from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2011, base_ring=CyclotomicField(2010))
M = H._module
chi = DirichletCharacter(H, M([1024]))
chi.galois_orbit()
[g,chi] = znchar(Mod(5,2011))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(2011\) | |
| Conductor: | \(2011\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(1005\) | 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: | $\Q(\zeta_{1005})$ |
| Fixed field: | Number field defined by a degree 1005 polynomial (not computed) |
First 31 of 528 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{2011}(5,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{164}{201}\right)\) | \(e\left(\frac{512}{1005}\right)\) | \(e\left(\frac{127}{201}\right)\) | \(e\left(\frac{683}{1005}\right)\) | \(e\left(\frac{109}{335}\right)\) | \(e\left(\frac{584}{1005}\right)\) | \(e\left(\frac{30}{67}\right)\) | \(e\left(\frac{19}{1005}\right)\) | \(e\left(\frac{166}{335}\right)\) | \(e\left(\frac{304}{1005}\right)\) |
| \(\chi_{2011}(9,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{49}{201}\right)\) | \(e\left(\frac{1}{1005}\right)\) | \(e\left(\frac{98}{201}\right)\) | \(e\left(\frac{19}{1005}\right)\) | \(e\left(\frac{82}{335}\right)\) | \(e\left(\frac{802}{1005}\right)\) | \(e\left(\frac{49}{67}\right)\) | \(e\left(\frac{2}{1005}\right)\) | \(e\left(\frac{88}{335}\right)\) | \(e\left(\frac{32}{1005}\right)\) |
| \(\chi_{2011}(20,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{109}{201}\right)\) | \(e\left(\frac{757}{1005}\right)\) | \(e\left(\frac{17}{201}\right)\) | \(e\left(\frac{313}{1005}\right)\) | \(e\left(\frac{99}{335}\right)\) | \(e\left(\frac{94}{1005}\right)\) | \(e\left(\frac{42}{67}\right)\) | \(e\left(\frac{509}{1005}\right)\) | \(e\left(\frac{286}{335}\right)\) | \(e\left(\frac{104}{1005}\right)\) |
| \(\chi_{2011}(21,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{76}{201}\right)\) | \(e\left(\frac{904}{1005}\right)\) | \(e\left(\frac{152}{201}\right)\) | \(e\left(\frac{91}{1005}\right)\) | \(e\left(\frac{93}{335}\right)\) | \(e\left(\frac{403}{1005}\right)\) | \(e\left(\frac{9}{67}\right)\) | \(e\left(\frac{803}{1005}\right)\) | \(e\left(\frac{157}{335}\right)\) | \(e\left(\frac{788}{1005}\right)\) |
| \(\chi_{2011}(22,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{53}{201}\right)\) | \(e\left(\frac{641}{1005}\right)\) | \(e\left(\frac{106}{201}\right)\) | \(e\left(\frac{119}{1005}\right)\) | \(e\left(\frac{302}{335}\right)\) | \(e\left(\frac{527}{1005}\right)\) | \(e\left(\frac{53}{67}\right)\) | \(e\left(\frac{277}{1005}\right)\) | \(e\left(\frac{128}{335}\right)\) | \(e\left(\frac{412}{1005}\right)\) |
| \(\chi_{2011}(23,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{47}{201}\right)\) | \(e\left(\frac{83}{1005}\right)\) | \(e\left(\frac{94}{201}\right)\) | \(e\left(\frac{572}{1005}\right)\) | \(e\left(\frac{106}{335}\right)\) | \(e\left(\frac{236}{1005}\right)\) | \(e\left(\frac{47}{67}\right)\) | \(e\left(\frac{166}{1005}\right)\) | \(e\left(\frac{269}{335}\right)\) | \(e\left(\frac{646}{1005}\right)\) |
| \(\chi_{2011}(24,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{143}{201}\right)\) | \(e\left(\frac{368}{1005}\right)\) | \(e\left(\frac{85}{201}\right)\) | \(e\left(\frac{962}{1005}\right)\) | \(e\left(\frac{26}{335}\right)\) | \(e\left(\frac{671}{1005}\right)\) | \(e\left(\frac{9}{67}\right)\) | \(e\left(\frac{736}{1005}\right)\) | \(e\left(\frac{224}{335}\right)\) | \(e\left(\frac{721}{1005}\right)\) |
| \(\chi_{2011}(25,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{127}{201}\right)\) | \(e\left(\frac{19}{1005}\right)\) | \(e\left(\frac{53}{201}\right)\) | \(e\left(\frac{361}{1005}\right)\) | \(e\left(\frac{218}{335}\right)\) | \(e\left(\frac{163}{1005}\right)\) | \(e\left(\frac{60}{67}\right)\) | \(e\left(\frac{38}{1005}\right)\) | \(e\left(\frac{332}{335}\right)\) | \(e\left(\frac{608}{1005}\right)\) |
| \(\chi_{2011}(34,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{32}{201}\right)\) | \(e\left(\frac{296}{1005}\right)\) | \(e\left(\frac{64}{201}\right)\) | \(e\left(\frac{599}{1005}\right)\) | \(e\left(\frac{152}{335}\right)\) | \(e\left(\frac{212}{1005}\right)\) | \(e\left(\frac{32}{67}\right)\) | \(e\left(\frac{592}{1005}\right)\) | \(e\left(\frac{253}{335}\right)\) | \(e\left(\frac{427}{1005}\right)\) |
| \(\chi_{2011}(38,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{125}{201}\right)\) | \(e\left(\frac{704}{1005}\right)\) | \(e\left(\frac{49}{201}\right)\) | \(e\left(\frac{311}{1005}\right)\) | \(e\left(\frac{108}{335}\right)\) | \(e\left(\frac{803}{1005}\right)\) | \(e\left(\frac{58}{67}\right)\) | \(e\left(\frac{403}{1005}\right)\) | \(e\left(\frac{312}{335}\right)\) | \(e\left(\frac{418}{1005}\right)\) |
| \(\chi_{2011}(49,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{103}{201}\right)\) | \(e\left(\frac{802}{1005}\right)\) | \(e\left(\frac{5}{201}\right)\) | \(e\left(\frac{163}{1005}\right)\) | \(e\left(\frac{104}{335}\right)\) | \(e\left(\frac{4}{1005}\right)\) | \(e\left(\frac{36}{67}\right)\) | \(e\left(\frac{599}{1005}\right)\) | \(e\left(\frac{226}{335}\right)\) | \(e\left(\frac{539}{1005}\right)\) |
| \(\chi_{2011}(52,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{191}{201}\right)\) | \(e\left(\frac{209}{1005}\right)\) | \(e\left(\frac{181}{201}\right)\) | \(e\left(\frac{956}{1005}\right)\) | \(e\left(\frac{53}{335}\right)\) | \(e\left(\frac{788}{1005}\right)\) | \(e\left(\frac{57}{67}\right)\) | \(e\left(\frac{418}{1005}\right)\) | \(e\left(\frac{302}{335}\right)\) | \(e\left(\frac{658}{1005}\right)\) |
| \(\chi_{2011}(54,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{46}{201}\right)\) | \(e\left(\frac{124}{1005}\right)\) | \(e\left(\frac{92}{201}\right)\) | \(e\left(\frac{346}{1005}\right)\) | \(e\left(\frac{118}{335}\right)\) | \(e\left(\frac{958}{1005}\right)\) | \(e\left(\frac{46}{67}\right)\) | \(e\left(\frac{248}{1005}\right)\) | \(e\left(\frac{192}{335}\right)\) | \(e\left(\frac{953}{1005}\right)\) |
| \(\chi_{2011}(56,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{170}{201}\right)\) | \(e\left(\frac{266}{1005}\right)\) | \(e\left(\frac{139}{201}\right)\) | \(e\left(\frac{29}{1005}\right)\) | \(e\left(\frac{37}{335}\right)\) | \(e\left(\frac{272}{1005}\right)\) | \(e\left(\frac{36}{67}\right)\) | \(e\left(\frac{532}{1005}\right)\) | \(e\left(\frac{293}{335}\right)\) | \(e\left(\frac{472}{1005}\right)\) |
| \(\chi_{2011}(65,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{201}\right)\) | \(e\left(\frac{476}{1005}\right)\) | \(e\left(\frac{16}{201}\right)\) | \(e\left(\frac{1004}{1005}\right)\) | \(e\left(\frac{172}{335}\right)\) | \(e\left(\frac{857}{1005}\right)\) | \(e\left(\frac{8}{67}\right)\) | \(e\left(\frac{952}{1005}\right)\) | \(e\left(\frac{13}{335}\right)\) | \(e\left(\frac{157}{1005}\right)\) |
| \(\chi_{2011}(70,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{188}{201}\right)\) | \(e\left(\frac{533}{1005}\right)\) | \(e\left(\frac{175}{201}\right)\) | \(e\left(\frac{77}{1005}\right)\) | \(e\left(\frac{156}{335}\right)\) | \(e\left(\frac{341}{1005}\right)\) | \(e\left(\frac{54}{67}\right)\) | \(e\left(\frac{61}{1005}\right)\) | \(e\left(\frac{4}{335}\right)\) | \(e\left(\frac{976}{1005}\right)\) |
| \(\chi_{2011}(71,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{151}{201}\right)\) | \(e\left(\frac{442}{1005}\right)\) | \(e\left(\frac{101}{201}\right)\) | \(e\left(\frac{358}{1005}\right)\) | \(e\left(\frac{64}{335}\right)\) | \(e\left(\frac{724}{1005}\right)\) | \(e\left(\frac{17}{67}\right)\) | \(e\left(\frac{884}{1005}\right)\) | \(e\left(\frac{36}{335}\right)\) | \(e\left(\frac{74}{1005}\right)\) |
| \(\chi_{2011}(81,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{98}{201}\right)\) | \(e\left(\frac{2}{1005}\right)\) | \(e\left(\frac{196}{201}\right)\) | \(e\left(\frac{38}{1005}\right)\) | \(e\left(\frac{164}{335}\right)\) | \(e\left(\frac{599}{1005}\right)\) | \(e\left(\frac{31}{67}\right)\) | \(e\left(\frac{4}{1005}\right)\) | \(e\left(\frac{176}{335}\right)\) | \(e\left(\frac{64}{1005}\right)\) |
| \(\chi_{2011}(83,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{110}{201}\right)\) | \(e\left(\frac{716}{1005}\right)\) | \(e\left(\frac{19}{201}\right)\) | \(e\left(\frac{539}{1005}\right)\) | \(e\left(\frac{87}{335}\right)\) | \(e\left(\frac{377}{1005}\right)\) | \(e\left(\frac{43}{67}\right)\) | \(e\left(\frac{427}{1005}\right)\) | \(e\left(\frac{28}{335}\right)\) | \(e\left(\frac{802}{1005}\right)\) |
| \(\chi_{2011}(87,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{190}{201}\right)\) | \(e\left(\frac{49}{1005}\right)\) | \(e\left(\frac{179}{201}\right)\) | \(e\left(\frac{931}{1005}\right)\) | \(e\left(\frac{333}{335}\right)\) | \(e\left(\frac{103}{1005}\right)\) | \(e\left(\frac{56}{67}\right)\) | \(e\left(\frac{98}{1005}\right)\) | \(e\left(\frac{292}{335}\right)\) | \(e\left(\frac{563}{1005}\right)\) |
| \(\chi_{2011}(88,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{199}{201}\right)\) | \(e\left(\frac{886}{1005}\right)\) | \(e\left(\frac{197}{201}\right)\) | \(e\left(\frac{754}{1005}\right)\) | \(e\left(\frac{292}{335}\right)\) | \(e\left(\frac{37}{1005}\right)\) | \(e\left(\frac{65}{67}\right)\) | \(e\left(\frac{767}{1005}\right)\) | \(e\left(\frac{248}{335}\right)\) | \(e\left(\frac{212}{1005}\right)\) |
| \(\chi_{2011}(89,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{148}{201}\right)\) | \(e\left(\frac{163}{1005}\right)\) | \(e\left(\frac{95}{201}\right)\) | \(e\left(\frac{82}{1005}\right)\) | \(e\left(\frac{301}{335}\right)\) | \(e\left(\frac{76}{1005}\right)\) | \(e\left(\frac{14}{67}\right)\) | \(e\left(\frac{326}{1005}\right)\) | \(e\left(\frac{274}{335}\right)\) | \(e\left(\frac{191}{1005}\right)\) |
| \(\chi_{2011}(92,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{193}{201}\right)\) | \(e\left(\frac{328}{1005}\right)\) | \(e\left(\frac{185}{201}\right)\) | \(e\left(\frac{202}{1005}\right)\) | \(e\left(\frac{96}{335}\right)\) | \(e\left(\frac{751}{1005}\right)\) | \(e\left(\frac{59}{67}\right)\) | \(e\left(\frac{656}{1005}\right)\) | \(e\left(\frac{54}{335}\right)\) | \(e\left(\frac{446}{1005}\right)\) |
| \(\chi_{2011}(94,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{124}{201}\right)\) | \(e\left(\frac{946}{1005}\right)\) | \(e\left(\frac{47}{201}\right)\) | \(e\left(\frac{889}{1005}\right)\) | \(e\left(\frac{187}{335}\right)\) | \(e\left(\frac{922}{1005}\right)\) | \(e\left(\frac{57}{67}\right)\) | \(e\left(\frac{887}{1005}\right)\) | \(e\left(\frac{168}{335}\right)\) | \(e\left(\frac{122}{1005}\right)\) |
| \(\chi_{2011}(96,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{88}{201}\right)\) | \(e\left(\frac{613}{1005}\right)\) | \(e\left(\frac{176}{201}\right)\) | \(e\left(\frac{592}{1005}\right)\) | \(e\left(\frac{16}{335}\right)\) | \(e\left(\frac{181}{1005}\right)\) | \(e\left(\frac{21}{67}\right)\) | \(e\left(\frac{221}{1005}\right)\) | \(e\left(\frac{9}{335}\right)\) | \(e\left(\frac{521}{1005}\right)\) |
| \(\chi_{2011}(106,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{73}{201}\right)\) | \(e\left(\frac{424}{1005}\right)\) | \(e\left(\frac{146}{201}\right)\) | \(e\left(\frac{16}{1005}\right)\) | \(e\left(\frac{263}{335}\right)\) | \(e\left(\frac{358}{1005}\right)\) | \(e\left(\frac{6}{67}\right)\) | \(e\left(\frac{848}{1005}\right)\) | \(e\left(\frac{127}{335}\right)\) | \(e\left(\frac{503}{1005}\right)\) |
| \(\chi_{2011}(109,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{182}{201}\right)\) | \(e\left(\frac{176}{1005}\right)\) | \(e\left(\frac{163}{201}\right)\) | \(e\left(\frac{329}{1005}\right)\) | \(e\left(\frac{27}{335}\right)\) | \(e\left(\frac{452}{1005}\right)\) | \(e\left(\frac{48}{67}\right)\) | \(e\left(\frac{352}{1005}\right)\) | \(e\left(\frac{78}{335}\right)\) | \(e\left(\frac{607}{1005}\right)\) |
| \(\chi_{2011}(110,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{16}{201}\right)\) | \(e\left(\frac{148}{1005}\right)\) | \(e\left(\frac{32}{201}\right)\) | \(e\left(\frac{802}{1005}\right)\) | \(e\left(\frac{76}{335}\right)\) | \(e\left(\frac{106}{1005}\right)\) | \(e\left(\frac{16}{67}\right)\) | \(e\left(\frac{296}{1005}\right)\) | \(e\left(\frac{294}{335}\right)\) | \(e\left(\frac{716}{1005}\right)\) |
| \(\chi_{2011}(111,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{124}{201}\right)\) | \(e\left(\frac{343}{1005}\right)\) | \(e\left(\frac{47}{201}\right)\) | \(e\left(\frac{487}{1005}\right)\) | \(e\left(\frac{321}{335}\right)\) | \(e\left(\frac{721}{1005}\right)\) | \(e\left(\frac{57}{67}\right)\) | \(e\left(\frac{686}{1005}\right)\) | \(e\left(\frac{34}{335}\right)\) | \(e\left(\frac{926}{1005}\right)\) |
| \(\chi_{2011}(118,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{201}\right)\) | \(e\left(\frac{841}{1005}\right)\) | \(e\left(\frac{8}{201}\right)\) | \(e\left(\frac{904}{1005}\right)\) | \(e\left(\frac{287}{335}\right)\) | \(e\left(\frac{127}{1005}\right)\) | \(e\left(\frac{4}{67}\right)\) | \(e\left(\frac{677}{1005}\right)\) | \(e\left(\frac{308}{335}\right)\) | \(e\left(\frac{782}{1005}\right)\) |
| \(\chi_{2011}(121,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{161}{201}\right)\) | \(e\left(\frac{32}{1005}\right)\) | \(e\left(\frac{121}{201}\right)\) | \(e\left(\frac{608}{1005}\right)\) | \(e\left(\frac{279}{335}\right)\) | \(e\left(\frac{539}{1005}\right)\) | \(e\left(\frac{27}{67}\right)\) | \(e\left(\frac{64}{1005}\right)\) | \(e\left(\frac{136}{335}\right)\) | \(e\left(\frac{19}{1005}\right)\) |