from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1011, base_ring=CyclotomicField(168))
M = H._module
chi = DirichletCharacter(H, M([0,127]))
chi.galois_orbit()
[g,chi] = znchar(Mod(28,1011))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(1011\) | |
| Conductor: | \(337\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(168\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 337.s | 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: | $\Q(\zeta_{168})$ |
| Fixed field: | Number field defined by a degree 168 polynomial (not computed) |
First 31 of 48 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{1011}(28,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{53}{56}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{127}{168}\right)\) | \(e\left(\frac{15}{56}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{17}{84}\right)\) | \(e\left(\frac{5}{21}\right)\) |
| \(\chi_{1011}(91,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{45}{56}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{167}{168}\right)\) | \(e\left(\frac{55}{56}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{25}{84}\right)\) | \(e\left(\frac{16}{21}\right)\) |
| \(\chi_{1011}(94,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{39}{56}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{29}{168}\right)\) | \(e\left(\frac{29}{56}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{31}{84}\right)\) | \(e\left(\frac{19}{21}\right)\) |
| \(\chi_{1011}(100,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{11}{56}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{1}{168}\right)\) | \(e\left(\frac{1}{56}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{59}{84}\right)\) | \(e\left(\frac{5}{21}\right)\) |
| \(\chi_{1011}(112,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{37}{56}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{95}{168}\right)\) | \(e\left(\frac{39}{56}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{61}{84}\right)\) | \(e\left(\frac{13}{21}\right)\) |
| \(\chi_{1011}(115,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{41}{56}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{131}{168}\right)\) | \(e\left(\frac{19}{56}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{1}{84}\right)\) | \(e\left(\frac{4}{21}\right)\) |
| \(\chi_{1011}(145,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{27}{56}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{145}{168}\right)\) | \(e\left(\frac{33}{56}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{71}{84}\right)\) | \(e\left(\frac{11}{21}\right)\) |
| \(\chi_{1011}(190,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{45}{56}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{55}{168}\right)\) | \(e\left(\frac{55}{56}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{53}{84}\right)\) | \(e\left(\frac{2}{21}\right)\) |
| \(\chi_{1011}(211,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{15}{56}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{37}{168}\right)\) | \(e\left(\frac{37}{56}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{83}{84}\right)\) | \(e\left(\frac{17}{21}\right)\) |
| \(\chi_{1011}(229,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{41}{56}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{19}{168}\right)\) | \(e\left(\frac{19}{56}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{29}{84}\right)\) | \(e\left(\frac{11}{21}\right)\) |
| \(\chi_{1011}(259,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{43}{56}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{65}{168}\right)\) | \(e\left(\frac{9}{56}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{55}{84}\right)\) | \(e\left(\frac{10}{21}\right)\) |
| \(\chi_{1011}(274,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{23}{56}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{53}{168}\right)\) | \(e\left(\frac{53}{56}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{19}{84}\right)\) | \(e\left(\frac{13}{21}\right)\) |
| \(\chi_{1011}(313,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{51}{56}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{25}{168}\right)\) | \(e\left(\frac{25}{56}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{47}{84}\right)\) | \(e\left(\frac{20}{21}\right)\) |
| \(\chi_{1011}(325,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{3}{56}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{41}{168}\right)\) | \(e\left(\frac{41}{56}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{67}{84}\right)\) | \(e\left(\frac{16}{21}\right)\) |
| \(\chi_{1011}(334,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{19}{56}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{73}{168}\right)\) | \(e\left(\frac{17}{56}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{23}{84}\right)\) | \(e\left(\frac{8}{21}\right)\) |
| \(\chi_{1011}(340,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{47}{56}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{157}{168}\right)\) | \(e\left(\frac{45}{56}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{23}{84}\right)\) | \(e\left(\frac{8}{21}\right)\) |
| \(\chi_{1011}(349,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{31}{56}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{125}{168}\right)\) | \(e\left(\frac{13}{56}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{67}{84}\right)\) | \(e\left(\frac{16}{21}\right)\) |
| \(\chi_{1011}(361,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{23}{56}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{109}{168}\right)\) | \(e\left(\frac{53}{56}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{47}{84}\right)\) | \(e\left(\frac{20}{21}\right)\) |
| \(\chi_{1011}(400,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{51}{56}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{137}{168}\right)\) | \(e\left(\frac{25}{56}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{19}{84}\right)\) | \(e\left(\frac{13}{21}\right)\) |
| \(\chi_{1011}(415,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{15}{56}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{149}{168}\right)\) | \(e\left(\frac{37}{56}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{55}{84}\right)\) | \(e\left(\frac{10}{21}\right)\) |
| \(\chi_{1011}(445,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{13}{56}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{103}{168}\right)\) | \(e\left(\frac{47}{56}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{29}{84}\right)\) | \(e\left(\frac{11}{21}\right)\) |
| \(\chi_{1011}(463,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{43}{56}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{121}{168}\right)\) | \(e\left(\frac{9}{56}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{83}{84}\right)\) | \(e\left(\frac{17}{21}\right)\) |
| \(\chi_{1011}(484,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{17}{56}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{139}{168}\right)\) | \(e\left(\frac{27}{56}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{53}{84}\right)\) | \(e\left(\frac{2}{21}\right)\) |
| \(\chi_{1011}(529,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{55}{56}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{61}{168}\right)\) | \(e\left(\frac{5}{56}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{71}{84}\right)\) | \(e\left(\frac{11}{21}\right)\) |
| \(\chi_{1011}(559,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{13}{56}\right)\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{47}{168}\right)\) | \(e\left(\frac{47}{56}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{1}{84}\right)\) | \(e\left(\frac{4}{21}\right)\) |
| \(\chi_{1011}(562,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{9}{56}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{11}{168}\right)\) | \(e\left(\frac{11}{56}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{61}{84}\right)\) | \(e\left(\frac{13}{21}\right)\) |
| \(\chi_{1011}(574,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{39}{56}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{85}{168}\right)\) | \(e\left(\frac{29}{56}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{59}{84}\right)\) | \(e\left(\frac{5}{21}\right)\) |
| \(\chi_{1011}(580,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{11}{56}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{113}{168}\right)\) | \(e\left(\frac{1}{56}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{31}{84}\right)\) | \(e\left(\frac{19}{21}\right)\) |
| \(\chi_{1011}(583,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{17}{56}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{83}{168}\right)\) | \(e\left(\frac{27}{56}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{25}{84}\right)\) | \(e\left(\frac{16}{21}\right)\) |
| \(\chi_{1011}(646,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{25}{56}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{43}{168}\right)\) | \(e\left(\frac{43}{56}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{17}{84}\right)\) | \(e\left(\frac{5}{21}\right)\) |
| \(\chi_{1011}(688,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{5}{56}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{143}{168}\right)\) | \(e\left(\frac{31}{56}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{37}{84}\right)\) | \(e\left(\frac{1}{21}\right)\) |