from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5241, base_ring=CyclotomicField(582))
M = H._module
chi = DirichletCharacter(H, M([291,88]))
chi.galois_orbit()
[g,chi] = znchar(Mod(35,5241))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(5241\) | |
| Conductor: | \(5241\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(582\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Related number fields
| Field of values: | $\Q(\zeta_{291})$ |
| Fixed field: | Number field defined by a degree 582 polynomial (not computed) |
First 31 of 192 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{5241}(35,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{379}{582}\right)\) | \(e\left(\frac{88}{291}\right)\) | \(e\left(\frac{581}{582}\right)\) | \(e\left(\frac{122}{291}\right)\) | \(e\left(\frac{185}{194}\right)\) | \(e\left(\frac{63}{97}\right)\) | \(e\left(\frac{1}{194}\right)\) | \(e\left(\frac{221}{291}\right)\) | \(e\left(\frac{41}{582}\right)\) | \(e\left(\frac{176}{291}\right)\) |
| \(\chi_{5241}(41,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{391}{582}\right)\) | \(e\left(\frac{100}{291}\right)\) | \(e\left(\frac{65}{582}\right)\) | \(e\left(\frac{218}{291}\right)\) | \(e\left(\frac{3}{194}\right)\) | \(e\left(\frac{76}{97}\right)\) | \(e\left(\frac{129}{194}\right)\) | \(e\left(\frac{185}{291}\right)\) | \(e\left(\frac{245}{582}\right)\) | \(e\left(\frac{200}{291}\right)\) |
| \(\chi_{5241}(65,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{113}{582}\right)\) | \(e\left(\frac{113}{291}\right)\) | \(e\left(\frac{379}{582}\right)\) | \(e\left(\frac{31}{291}\right)\) | \(e\left(\frac{113}{194}\right)\) | \(e\left(\frac{82}{97}\right)\) | \(e\left(\frac{9}{194}\right)\) | \(e\left(\frac{49}{291}\right)\) | \(e\left(\frac{175}{582}\right)\) | \(e\left(\frac{226}{291}\right)\) |
| \(\chi_{5241}(92,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{401}{582}\right)\) | \(e\left(\frac{110}{291}\right)\) | \(e\left(\frac{217}{582}\right)\) | \(e\left(\frac{7}{291}\right)\) | \(e\left(\frac{13}{194}\right)\) | \(e\left(\frac{6}{97}\right)\) | \(e\left(\frac{171}{194}\right)\) | \(e\left(\frac{58}{291}\right)\) | \(e\left(\frac{415}{582}\right)\) | \(e\left(\frac{220}{291}\right)\) |
| \(\chi_{5241}(185,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{563}{582}\right)\) | \(e\left(\frac{272}{291}\right)\) | \(e\left(\frac{235}{582}\right)\) | \(e\left(\frac{139}{291}\right)\) | \(e\left(\frac{175}{194}\right)\) | \(e\left(\frac{36}{97}\right)\) | \(e\left(\frac{153}{194}\right)\) | \(e\left(\frac{154}{291}\right)\) | \(e\left(\frac{259}{582}\right)\) | \(e\left(\frac{253}{291}\right)\) |
| \(\chi_{5241}(224,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{395}{582}\right)\) | \(e\left(\frac{104}{291}\right)\) | \(e\left(\frac{475}{582}\right)\) | \(e\left(\frac{250}{291}\right)\) | \(e\left(\frac{7}{194}\right)\) | \(e\left(\frac{48}{97}\right)\) | \(e\left(\frac{107}{194}\right)\) | \(e\left(\frac{76}{291}\right)\) | \(e\left(\frac{313}{582}\right)\) | \(e\left(\frac{208}{291}\right)\) |
| \(\chi_{5241}(272,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{35}{582}\right)\) | \(e\left(\frac{35}{291}\right)\) | \(e\left(\frac{241}{582}\right)\) | \(e\left(\frac{280}{291}\right)\) | \(e\left(\frac{35}{194}\right)\) | \(e\left(\frac{46}{97}\right)\) | \(e\left(\frac{147}{194}\right)\) | \(e\left(\frac{283}{291}\right)\) | \(e\left(\frac{13}{582}\right)\) | \(e\left(\frac{70}{291}\right)\) |
| \(\chi_{5241}(302,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{53}{582}\right)\) | \(e\left(\frac{53}{291}\right)\) | \(e\left(\frac{49}{582}\right)\) | \(e\left(\frac{133}{291}\right)\) | \(e\left(\frac{53}{194}\right)\) | \(e\left(\frac{17}{97}\right)\) | \(e\left(\frac{145}{194}\right)\) | \(e\left(\frac{229}{291}\right)\) | \(e\left(\frac{319}{582}\right)\) | \(e\left(\frac{106}{291}\right)\) |
| \(\chi_{5241}(350,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{365}{582}\right)\) | \(e\left(\frac{74}{291}\right)\) | \(e\left(\frac{19}{582}\right)\) | \(e\left(\frac{10}{291}\right)\) | \(e\left(\frac{171}{194}\right)\) | \(e\left(\frac{64}{97}\right)\) | \(e\left(\frac{175}{194}\right)\) | \(e\left(\frac{166}{291}\right)\) | \(e\left(\frac{385}{582}\right)\) | \(e\left(\frac{148}{291}\right)\) |
| \(\chi_{5241}(353,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{425}{582}\right)\) | \(e\left(\frac{134}{291}\right)\) | \(e\left(\frac{349}{582}\right)\) | \(e\left(\frac{199}{291}\right)\) | \(e\left(\frac{37}{194}\right)\) | \(e\left(\frac{32}{97}\right)\) | \(e\left(\frac{39}{194}\right)\) | \(e\left(\frac{277}{291}\right)\) | \(e\left(\frac{241}{582}\right)\) | \(e\left(\frac{268}{291}\right)\) |
| \(\chi_{5241}(362,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{527}{582}\right)\) | \(e\left(\frac{236}{291}\right)\) | \(e\left(\frac{37}{582}\right)\) | \(e\left(\frac{142}{291}\right)\) | \(e\left(\frac{139}{194}\right)\) | \(e\left(\frac{94}{97}\right)\) | \(e\left(\frac{157}{194}\right)\) | \(e\left(\frac{262}{291}\right)\) | \(e\left(\frac{229}{582}\right)\) | \(e\left(\frac{181}{291}\right)\) |
| \(\chi_{5241}(386,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{449}{582}\right)\) | \(e\left(\frac{158}{291}\right)\) | \(e\left(\frac{481}{582}\right)\) | \(e\left(\frac{100}{291}\right)\) | \(e\left(\frac{61}{194}\right)\) | \(e\left(\frac{58}{97}\right)\) | \(e\left(\frac{101}{194}\right)\) | \(e\left(\frac{205}{291}\right)\) | \(e\left(\frac{67}{582}\right)\) | \(e\left(\frac{25}{291}\right)\) |
| \(\chi_{5241}(410,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{377}{582}\right)\) | \(e\left(\frac{86}{291}\right)\) | \(e\left(\frac{85}{582}\right)\) | \(e\left(\frac{106}{291}\right)\) | \(e\left(\frac{183}{194}\right)\) | \(e\left(\frac{77}{97}\right)\) | \(e\left(\frac{109}{194}\right)\) | \(e\left(\frac{130}{291}\right)\) | \(e\left(\frac{7}{582}\right)\) | \(e\left(\frac{172}{291}\right)\) |
| \(\chi_{5241}(413,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{457}{582}\right)\) | \(e\left(\frac{166}{291}\right)\) | \(e\left(\frac{137}{582}\right)\) | \(e\left(\frac{164}{291}\right)\) | \(e\left(\frac{69}{194}\right)\) | \(e\left(\frac{2}{97}\right)\) | \(e\left(\frac{57}{194}\right)\) | \(e\left(\frac{278}{291}\right)\) | \(e\left(\frac{203}{582}\right)\) | \(e\left(\frac{41}{291}\right)\) |
| \(\chi_{5241}(419,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{131}{582}\right)\) | \(e\left(\frac{131}{291}\right)\) | \(e\left(\frac{187}{582}\right)\) | \(e\left(\frac{175}{291}\right)\) | \(e\left(\frac{131}{194}\right)\) | \(e\left(\frac{53}{97}\right)\) | \(e\left(\frac{7}{194}\right)\) | \(e\left(\frac{286}{291}\right)\) | \(e\left(\frac{481}{582}\right)\) | \(e\left(\frac{262}{291}\right)\) |
| \(\chi_{5241}(425,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{582}\right)\) | \(e\left(\frac{5}{291}\right)\) | \(e\left(\frac{367}{582}\right)\) | \(e\left(\frac{40}{291}\right)\) | \(e\left(\frac{5}{194}\right)\) | \(e\left(\frac{62}{97}\right)\) | \(e\left(\frac{21}{194}\right)\) | \(e\left(\frac{82}{291}\right)\) | \(e\left(\frac{85}{582}\right)\) | \(e\left(\frac{10}{291}\right)\) |
| \(\chi_{5241}(479,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{545}{582}\right)\) | \(e\left(\frac{254}{291}\right)\) | \(e\left(\frac{427}{582}\right)\) | \(e\left(\frac{286}{291}\right)\) | \(e\left(\frac{157}{194}\right)\) | \(e\left(\frac{65}{97}\right)\) | \(e\left(\frac{155}{194}\right)\) | \(e\left(\frac{208}{291}\right)\) | \(e\left(\frac{535}{582}\right)\) | \(e\left(\frac{217}{291}\right)\) |
| \(\chi_{5241}(557,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{413}{582}\right)\) | \(e\left(\frac{122}{291}\right)\) | \(e\left(\frac{283}{582}\right)\) | \(e\left(\frac{103}{291}\right)\) | \(e\left(\frac{25}{194}\right)\) | \(e\left(\frac{19}{97}\right)\) | \(e\left(\frac{105}{194}\right)\) | \(e\left(\frac{22}{291}\right)\) | \(e\left(\frac{37}{582}\right)\) | \(e\left(\frac{244}{291}\right)\) |
| \(\chi_{5241}(566,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{145}{582}\right)\) | \(e\left(\frac{145}{291}\right)\) | \(e\left(\frac{167}{582}\right)\) | \(e\left(\frac{287}{291}\right)\) | \(e\left(\frac{145}{194}\right)\) | \(e\left(\frac{52}{97}\right)\) | \(e\left(\frac{27}{194}\right)\) | \(e\left(\frac{50}{291}\right)\) | \(e\left(\frac{137}{582}\right)\) | \(e\left(\frac{290}{291}\right)\) |
| \(\chi_{5241}(569,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{509}{582}\right)\) | \(e\left(\frac{218}{291}\right)\) | \(e\left(\frac{229}{582}\right)\) | \(e\left(\frac{289}{291}\right)\) | \(e\left(\frac{121}{194}\right)\) | \(e\left(\frac{26}{97}\right)\) | \(e\left(\frac{159}{194}\right)\) | \(e\left(\frac{25}{291}\right)\) | \(e\left(\frac{505}{582}\right)\) | \(e\left(\frac{145}{291}\right)\) |
| \(\chi_{5241}(572,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{559}{582}\right)\) | \(e\left(\frac{268}{291}\right)\) | \(e\left(\frac{407}{582}\right)\) | \(e\left(\frac{107}{291}\right)\) | \(e\left(\frac{171}{194}\right)\) | \(e\left(\frac{64}{97}\right)\) | \(e\left(\frac{175}{194}\right)\) | \(e\left(\frac{263}{291}\right)\) | \(e\left(\frac{191}{582}\right)\) | \(e\left(\frac{245}{291}\right)\) |
| \(\chi_{5241}(581,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{523}{582}\right)\) | \(e\left(\frac{232}{291}\right)\) | \(e\left(\frac{209}{582}\right)\) | \(e\left(\frac{110}{291}\right)\) | \(e\left(\frac{135}{194}\right)\) | \(e\left(\frac{25}{97}\right)\) | \(e\left(\frac{179}{194}\right)\) | \(e\left(\frac{80}{291}\right)\) | \(e\left(\frac{161}{582}\right)\) | \(e\left(\frac{173}{291}\right)\) |
| \(\chi_{5241}(584,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{217}{582}\right)\) | \(e\left(\frac{217}{291}\right)\) | \(e\left(\frac{563}{582}\right)\) | \(e\left(\frac{281}{291}\right)\) | \(e\left(\frac{23}{194}\right)\) | \(e\left(\frac{33}{97}\right)\) | \(e\left(\frac{19}{194}\right)\) | \(e\left(\frac{125}{291}\right)\) | \(e\left(\frac{197}{582}\right)\) | \(e\left(\frac{143}{291}\right)\) |
| \(\chi_{5241}(593,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{233}{582}\right)\) | \(e\left(\frac{233}{291}\right)\) | \(e\left(\frac{457}{582}\right)\) | \(e\left(\frac{118}{291}\right)\) | \(e\left(\frac{39}{194}\right)\) | \(e\left(\frac{18}{97}\right)\) | \(e\left(\frac{125}{194}\right)\) | \(e\left(\frac{271}{291}\right)\) | \(e\left(\frac{469}{582}\right)\) | \(e\left(\frac{175}{291}\right)\) |
| \(\chi_{5241}(599,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{203}{582}\right)\) | \(e\left(\frac{203}{291}\right)\) | \(e\left(\frac{1}{582}\right)\) | \(e\left(\frac{169}{291}\right)\) | \(e\left(\frac{9}{194}\right)\) | \(e\left(\frac{34}{97}\right)\) | \(e\left(\frac{193}{194}\right)\) | \(e\left(\frac{70}{291}\right)\) | \(e\left(\frac{541}{582}\right)\) | \(e\left(\frac{115}{291}\right)\) |
| \(\chi_{5241}(602,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{295}{582}\right)\) | \(e\left(\frac{4}{291}\right)\) | \(e\left(\frac{119}{582}\right)\) | \(e\left(\frac{32}{291}\right)\) | \(e\left(\frac{101}{194}\right)\) | \(e\left(\frac{69}{97}\right)\) | \(e\left(\frac{75}{194}\right)\) | \(e\left(\frac{182}{291}\right)\) | \(e\left(\frac{359}{582}\right)\) | \(e\left(\frac{8}{291}\right)\) |
| \(\chi_{5241}(611,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{133}{582}\right)\) | \(e\left(\frac{133}{291}\right)\) | \(e\left(\frac{101}{582}\right)\) | \(e\left(\frac{191}{291}\right)\) | \(e\left(\frac{133}{194}\right)\) | \(e\left(\frac{39}{97}\right)\) | \(e\left(\frac{93}{194}\right)\) | \(e\left(\frac{86}{291}\right)\) | \(e\left(\frac{515}{582}\right)\) | \(e\left(\frac{266}{291}\right)\) |
| \(\chi_{5241}(614,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{107}{582}\right)\) | \(e\left(\frac{107}{291}\right)\) | \(e\left(\frac{55}{582}\right)\) | \(e\left(\frac{274}{291}\right)\) | \(e\left(\frac{107}{194}\right)\) | \(e\left(\frac{27}{97}\right)\) | \(e\left(\frac{139}{194}\right)\) | \(e\left(\frac{67}{291}\right)\) | \(e\left(\frac{73}{582}\right)\) | \(e\left(\frac{214}{291}\right)\) |
| \(\chi_{5241}(632,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{271}{582}\right)\) | \(e\left(\frac{271}{291}\right)\) | \(e\left(\frac{569}{582}\right)\) | \(e\left(\frac{131}{291}\right)\) | \(e\left(\frac{77}{194}\right)\) | \(e\left(\frac{43}{97}\right)\) | \(e\left(\frac{13}{194}\right)\) | \(e\left(\frac{254}{291}\right)\) | \(e\left(\frac{533}{582}\right)\) | \(e\left(\frac{251}{291}\right)\) |
| \(\chi_{5241}(647,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{403}{582}\right)\) | \(e\left(\frac{112}{291}\right)\) | \(e\left(\frac{131}{582}\right)\) | \(e\left(\frac{23}{291}\right)\) | \(e\left(\frac{15}{194}\right)\) | \(e\left(\frac{89}{97}\right)\) | \(e\left(\frac{63}{194}\right)\) | \(e\left(\frac{149}{291}\right)\) | \(e\left(\frac{449}{582}\right)\) | \(e\left(\frac{224}{291}\right)\) |
| \(\chi_{5241}(665,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{269}{582}\right)\) | \(e\left(\frac{269}{291}\right)\) | \(e\left(\frac{73}{582}\right)\) | \(e\left(\frac{115}{291}\right)\) | \(e\left(\frac{75}{194}\right)\) | \(e\left(\frac{57}{97}\right)\) | \(e\left(\frac{121}{194}\right)\) | \(e\left(\frac{163}{291}\right)\) | \(e\left(\frac{499}{582}\right)\) | \(e\left(\frac{247}{291}\right)\) |