from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1109, base_ring=CyclotomicField(1108))
M = H._module
chi = DirichletCharacter(H, M([1]))
chi.galois_orbit()
[g,chi] = znchar(Mod(2,1109))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(1109\) | |
| Conductor: | \(1109\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(1108\) | 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: | $\Q(\zeta_{1108})$ |
| Fixed field: | Number field defined by a degree 1108 polynomial (not computed) |
First 31 of 552 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{1109}(2,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{1108}\right)\) | \(e\left(\frac{219}{1108}\right)\) | \(e\left(\frac{1}{554}\right)\) | \(e\left(\frac{297}{554}\right)\) | \(e\left(\frac{55}{277}\right)\) | \(e\left(\frac{311}{1108}\right)\) | \(e\left(\frac{3}{1108}\right)\) | \(e\left(\frac{219}{554}\right)\) | \(e\left(\frac{595}{1108}\right)\) | \(e\left(\frac{455}{554}\right)\) |
| \(\chi_{1109}(3,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{219}{1108}\right)\) | \(e\left(\frac{317}{1108}\right)\) | \(e\left(\frac{219}{554}\right)\) | \(e\left(\frac{225}{554}\right)\) | \(e\left(\frac{134}{277}\right)\) | \(e\left(\frac{521}{1108}\right)\) | \(e\left(\frac{657}{1108}\right)\) | \(e\left(\frac{317}{554}\right)\) | \(e\left(\frac{669}{1108}\right)\) | \(e\left(\frac{479}{554}\right)\) |
| \(\chi_{1109}(7,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{311}{1108}\right)\) | \(e\left(\frac{521}{1108}\right)\) | \(e\left(\frac{311}{554}\right)\) | \(e\left(\frac{403}{554}\right)\) | \(e\left(\frac{208}{277}\right)\) | \(e\left(\frac{325}{1108}\right)\) | \(e\left(\frac{933}{1108}\right)\) | \(e\left(\frac{521}{554}\right)\) | \(e\left(\frac{9}{1108}\right)\) | \(e\left(\frac{235}{554}\right)\) |
| \(\chi_{1109}(8,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{1108}\right)\) | \(e\left(\frac{657}{1108}\right)\) | \(e\left(\frac{3}{554}\right)\) | \(e\left(\frac{337}{554}\right)\) | \(e\left(\frac{165}{277}\right)\) | \(e\left(\frac{933}{1108}\right)\) | \(e\left(\frac{9}{1108}\right)\) | \(e\left(\frac{103}{554}\right)\) | \(e\left(\frac{677}{1108}\right)\) | \(e\left(\frac{257}{554}\right)\) |
| \(\chi_{1109}(10,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{595}{1108}\right)\) | \(e\left(\frac{669}{1108}\right)\) | \(e\left(\frac{41}{554}\right)\) | \(e\left(\frac{543}{554}\right)\) | \(e\left(\frac{39}{277}\right)\) | \(e\left(\frac{9}{1108}\right)\) | \(e\left(\frac{677}{1108}\right)\) | \(e\left(\frac{115}{554}\right)\) | \(e\left(\frac{573}{1108}\right)\) | \(e\left(\frac{373}{554}\right)\) |
| \(\chi_{1109}(12,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{221}{1108}\right)\) | \(e\left(\frac{755}{1108}\right)\) | \(e\left(\frac{221}{554}\right)\) | \(e\left(\frac{265}{554}\right)\) | \(e\left(\frac{244}{277}\right)\) | \(e\left(\frac{35}{1108}\right)\) | \(e\left(\frac{663}{1108}\right)\) | \(e\left(\frac{201}{554}\right)\) | \(e\left(\frac{751}{1108}\right)\) | \(e\left(\frac{281}{554}\right)\) |
| \(\chi_{1109}(15,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{813}{1108}\right)\) | \(e\left(\frac{767}{1108}\right)\) | \(e\left(\frac{259}{554}\right)\) | \(e\left(\frac{471}{554}\right)\) | \(e\left(\frac{118}{277}\right)\) | \(e\left(\frac{219}{1108}\right)\) | \(e\left(\frac{223}{1108}\right)\) | \(e\left(\frac{213}{554}\right)\) | \(e\left(\frac{647}{1108}\right)\) | \(e\left(\frac{397}{554}\right)\) |
| \(\chi_{1109}(18,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{439}{1108}\right)\) | \(e\left(\frac{853}{1108}\right)\) | \(e\left(\frac{439}{554}\right)\) | \(e\left(\frac{193}{554}\right)\) | \(e\left(\frac{46}{277}\right)\) | \(e\left(\frac{245}{1108}\right)\) | \(e\left(\frac{209}{1108}\right)\) | \(e\left(\frac{299}{554}\right)\) | \(e\left(\frac{825}{1108}\right)\) | \(e\left(\frac{305}{554}\right)\) |
| \(\chi_{1109}(22,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{911}{1108}\right)\) | \(e\left(\frac{69}{1108}\right)\) | \(e\left(\frac{357}{554}\right)\) | \(e\left(\frac{215}{554}\right)\) | \(e\left(\frac{245}{277}\right)\) | \(e\left(\frac{781}{1108}\right)\) | \(e\left(\frac{517}{1108}\right)\) | \(e\left(\frac{69}{554}\right)\) | \(e\left(\frac{233}{1108}\right)\) | \(e\left(\frac{113}{554}\right)\) |
| \(\chi_{1109}(23,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{925}{1108}\right)\) | \(e\left(\frac{919}{1108}\right)\) | \(e\left(\frac{371}{554}\right)\) | \(e\left(\frac{495}{554}\right)\) | \(e\left(\frac{184}{277}\right)\) | \(e\left(\frac{703}{1108}\right)\) | \(e\left(\frac{559}{1108}\right)\) | \(e\left(\frac{365}{554}\right)\) | \(e\left(\frac{807}{1108}\right)\) | \(e\left(\frac{389}{554}\right)\) |
| \(\chi_{1109}(26,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1101}{1108}\right)\) | \(e\left(\frac{683}{1108}\right)\) | \(e\left(\frac{547}{554}\right)\) | \(e\left(\frac{137}{554}\right)\) | \(e\left(\frac{169}{277}\right)\) | \(e\left(\frac{39}{1108}\right)\) | \(e\left(\frac{1087}{1108}\right)\) | \(e\left(\frac{129}{554}\right)\) | \(e\left(\frac{267}{1108}\right)\) | \(e\left(\frac{139}{554}\right)\) |
| \(\chi_{1109}(27,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{657}{1108}\right)\) | \(e\left(\frac{951}{1108}\right)\) | \(e\left(\frac{103}{554}\right)\) | \(e\left(\frac{121}{554}\right)\) | \(e\left(\frac{125}{277}\right)\) | \(e\left(\frac{455}{1108}\right)\) | \(e\left(\frac{863}{1108}\right)\) | \(e\left(\frac{397}{554}\right)\) | \(e\left(\frac{899}{1108}\right)\) | \(e\left(\frac{329}{554}\right)\) |
| \(\chi_{1109}(28,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{313}{1108}\right)\) | \(e\left(\frac{959}{1108}\right)\) | \(e\left(\frac{313}{554}\right)\) | \(e\left(\frac{443}{554}\right)\) | \(e\left(\frac{41}{277}\right)\) | \(e\left(\frac{947}{1108}\right)\) | \(e\left(\frac{939}{1108}\right)\) | \(e\left(\frac{405}{554}\right)\) | \(e\left(\frac{91}{1108}\right)\) | \(e\left(\frac{37}{554}\right)\) |
| \(\chi_{1109}(31,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{979}{1108}\right)\) | \(e\left(\frac{557}{1108}\right)\) | \(e\left(\frac{425}{554}\right)\) | \(e\left(\frac{467}{554}\right)\) | \(e\left(\frac{107}{277}\right)\) | \(e\left(\frac{877}{1108}\right)\) | \(e\left(\frac{721}{1108}\right)\) | \(e\left(\frac{3}{554}\right)\) | \(e\left(\frac{805}{1108}\right)\) | \(e\left(\frac{29}{554}\right)\) |
| \(\chi_{1109}(32,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{1108}\right)\) | \(e\left(\frac{1095}{1108}\right)\) | \(e\left(\frac{5}{554}\right)\) | \(e\left(\frac{377}{554}\right)\) | \(e\left(\frac{275}{277}\right)\) | \(e\left(\frac{447}{1108}\right)\) | \(e\left(\frac{15}{1108}\right)\) | \(e\left(\frac{541}{554}\right)\) | \(e\left(\frac{759}{1108}\right)\) | \(e\left(\frac{59}{554}\right)\) |
| \(\chi_{1109}(33,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{21}{1108}\right)\) | \(e\left(\frac{167}{1108}\right)\) | \(e\left(\frac{21}{554}\right)\) | \(e\left(\frac{143}{554}\right)\) | \(e\left(\frac{47}{277}\right)\) | \(e\left(\frac{991}{1108}\right)\) | \(e\left(\frac{63}{1108}\right)\) | \(e\left(\frac{167}{554}\right)\) | \(e\left(\frac{307}{1108}\right)\) | \(e\left(\frac{137}{554}\right)\) |
| \(\chi_{1109}(34,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1079}{1108}\right)\) | \(e\left(\frac{297}{1108}\right)\) | \(e\left(\frac{525}{554}\right)\) | \(e\left(\frac{251}{554}\right)\) | \(e\left(\frac{67}{277}\right)\) | \(e\left(\frac{953}{1108}\right)\) | \(e\left(\frac{1021}{1108}\right)\) | \(e\left(\frac{297}{554}\right)\) | \(e\left(\frac{473}{1108}\right)\) | \(e\left(\frac{101}{554}\right)\) |
| \(\chi_{1109}(35,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{905}{1108}\right)\) | \(e\left(\frac{971}{1108}\right)\) | \(e\left(\frac{351}{554}\right)\) | \(e\left(\frac{95}{554}\right)\) | \(e\left(\frac{192}{277}\right)\) | \(e\left(\frac{23}{1108}\right)\) | \(e\left(\frac{499}{1108}\right)\) | \(e\left(\frac{417}{554}\right)\) | \(e\left(\frac{1095}{1108}\right)\) | \(e\left(\frac{153}{554}\right)\) |
| \(\chi_{1109}(38,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{165}{1108}\right)\) | \(e\left(\frac{679}{1108}\right)\) | \(e\left(\frac{165}{554}\right)\) | \(e\left(\frac{253}{554}\right)\) | \(e\left(\frac{211}{277}\right)\) | \(e\left(\frac{347}{1108}\right)\) | \(e\left(\frac{495}{1108}\right)\) | \(e\left(\frac{125}{554}\right)\) | \(e\left(\frac{671}{1108}\right)\) | \(e\left(\frac{285}{554}\right)\) |
| \(\chi_{1109}(39,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{211}{1108}\right)\) | \(e\left(\frac{781}{1108}\right)\) | \(e\left(\frac{211}{554}\right)\) | \(e\left(\frac{65}{554}\right)\) | \(e\left(\frac{248}{277}\right)\) | \(e\left(\frac{249}{1108}\right)\) | \(e\left(\frac{633}{1108}\right)\) | \(e\left(\frac{227}{554}\right)\) | \(e\left(\frac{341}{1108}\right)\) | \(e\left(\frac{163}{554}\right)\) |
| \(\chi_{1109}(40,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{597}{1108}\right)\) | \(e\left(\frac{1107}{1108}\right)\) | \(e\left(\frac{43}{554}\right)\) | \(e\left(\frac{29}{554}\right)\) | \(e\left(\frac{149}{277}\right)\) | \(e\left(\frac{631}{1108}\right)\) | \(e\left(\frac{683}{1108}\right)\) | \(e\left(\frac{553}{554}\right)\) | \(e\left(\frac{655}{1108}\right)\) | \(e\left(\frac{175}{554}\right)\) |
| \(\chi_{1109}(42,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{531}{1108}\right)\) | \(e\left(\frac{1057}{1108}\right)\) | \(e\left(\frac{531}{554}\right)\) | \(e\left(\frac{371}{554}\right)\) | \(e\left(\frac{120}{277}\right)\) | \(e\left(\frac{49}{1108}\right)\) | \(e\left(\frac{485}{1108}\right)\) | \(e\left(\frac{503}{554}\right)\) | \(e\left(\frac{165}{1108}\right)\) | \(e\left(\frac{61}{554}\right)\) |
| \(\chi_{1109}(43,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{445}{1108}\right)\) | \(e\left(\frac{1059}{1108}\right)\) | \(e\left(\frac{445}{554}\right)\) | \(e\left(\frac{313}{554}\right)\) | \(e\left(\frac{99}{277}\right)\) | \(e\left(\frac{1003}{1108}\right)\) | \(e\left(\frac{227}{1108}\right)\) | \(e\left(\frac{505}{554}\right)\) | \(e\left(\frac{1071}{1108}\right)\) | \(e\left(\frac{265}{554}\right)\) |
| \(\chi_{1109}(48,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{223}{1108}\right)\) | \(e\left(\frac{85}{1108}\right)\) | \(e\left(\frac{223}{554}\right)\) | \(e\left(\frac{305}{554}\right)\) | \(e\left(\frac{77}{277}\right)\) | \(e\left(\frac{657}{1108}\right)\) | \(e\left(\frac{669}{1108}\right)\) | \(e\left(\frac{85}{554}\right)\) | \(e\left(\frac{833}{1108}\right)\) | \(e\left(\frac{83}{554}\right)\) |
| \(\chi_{1109}(50,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{81}{1108}\right)\) | \(e\left(\frac{11}{1108}\right)\) | \(e\left(\frac{81}{554}\right)\) | \(e\left(\frac{235}{554}\right)\) | \(e\left(\frac{23}{277}\right)\) | \(e\left(\frac{815}{1108}\right)\) | \(e\left(\frac{243}{1108}\right)\) | \(e\left(\frac{11}{554}\right)\) | \(e\left(\frac{551}{1108}\right)\) | \(e\left(\frac{291}{554}\right)\) |
| \(\chi_{1109}(51,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{189}{1108}\right)\) | \(e\left(\frac{395}{1108}\right)\) | \(e\left(\frac{189}{554}\right)\) | \(e\left(\frac{179}{554}\right)\) | \(e\left(\frac{146}{277}\right)\) | \(e\left(\frac{55}{1108}\right)\) | \(e\left(\frac{567}{1108}\right)\) | \(e\left(\frac{395}{554}\right)\) | \(e\left(\frac{547}{1108}\right)\) | \(e\left(\frac{125}{554}\right)\) |
| \(\chi_{1109}(57,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{383}{1108}\right)\) | \(e\left(\frac{777}{1108}\right)\) | \(e\left(\frac{383}{554}\right)\) | \(e\left(\frac{181}{554}\right)\) | \(e\left(\frac{13}{277}\right)\) | \(e\left(\frac{557}{1108}\right)\) | \(e\left(\frac{41}{1108}\right)\) | \(e\left(\frac{223}{554}\right)\) | \(e\left(\frac{745}{1108}\right)\) | \(e\left(\frac{309}{554}\right)\) |
| \(\chi_{1109}(58,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{701}{1108}\right)\) | \(e\left(\frac{615}{1108}\right)\) | \(e\left(\frac{147}{554}\right)\) | \(e\left(\frac{447}{554}\right)\) | \(e\left(\frac{52}{277}\right)\) | \(e\left(\frac{843}{1108}\right)\) | \(e\left(\frac{995}{1108}\right)\) | \(e\left(\frac{61}{554}\right)\) | \(e\left(\frac{487}{1108}\right)\) | \(e\left(\frac{405}{554}\right)\) |
| \(\chi_{1109}(59,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{57}{1108}\right)\) | \(e\left(\frac{295}{1108}\right)\) | \(e\left(\frac{57}{554}\right)\) | \(e\left(\frac{309}{554}\right)\) | \(e\left(\frac{88}{277}\right)\) | \(e\left(\frac{1107}{1108}\right)\) | \(e\left(\frac{171}{1108}\right)\) | \(e\left(\frac{295}{554}\right)\) | \(e\left(\frac{675}{1108}\right)\) | \(e\left(\frac{451}{554}\right)\) |
| \(\chi_{1109}(60,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{815}{1108}\right)\) | \(e\left(\frac{97}{1108}\right)\) | \(e\left(\frac{261}{554}\right)\) | \(e\left(\frac{511}{554}\right)\) | \(e\left(\frac{228}{277}\right)\) | \(e\left(\frac{841}{1108}\right)\) | \(e\left(\frac{229}{1108}\right)\) | \(e\left(\frac{97}{554}\right)\) | \(e\left(\frac{729}{1108}\right)\) | \(e\left(\frac{199}{554}\right)\) |
| \(\chi_{1109}(61,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1025}{1108}\right)\) | \(e\left(\frac{659}{1108}\right)\) | \(e\left(\frac{471}{554}\right)\) | \(e\left(\frac{279}{554}\right)\) | \(e\left(\frac{144}{277}\right)\) | \(e\left(\frac{779}{1108}\right)\) | \(e\left(\frac{859}{1108}\right)\) | \(e\left(\frac{105}{554}\right)\) | \(e\left(\frac{475}{1108}\right)\) | \(e\left(\frac{461}{554}\right)\) |