right)\) | \(e\left(\frac{19}{23}\right)\) | \(e\left(\frac{37}{46}\right)\) | \(e\left(\frac{11}{23}\right)\) | \(e\left(\frac{3}{23}\right)\) | \(e\left(\frac{9}{46}\right)\) | \(e\left(\frac{22}{23}\right)\) |
| \(\chi_{987}(748,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{23}\right)\) | \(e\left(\frac{4}{23}\right)\) | \(e\left(\frac{18}{23}\right)\) | \(e\left(\frac{6}{23}\right)\) | \(e\left(\frac{20}{23}\right)\) | \(e\left(\frac{45}{46}\right)\) | \(e\left(\frac{14}{23}\right)\) | \(e\left(\frac{8}{23}\right)\) | \(e\left(\frac{1}{46}\right)\) | \(e\left(\frac{5}{23}\right)\) |
| \(\chi_{987}(790,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{15}{23}\right)\) | \(e\left(\frac{7}{23}\right)\) | \(e\left(\frac{20}{23}\right)\) | \(e\left(\frac{22}{23}\right)\) | \(e\left(\frac{12}{23}\right)\) | \(e\left(\frac{27}{46}\right)\) | \(e\left(\frac{13}{23}\right)\) | \(e\left(\frac{14}{23}\right)\) | \(e\left(\frac{19}{46}\right)\) | \(e\left(\frac{3}{23}\right)\) |
| \(\chi_{987}(832,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{23}\right)\) | \(e\left(\frac{3}{23}\right)\) | \(e\left(\frac{2}{23}\right)\) | \(e\left(\frac{16}{23}\right)\) | \(e\left(\frac{15}{23}\right)\) | \(e\left(\frac{5}{46}\right)\) | \(e\left(\frac{22}{23}\right)\) | \(e\left(\frac{6}{23}\right)\) | \(e\left(\frac{41}{46}\right)\) | \(e\left(\frac{21}{23}\right)\) |
| \(\chi_{987}(916,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{22}{23}\right)\) | \(e\left(\frac{21}{23}\right)\) | \(e\left(\frac{14}{23}\right)\) | \(e\left(\frac{20}{23}\right)\) | \(e\left(\frac{13}{23}\right)\) | \(e\left(\frac{35}{46}\right)\) | \(e\left(\frac{16}{23}\right)\) | \(e\left(\frac{19}{23}\right)\) | \(e\left(\frac{11}{46}\right)\) | \(e\left(\frac{9}{23}\right)\) |
| \(\chi_{987}(937,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{23}\right)\) | \(e\left(\frac{15}{23}\right)\) | \(e\left(\frac{10}{23}\right)\) | \(e\left(\frac{11}{23}\right)\) | \(e\left(\frac{6}{23}\right)\) | \(e\left(\frac{25}{46}\right)\) | \(e\left(\frac{18}{23}\right)\) | \(e\left(\frac{7}{23}\right)\) | \(e\left(\frac{21}{46}\right)\) | \(e\left(\frac{13}{23}\right)\) |
| \(\chi_{987}(979,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{23}\right)\) | \(e\left(\frac{6}{23}\right)\) | \(e\left(\frac{4}{23}\right)\) | \(e\left(\frac{9}{23}\right)\) | \(e\left(\frac{7}{23}\right)\) | \(e\left(\frac{33}{46}\right)\) | \(e\left(\frac{21}{23}\right)\) | \(e\left(\frac{12}{23}\right)\) | \(e\left(\frac{13}{46}\right)\) | \(e\left(\frac{19}{23}\right)\) |