from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1157, base_ring=CyclotomicField(264))
M = H._module
chi = DirichletCharacter(H, M([110,51]))
chi.galois_orbit()
[g,chi] = znchar(Mod(6,1157))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(1157\) | |
| Conductor: | \(1157\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(264\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | 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: | $\Q(\zeta_{264})$ |
| Fixed field: | Number field defined by a degree 264 polynomial (not computed) |
First 31 of 80 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{1157}(6,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{67}{132}\right)\) | \(e\left(\frac{227}{264}\right)\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{97}{264}\right)\) | \(e\left(\frac{61}{264}\right)\) | \(e\left(\frac{23}{44}\right)\) | \(e\left(\frac{95}{132}\right)\) | \(e\left(\frac{103}{132}\right)\) | \(e\left(\frac{19}{132}\right)\) |
| \(\chi_{1157}(28,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{83}{132}\right)\) | \(e\left(\frac{163}{264}\right)\) | \(e\left(\frac{17}{66}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{65}{264}\right)\) | \(e\left(\frac{245}{264}\right)\) | \(e\left(\frac{39}{44}\right)\) | \(e\left(\frac{31}{132}\right)\) | \(e\left(\frac{35}{132}\right)\) | \(e\left(\frac{59}{132}\right)\) |
| \(\chi_{1157}(41,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{119}{132}\right)\) | \(e\left(\frac{151}{264}\right)\) | \(e\left(\frac{53}{66}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{125}{264}\right)\) | \(e\left(\frac{65}{264}\right)\) | \(e\left(\frac{31}{44}\right)\) | \(e\left(\frac{19}{132}\right)\) | \(e\left(\frac{47}{132}\right)\) | \(e\left(\frac{83}{132}\right)\) |
| \(\chi_{1157}(59,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{97}{132}\right)\) | \(e\left(\frac{41}{264}\right)\) | \(e\left(\frac{31}{66}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{235}{264}\right)\) | \(e\left(\frac{175}{264}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{41}{132}\right)\) | \(e\left(\frac{25}{132}\right)\) | \(e\left(\frac{61}{132}\right)\) |
| \(\chi_{1157}(63,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{89}{132}\right)\) | \(e\left(\frac{73}{264}\right)\) | \(e\left(\frac{23}{66}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{251}{264}\right)\) | \(e\left(\frac{215}{264}\right)\) | \(e\left(\frac{1}{44}\right)\) | \(e\left(\frac{73}{132}\right)\) | \(e\left(\frac{125}{132}\right)\) | \(e\left(\frac{41}{132}\right)\) |
| \(\chi_{1157}(76,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{101}{132}\right)\) | \(e\left(\frac{25}{264}\right)\) | \(e\left(\frac{35}{66}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{227}{264}\right)\) | \(e\left(\frac{23}{264}\right)\) | \(e\left(\frac{13}{44}\right)\) | \(e\left(\frac{25}{132}\right)\) | \(e\left(\frac{41}{132}\right)\) | \(e\left(\frac{5}{132}\right)\) |
| \(\chi_{1157}(102,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{101}{132}\right)\) | \(e\left(\frac{157}{264}\right)\) | \(e\left(\frac{35}{66}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{95}{264}\right)\) | \(e\left(\frac{155}{264}\right)\) | \(e\left(\frac{13}{44}\right)\) | \(e\left(\frac{25}{132}\right)\) | \(e\left(\frac{41}{132}\right)\) | \(e\left(\frac{5}{132}\right)\) |
| \(\chi_{1157}(115,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{89}{132}\right)\) | \(e\left(\frac{205}{264}\right)\) | \(e\left(\frac{23}{66}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{119}{264}\right)\) | \(e\left(\frac{83}{264}\right)\) | \(e\left(\frac{1}{44}\right)\) | \(e\left(\frac{73}{132}\right)\) | \(e\left(\frac{125}{132}\right)\) | \(e\left(\frac{41}{132}\right)\) |
| \(\chi_{1157}(124,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{49}{132}\right)\) | \(e\left(\frac{101}{264}\right)\) | \(e\left(\frac{49}{66}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{199}{264}\right)\) | \(e\left(\frac{19}{264}\right)\) | \(e\left(\frac{5}{44}\right)\) | \(e\left(\frac{101}{132}\right)\) | \(e\left(\frac{97}{132}\right)\) | \(e\left(\frac{73}{132}\right)\) |
| \(\chi_{1157}(132,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{47}{132}\right)\) | \(e\left(\frac{175}{264}\right)\) | \(e\left(\frac{47}{66}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{5}{264}\right)\) | \(e\left(\frac{161}{264}\right)\) | \(e\left(\frac{3}{44}\right)\) | \(e\left(\frac{43}{132}\right)\) | \(e\left(\frac{23}{132}\right)\) | \(e\left(\frac{35}{132}\right)\) |
| \(\chi_{1157}(145,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{71}{132}\right)\) | \(e\left(\frac{211}{264}\right)\) | \(e\left(\frac{5}{66}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{89}{264}\right)\) | \(e\left(\frac{173}{264}\right)\) | \(e\left(\frac{27}{44}\right)\) | \(e\left(\frac{79}{132}\right)\) | \(e\left(\frac{119}{132}\right)\) | \(e\left(\frac{95}{132}\right)\) |
| \(\chi_{1157}(163,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{109}{132}\right)\) | \(e\left(\frac{257}{264}\right)\) | \(e\left(\frac{43}{66}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{211}{264}\right)\) | \(e\left(\frac{247}{264}\right)\) | \(e\left(\frac{21}{44}\right)\) | \(e\left(\frac{125}{132}\right)\) | \(e\left(\frac{73}{132}\right)\) | \(e\left(\frac{25}{132}\right)\) |
| \(\chi_{1157}(171,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{107}{132}\right)\) | \(e\left(\frac{199}{264}\right)\) | \(e\left(\frac{41}{66}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{149}{264}\right)\) | \(e\left(\frac{257}{264}\right)\) | \(e\left(\frac{19}{44}\right)\) | \(e\left(\frac{67}{132}\right)\) | \(e\left(\frac{131}{132}\right)\) | \(e\left(\frac{119}{132}\right)\) |
| \(\chi_{1157}(175,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{79}{132}\right)\) | \(e\left(\frac{47}{264}\right)\) | \(e\left(\frac{13}{66}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{205}{264}\right)\) | \(e\left(\frac{1}{264}\right)\) | \(e\left(\frac{35}{44}\right)\) | \(e\left(\frac{47}{132}\right)\) | \(e\left(\frac{19}{132}\right)\) | \(e\left(\frac{115}{132}\right)\) |
| \(\chi_{1157}(184,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{132}\right)\) | \(e\left(\frac{139}{264}\right)\) | \(e\left(\frac{23}{66}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{185}{264}\right)\) | \(e\left(\frac{149}{264}\right)\) | \(e\left(\frac{23}{44}\right)\) | \(e\left(\frac{7}{132}\right)\) | \(e\left(\frac{59}{132}\right)\) | \(e\left(\frac{107}{132}\right)\) |
| \(\chi_{1157}(193,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{65}{132}\right)\) | \(e\left(\frac{37}{264}\right)\) | \(e\left(\frac{65}{66}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{167}{264}\right)\) | \(e\left(\frac{203}{264}\right)\) | \(e\left(\frac{21}{44}\right)\) | \(e\left(\frac{37}{132}\right)\) | \(e\left(\frac{29}{132}\right)\) | \(e\left(\frac{113}{132}\right)\) |
| \(\chi_{1157}(201,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{103}{132}\right)\) | \(e\left(\frac{83}{264}\right)\) | \(e\left(\frac{37}{66}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{25}{264}\right)\) | \(e\left(\frac{13}{264}\right)\) | \(e\left(\frac{15}{44}\right)\) | \(e\left(\frac{83}{132}\right)\) | \(e\left(\frac{115}{132}\right)\) | \(e\left(\frac{43}{132}\right)\) |
| \(\chi_{1157}(232,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{132}\right)\) | \(e\left(\frac{145}{264}\right)\) | \(e\left(\frac{5}{66}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{155}{264}\right)\) | \(e\left(\frac{239}{264}\right)\) | \(e\left(\frac{5}{44}\right)\) | \(e\left(\frac{13}{132}\right)\) | \(e\left(\frac{53}{132}\right)\) | \(e\left(\frac{29}{132}\right)\) |
| \(\chi_{1157}(241,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{132}\right)\) | \(e\left(\frac{161}{264}\right)\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{163}{264}\right)\) | \(e\left(\frac{127}{264}\right)\) | \(e\left(\frac{1}{44}\right)\) | \(e\left(\frac{29}{132}\right)\) | \(e\left(\frac{37}{132}\right)\) | \(e\left(\frac{85}{132}\right)\) |
| \(\chi_{1157}(253,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{132}\right)\) | \(e\left(\frac{71}{264}\right)\) | \(e\left(\frac{7}{66}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{85}{264}\right)\) | \(e\left(\frac{97}{264}\right)\) | \(e\left(\frac{7}{44}\right)\) | \(e\left(\frac{71}{132}\right)\) | \(e\left(\frac{127}{132}\right)\) | \(e\left(\frac{67}{132}\right)\) |
| \(\chi_{1157}(254,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{132}\right)\) | \(e\left(\frac{113}{264}\right)\) | \(e\left(\frac{13}{66}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{139}{264}\right)\) | \(e\left(\frac{199}{264}\right)\) | \(e\left(\frac{13}{44}\right)\) | \(e\left(\frac{113}{132}\right)\) | \(e\left(\frac{85}{132}\right)\) | \(e\left(\frac{49}{132}\right)\) |
| \(\chi_{1157}(280,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{132}\right)\) | \(e\left(\frac{245}{264}\right)\) | \(e\left(\frac{13}{66}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{7}{264}\right)\) | \(e\left(\frac{67}{264}\right)\) | \(e\left(\frac{13}{44}\right)\) | \(e\left(\frac{113}{132}\right)\) | \(e\left(\frac{85}{132}\right)\) | \(e\left(\frac{49}{132}\right)\) |
| \(\chi_{1157}(293,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{132}\right)\) | \(e\left(\frac{29}{264}\right)\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{31}{264}\right)\) | \(e\left(\frac{259}{264}\right)\) | \(e\left(\frac{1}{44}\right)\) | \(e\left(\frac{29}{132}\right)\) | \(e\left(\frac{37}{132}\right)\) | \(e\left(\frac{85}{132}\right)\) |
| \(\chi_{1157}(297,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{53}{132}\right)\) | \(e\left(\frac{85}{264}\right)\) | \(e\left(\frac{53}{66}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{191}{264}\right)\) | \(e\left(\frac{131}{264}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{85}{132}\right)\) | \(e\left(\frac{113}{132}\right)\) | \(e\left(\frac{17}{132}\right)\) |
| \(\chi_{1157}(353,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{35}{132}\right)\) | \(e\left(\frac{223}{264}\right)\) | \(e\left(\frac{35}{66}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{29}{264}\right)\) | \(e\left(\frac{89}{264}\right)\) | \(e\left(\frac{35}{44}\right)\) | \(e\left(\frac{91}{132}\right)\) | \(e\left(\frac{107}{132}\right)\) | \(e\left(\frac{71}{132}\right)\) |
| \(\chi_{1157}(370,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{132}\right)\) | \(e\left(\frac{203}{264}\right)\) | \(e\left(\frac{7}{66}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{217}{264}\right)\) | \(e\left(\frac{229}{264}\right)\) | \(e\left(\frac{7}{44}\right)\) | \(e\left(\frac{71}{132}\right)\) | \(e\left(\frac{127}{132}\right)\) | \(e\left(\frac{67}{132}\right)\) |
| \(\chi_{1157}(371,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{109}{132}\right)\) | \(e\left(\frac{125}{264}\right)\) | \(e\left(\frac{43}{66}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{79}{264}\right)\) | \(e\left(\frac{115}{264}\right)\) | \(e\left(\frac{21}{44}\right)\) | \(e\left(\frac{125}{132}\right)\) | \(e\left(\frac{73}{132}\right)\) | \(e\left(\frac{25}{132}\right)\) |
| \(\chi_{1157}(375,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{125}{132}\right)\) | \(e\left(\frac{193}{264}\right)\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{179}{264}\right)\) | \(e\left(\frac{167}{264}\right)\) | \(e\left(\frac{37}{44}\right)\) | \(e\left(\frac{61}{132}\right)\) | \(e\left(\frac{5}{132}\right)\) | \(e\left(\frac{65}{132}\right)\) |
| \(\chi_{1157}(379,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{59}{132}\right)\) | \(e\left(\frac{259}{264}\right)\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{113}{264}\right)\) | \(e\left(\frac{101}{264}\right)\) | \(e\left(\frac{15}{44}\right)\) | \(e\left(\frac{127}{132}\right)\) | \(e\left(\frac{71}{132}\right)\) | \(e\left(\frac{131}{132}\right)\) |
| \(\chi_{1157}(410,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{49}{132}\right)\) | \(e\left(\frac{233}{264}\right)\) | \(e\left(\frac{49}{66}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{67}{264}\right)\) | \(e\left(\frac{151}{264}\right)\) | \(e\left(\frac{5}{44}\right)\) | \(e\left(\frac{101}{132}\right)\) | \(e\left(\frac{97}{132}\right)\) | \(e\left(\frac{73}{132}\right)\) |
| \(\chi_{1157}(414,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{132}\right)\) | \(e\left(\frac{49}{264}\right)\) | \(e\left(\frac{29}{66}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{107}{264}\right)\) | \(e\left(\frac{119}{264}\right)\) | \(e\left(\frac{29}{44}\right)\) | \(e\left(\frac{49}{132}\right)\) | \(e\left(\frac{17}{132}\right)\) | \(e\left(\frac{89}{132}\right)\) |