from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2011, base_ring=CyclotomicField(670))
M = H._module
chi = DirichletCharacter(H, M([82]))
chi.galois_orbit()
[g,chi] = znchar(Mod(6,2011))
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: | \(335\) | 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: | $\Q(\zeta_{335})$ |
| Fixed field: | Number field defined by a degree 335 polynomial (not computed) |
First 31 of 264 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{2011}(6,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{66}{67}\right)\) | \(e\left(\frac{41}{335}\right)\) | \(e\left(\frac{65}{67}\right)\) | \(e\left(\frac{109}{335}\right)\) | \(e\left(\frac{36}{335}\right)\) | \(e\left(\frac{52}{335}\right)\) | \(e\left(\frac{64}{67}\right)\) | \(e\left(\frac{82}{335}\right)\) | \(e\left(\frac{104}{335}\right)\) | \(e\left(\frac{307}{335}\right)\) |
| \(\chi_{2011}(13,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{15}{67}\right)\) | \(e\left(\frac{323}{335}\right)\) | \(e\left(\frac{30}{67}\right)\) | \(e\left(\frac{107}{335}\right)\) | \(e\left(\frac{63}{335}\right)\) | \(e\left(\frac{91}{335}\right)\) | \(e\left(\frac{45}{67}\right)\) | \(e\left(\frac{311}{335}\right)\) | \(e\left(\frac{182}{335}\right)\) | \(e\left(\frac{286}{335}\right)\) |
| \(\chi_{2011}(14,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{67}\right)\) | \(e\left(\frac{7}{335}\right)\) | \(e\left(\frac{16}{67}\right)\) | \(e\left(\frac{133}{335}\right)\) | \(e\left(\frac{47}{335}\right)\) | \(e\left(\frac{254}{335}\right)\) | \(e\left(\frac{24}{67}\right)\) | \(e\left(\frac{14}{335}\right)\) | \(e\left(\frac{173}{335}\right)\) | \(e\left(\frac{224}{335}\right)\) |
| \(\chi_{2011}(31,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{46}{67}\right)\) | \(e\left(\frac{258}{335}\right)\) | \(e\left(\frac{25}{67}\right)\) | \(e\left(\frac{212}{335}\right)\) | \(e\left(\frac{153}{335}\right)\) | \(e\left(\frac{221}{335}\right)\) | \(e\left(\frac{4}{67}\right)\) | \(e\left(\frac{181}{335}\right)\) | \(e\left(\frac{107}{335}\right)\) | \(e\left(\frac{216}{335}\right)\) |
| \(\chi_{2011}(33,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{35}{67}\right)\) | \(e\left(\frac{173}{335}\right)\) | \(e\left(\frac{3}{67}\right)\) | \(e\left(\frac{272}{335}\right)\) | \(e\left(\frac{13}{335}\right)\) | \(e\left(\frac{56}{335}\right)\) | \(e\left(\frac{38}{67}\right)\) | \(e\left(\frac{11}{335}\right)\) | \(e\left(\frac{112}{335}\right)\) | \(e\left(\frac{176}{335}\right)\) |
| \(\chi_{2011}(36,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{65}{67}\right)\) | \(e\left(\frac{82}{335}\right)\) | \(e\left(\frac{63}{67}\right)\) | \(e\left(\frac{218}{335}\right)\) | \(e\left(\frac{72}{335}\right)\) | \(e\left(\frac{104}{335}\right)\) | \(e\left(\frac{61}{67}\right)\) | \(e\left(\frac{164}{335}\right)\) | \(e\left(\frac{208}{335}\right)\) | \(e\left(\frac{279}{335}\right)\) |
| \(\chi_{2011}(41,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{57}{67}\right)\) | \(e\left(\frac{276}{335}\right)\) | \(e\left(\frac{47}{67}\right)\) | \(e\left(\frac{219}{335}\right)\) | \(e\left(\frac{226}{335}\right)\) | \(e\left(\frac{252}{335}\right)\) | \(e\left(\frac{37}{67}\right)\) | \(e\left(\frac{217}{335}\right)\) | \(e\left(\frac{169}{335}\right)\) | \(e\left(\frac{122}{335}\right)\) |
| \(\chi_{2011}(43,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{62}{67}\right)\) | \(e\left(\frac{272}{335}\right)\) | \(e\left(\frac{57}{67}\right)\) | \(e\left(\frac{143}{335}\right)\) | \(e\left(\frac{247}{335}\right)\) | \(e\left(\frac{59}{335}\right)\) | \(e\left(\frac{52}{67}\right)\) | \(e\left(\frac{209}{335}\right)\) | \(e\left(\frac{118}{335}\right)\) | \(e\left(\frac{329}{335}\right)\) |
| \(\chi_{2011}(45,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{67}\right)\) | \(e\left(\frac{171}{335}\right)\) | \(e\left(\frac{8}{67}\right)\) | \(e\left(\frac{234}{335}\right)\) | \(e\left(\frac{191}{335}\right)\) | \(e\left(\frac{127}{335}\right)\) | \(e\left(\frac{12}{67}\right)\) | \(e\left(\frac{7}{335}\right)\) | \(e\left(\frac{254}{335}\right)\) | \(e\left(\frac{112}{335}\right)\) |
| \(\chi_{2011}(51,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{28}{67}\right)\) | \(e\left(\frac{58}{335}\right)\) | \(e\left(\frac{56}{67}\right)\) | \(e\left(\frac{97}{335}\right)\) | \(e\left(\frac{198}{335}\right)\) | \(e\left(\frac{286}{335}\right)\) | \(e\left(\frac{17}{67}\right)\) | \(e\left(\frac{116}{335}\right)\) | \(e\left(\frac{237}{335}\right)\) | \(e\left(\frac{181}{335}\right)\) |
| \(\chi_{2011}(57,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{59}{67}\right)\) | \(e\left(\frac{194}{335}\right)\) | \(e\left(\frac{51}{67}\right)\) | \(e\left(\frac{1}{335}\right)\) | \(e\left(\frac{154}{335}\right)\) | \(e\left(\frac{148}{335}\right)\) | \(e\left(\frac{43}{67}\right)\) | \(e\left(\frac{53}{335}\right)\) | \(e\left(\frac{296}{335}\right)\) | \(e\left(\frac{178}{335}\right)\) |
| \(\chi_{2011}(58,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{46}{67}\right)\) | \(e\left(\frac{57}{335}\right)\) | \(e\left(\frac{25}{67}\right)\) | \(e\left(\frac{78}{335}\right)\) | \(e\left(\frac{287}{335}\right)\) | \(e\left(\frac{154}{335}\right)\) | \(e\left(\frac{4}{67}\right)\) | \(e\left(\frac{114}{335}\right)\) | \(e\left(\frac{308}{335}\right)\) | \(e\left(\frac{149}{335}\right)\) |
| \(\chi_{2011}(77,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{44}{67}\right)\) | \(e\left(\frac{139}{335}\right)\) | \(e\left(\frac{21}{67}\right)\) | \(e\left(\frac{296}{335}\right)\) | \(e\left(\frac{24}{335}\right)\) | \(e\left(\frac{258}{335}\right)\) | \(e\left(\frac{65}{67}\right)\) | \(e\left(\frac{278}{335}\right)\) | \(e\left(\frac{181}{335}\right)\) | \(e\left(\frac{93}{335}\right)\) |
| \(\chi_{2011}(78,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{14}{67}\right)\) | \(e\left(\frac{29}{335}\right)\) | \(e\left(\frac{28}{67}\right)\) | \(e\left(\frac{216}{335}\right)\) | \(e\left(\frac{99}{335}\right)\) | \(e\left(\frac{143}{335}\right)\) | \(e\left(\frac{42}{67}\right)\) | \(e\left(\frac{58}{335}\right)\) | \(e\left(\frac{286}{335}\right)\) | \(e\left(\frac{258}{335}\right)\) |
| \(\chi_{2011}(80,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{18}{67}\right)\) | \(e\left(\frac{334}{335}\right)\) | \(e\left(\frac{36}{67}\right)\) | \(e\left(\frac{316}{335}\right)\) | \(e\left(\frac{89}{335}\right)\) | \(e\left(\frac{203}{335}\right)\) | \(e\left(\frac{54}{67}\right)\) | \(e\left(\frac{333}{335}\right)\) | \(e\left(\frac{71}{335}\right)\) | \(e\left(\frac{303}{335}\right)\) |
| \(\chi_{2011}(84,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{67}\right)\) | \(e\left(\frac{48}{335}\right)\) | \(e\left(\frac{14}{67}\right)\) | \(e\left(\frac{242}{335}\right)\) | \(e\left(\frac{83}{335}\right)\) | \(e\left(\frac{306}{335}\right)\) | \(e\left(\frac{21}{67}\right)\) | \(e\left(\frac{96}{335}\right)\) | \(e\left(\frac{277}{335}\right)\) | \(e\left(\frac{196}{335}\right)\) |
| \(\chi_{2011}(100,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{24}{67}\right)\) | \(e\left(\frac{88}{335}\right)\) | \(e\left(\frac{48}{67}\right)\) | \(e\left(\frac{332}{335}\right)\) | \(e\left(\frac{208}{335}\right)\) | \(e\left(\frac{226}{335}\right)\) | \(e\left(\frac{5}{67}\right)\) | \(e\left(\frac{176}{335}\right)\) | \(e\left(\frac{117}{335}\right)\) | \(e\left(\frac{136}{335}\right)\) |
| \(\chi_{2011}(101,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{47}{67}\right)\) | \(e\left(\frac{83}{335}\right)\) | \(e\left(\frac{27}{67}\right)\) | \(e\left(\frac{237}{335}\right)\) | \(e\left(\frac{318}{335}\right)\) | \(e\left(\frac{236}{335}\right)\) | \(e\left(\frac{7}{67}\right)\) | \(e\left(\frac{166}{335}\right)\) | \(e\left(\frac{137}{335}\right)\) | \(e\left(\frac{311}{335}\right)\) |
| \(\chi_{2011}(105,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{67}\right)\) | \(e\left(\frac{137}{335}\right)\) | \(e\left(\frac{26}{67}\right)\) | \(e\left(\frac{258}{335}\right)\) | \(e\left(\frac{202}{335}\right)\) | \(e\left(\frac{329}{335}\right)\) | \(e\left(\frac{39}{67}\right)\) | \(e\left(\frac{274}{335}\right)\) | \(e\left(\frac{323}{335}\right)\) | \(e\left(\frac{29}{335}\right)\) |
| \(\chi_{2011}(119,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{67}\right)\) | \(e\left(\frac{24}{335}\right)\) | \(e\left(\frac{7}{67}\right)\) | \(e\left(\frac{121}{335}\right)\) | \(e\left(\frac{209}{335}\right)\) | \(e\left(\frac{153}{335}\right)\) | \(e\left(\frac{44}{67}\right)\) | \(e\left(\frac{48}{335}\right)\) | \(e\left(\frac{306}{335}\right)\) | \(e\left(\frac{98}{335}\right)\) |
| \(\chi_{2011}(125,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{30}{67}\right)\) | \(e\left(\frac{177}{335}\right)\) | \(e\left(\frac{60}{67}\right)\) | \(e\left(\frac{13}{335}\right)\) | \(e\left(\frac{327}{335}\right)\) | \(e\left(\frac{249}{335}\right)\) | \(e\left(\frac{23}{67}\right)\) | \(e\left(\frac{19}{335}\right)\) | \(e\left(\frac{163}{335}\right)\) | \(e\left(\frac{304}{335}\right)\) |
| \(\chi_{2011}(127,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{62}{67}\right)\) | \(e\left(\frac{138}{335}\right)\) | \(e\left(\frac{57}{67}\right)\) | \(e\left(\frac{277}{335}\right)\) | \(e\left(\frac{113}{335}\right)\) | \(e\left(\frac{126}{335}\right)\) | \(e\left(\frac{52}{67}\right)\) | \(e\left(\frac{276}{335}\right)\) | \(e\left(\frac{252}{335}\right)\) | \(e\left(\frac{61}{335}\right)\) |
| \(\chi_{2011}(146,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{27}{67}\right)\) | \(e\left(\frac{166}{335}\right)\) | \(e\left(\frac{54}{67}\right)\) | \(e\left(\frac{139}{335}\right)\) | \(e\left(\frac{301}{335}\right)\) | \(e\left(\frac{137}{335}\right)\) | \(e\left(\frac{14}{67}\right)\) | \(e\left(\frac{332}{335}\right)\) | \(e\left(\frac{274}{335}\right)\) | \(e\left(\frac{287}{335}\right)\) |
| \(\chi_{2011}(151,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{57}{67}\right)\) | \(e\left(\frac{8}{335}\right)\) | \(e\left(\frac{47}{67}\right)\) | \(e\left(\frac{152}{335}\right)\) | \(e\left(\frac{293}{335}\right)\) | \(e\left(\frac{51}{335}\right)\) | \(e\left(\frac{37}{67}\right)\) | \(e\left(\frac{16}{335}\right)\) | \(e\left(\frac{102}{335}\right)\) | \(e\left(\frac{256}{335}\right)\) |
| \(\chi_{2011}(158,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{47}{67}\right)\) | \(e\left(\frac{284}{335}\right)\) | \(e\left(\frac{27}{67}\right)\) | \(e\left(\frac{36}{335}\right)\) | \(e\left(\frac{184}{335}\right)\) | \(e\left(\frac{303}{335}\right)\) | \(e\left(\frac{7}{67}\right)\) | \(e\left(\frac{233}{335}\right)\) | \(e\left(\frac{271}{335}\right)\) | \(e\left(\frac{43}{335}\right)\) |
| \(\chi_{2011}(169,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{30}{67}\right)\) | \(e\left(\frac{311}{335}\right)\) | \(e\left(\frac{60}{67}\right)\) | \(e\left(\frac{214}{335}\right)\) | \(e\left(\frac{126}{335}\right)\) | \(e\left(\frac{182}{335}\right)\) | \(e\left(\frac{23}{67}\right)\) | \(e\left(\frac{287}{335}\right)\) | \(e\left(\frac{29}{335}\right)\) | \(e\left(\frac{237}{335}\right)\) |
| \(\chi_{2011}(181,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{67}\right)\) | \(e\left(\frac{247}{335}\right)\) | \(e\left(\frac{19}{67}\right)\) | \(e\left(\frac{3}{335}\right)\) | \(e\left(\frac{127}{335}\right)\) | \(e\left(\frac{109}{335}\right)\) | \(e\left(\frac{62}{67}\right)\) | \(e\left(\frac{159}{335}\right)\) | \(e\left(\frac{218}{335}\right)\) | \(e\left(\frac{199}{335}\right)\) |
| \(\chi_{2011}(183,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{67}\right)\) | \(e\left(\frac{203}{335}\right)\) | \(e\left(\frac{62}{67}\right)\) | \(e\left(\frac{172}{335}\right)\) | \(e\left(\frac{23}{335}\right)\) | \(e\left(\frac{331}{335}\right)\) | \(e\left(\frac{26}{67}\right)\) | \(e\left(\frac{71}{335}\right)\) | \(e\left(\frac{327}{335}\right)\) | \(e\left(\frac{131}{335}\right)\) |
| \(\chi_{2011}(186,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{45}{67}\right)\) | \(e\left(\frac{299}{335}\right)\) | \(e\left(\frac{23}{67}\right)\) | \(e\left(\frac{321}{335}\right)\) | \(e\left(\frac{189}{335}\right)\) | \(e\left(\frac{273}{335}\right)\) | \(e\left(\frac{1}{67}\right)\) | \(e\left(\frac{263}{335}\right)\) | \(e\left(\frac{211}{335}\right)\) | \(e\left(\frac{188}{335}\right)\) |
| \(\chi_{2011}(191,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{35}{67}\right)\) | \(e\left(\frac{39}{335}\right)\) | \(e\left(\frac{3}{67}\right)\) | \(e\left(\frac{71}{335}\right)\) | \(e\left(\frac{214}{335}\right)\) | \(e\left(\frac{123}{335}\right)\) | \(e\left(\frac{38}{67}\right)\) | \(e\left(\frac{78}{335}\right)\) | \(e\left(\frac{246}{335}\right)\) | \(e\left(\frac{243}{335}\right)\) |
| \(\chi_{2011}(196,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{16}{67}\right)\) | \(e\left(\frac{14}{335}\right)\) | \(e\left(\frac{32}{67}\right)\) | \(e\left(\frac{266}{335}\right)\) | \(e\left(\frac{94}{335}\right)\) | \(e\left(\frac{173}{335}\right)\) | \(e\left(\frac{48}{67}\right)\) | \(e\left(\frac{28}{335}\right)\) | \(e\left(\frac{11}{335}\right)\) | \(e\left(\frac{113}{335}\right)\) |