from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(529, base_ring=CyclotomicField(506))
M = H._module
chi = DirichletCharacter(H, M([1]))
chi.galois_orbit()
[g,chi] = znchar(Mod(5,529))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(529\) | |
| Conductor: | \(529\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(506\) | 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: | $\Q(\zeta_{253})$ |
| Fixed field: | Number field defined by a degree 506 polynomial (not computed) |
First 31 of 220 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{529}(5,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{100}{253}\right)\) | \(e\left(\frac{8}{253}\right)\) | \(e\left(\frac{200}{253}\right)\) | \(e\left(\frac{1}{506}\right)\) | \(e\left(\frac{108}{253}\right)\) | \(e\left(\frac{129}{506}\right)\) | \(e\left(\frac{47}{253}\right)\) | \(e\left(\frac{16}{253}\right)\) | \(e\left(\frac{201}{506}\right)\) | \(e\left(\frac{449}{506}\right)\) |
| \(\chi_{529}(7,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{250}{253}\right)\) | \(e\left(\frac{20}{253}\right)\) | \(e\left(\frac{247}{253}\right)\) | \(e\left(\frac{129}{506}\right)\) | \(e\left(\frac{17}{253}\right)\) | \(e\left(\frac{449}{506}\right)\) | \(e\left(\frac{244}{253}\right)\) | \(e\left(\frac{40}{253}\right)\) | \(e\left(\frac{123}{506}\right)\) | \(e\left(\frac{237}{506}\right)\) |
| \(\chi_{529}(10,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{113}{253}\right)\) | \(e\left(\frac{90}{253}\right)\) | \(e\left(\frac{226}{253}\right)\) | \(e\left(\frac{201}{506}\right)\) | \(e\left(\frac{203}{253}\right)\) | \(e\left(\frac{123}{506}\right)\) | \(e\left(\frac{86}{253}\right)\) | \(e\left(\frac{180}{253}\right)\) | \(e\left(\frac{427}{506}\right)\) | \(e\left(\frac{181}{506}\right)\) |
| \(\chi_{529}(11,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{119}{253}\right)\) | \(e\left(\frac{50}{253}\right)\) | \(e\left(\frac{238}{253}\right)\) | \(e\left(\frac{449}{506}\right)\) | \(e\left(\frac{169}{253}\right)\) | \(e\left(\frac{237}{506}\right)\) | \(e\left(\frac{104}{253}\right)\) | \(e\left(\frac{100}{253}\right)\) | \(e\left(\frac{181}{506}\right)\) | \(e\left(\frac{213}{506}\right)\) |
| \(\chi_{529}(14,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{10}{253}\right)\) | \(e\left(\frac{102}{253}\right)\) | \(e\left(\frac{20}{253}\right)\) | \(e\left(\frac{329}{506}\right)\) | \(e\left(\frac{112}{253}\right)\) | \(e\left(\frac{443}{506}\right)\) | \(e\left(\frac{30}{253}\right)\) | \(e\left(\frac{204}{253}\right)\) | \(e\left(\frac{349}{506}\right)\) | \(e\left(\frac{475}{506}\right)\) |
| \(\chi_{529}(15,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{182}{253}\right)\) | \(e\left(\frac{136}{253}\right)\) | \(e\left(\frac{111}{253}\right)\) | \(e\left(\frac{17}{506}\right)\) | \(e\left(\frac{65}{253}\right)\) | \(e\left(\frac{169}{506}\right)\) | \(e\left(\frac{40}{253}\right)\) | \(e\left(\frac{19}{253}\right)\) | \(e\left(\frac{381}{506}\right)\) | \(e\left(\frac{43}{506}\right)\) |
| \(\chi_{529}(17,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{117}{253}\right)\) | \(e\left(\frac{232}{253}\right)\) | \(e\left(\frac{234}{253}\right)\) | \(e\left(\frac{29}{506}\right)\) | \(e\left(\frac{96}{253}\right)\) | \(e\left(\frac{199}{506}\right)\) | \(e\left(\frac{98}{253}\right)\) | \(e\left(\frac{211}{253}\right)\) | \(e\left(\frac{263}{506}\right)\) | \(e\left(\frac{371}{506}\right)\) |
| \(\chi_{529}(19,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{253}\right)\) | \(e\left(\frac{197}{253}\right)\) | \(e\left(\frac{118}{253}\right)\) | \(e\left(\frac{499}{506}\right)\) | \(e\left(\frac{3}{253}\right)\) | \(e\left(\frac{109}{506}\right)\) | \(e\left(\frac{177}{253}\right)\) | \(e\left(\frac{141}{253}\right)\) | \(e\left(\frac{111}{506}\right)\) | \(e\left(\frac{399}{506}\right)\) |
| \(\chi_{529}(20,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{126}{253}\right)\) | \(e\left(\frac{172}{253}\right)\) | \(e\left(\frac{252}{253}\right)\) | \(e\left(\frac{401}{506}\right)\) | \(e\left(\frac{45}{253}\right)\) | \(e\left(\frac{117}{506}\right)\) | \(e\left(\frac{125}{253}\right)\) | \(e\left(\frac{91}{253}\right)\) | \(e\left(\frac{147}{506}\right)\) | \(e\left(\frac{419}{506}\right)\) |
| \(\chi_{529}(21,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{79}{253}\right)\) | \(e\left(\frac{148}{253}\right)\) | \(e\left(\frac{158}{253}\right)\) | \(e\left(\frac{145}{506}\right)\) | \(e\left(\frac{227}{253}\right)\) | \(e\left(\frac{489}{506}\right)\) | \(e\left(\frac{237}{253}\right)\) | \(e\left(\frac{43}{253}\right)\) | \(e\left(\frac{303}{506}\right)\) | \(e\left(\frac{337}{506}\right)\) |
| \(\chi_{529}(30,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{195}{253}\right)\) | \(e\left(\frac{218}{253}\right)\) | \(e\left(\frac{137}{253}\right)\) | \(e\left(\frac{217}{506}\right)\) | \(e\left(\frac{160}{253}\right)\) | \(e\left(\frac{163}{506}\right)\) | \(e\left(\frac{79}{253}\right)\) | \(e\left(\frac{183}{253}\right)\) | \(e\left(\frac{101}{506}\right)\) | \(e\left(\frac{281}{506}\right)\) |
| \(\chi_{529}(33,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{201}{253}\right)\) | \(e\left(\frac{178}{253}\right)\) | \(e\left(\frac{149}{253}\right)\) | \(e\left(\frac{465}{506}\right)\) | \(e\left(\frac{126}{253}\right)\) | \(e\left(\frac{277}{506}\right)\) | \(e\left(\frac{97}{253}\right)\) | \(e\left(\frac{103}{253}\right)\) | \(e\left(\frac{361}{506}\right)\) | \(e\left(\frac{313}{506}\right)\) |
| \(\chi_{529}(34,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{130}{253}\right)\) | \(e\left(\frac{61}{253}\right)\) | \(e\left(\frac{7}{253}\right)\) | \(e\left(\frac{229}{506}\right)\) | \(e\left(\frac{191}{253}\right)\) | \(e\left(\frac{193}{506}\right)\) | \(e\left(\frac{137}{253}\right)\) | \(e\left(\frac{122}{253}\right)\) | \(e\left(\frac{489}{506}\right)\) | \(e\left(\frac{103}{506}\right)\) |
| \(\chi_{529}(37,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{109}{253}\right)\) | \(e\left(\frac{201}{253}\right)\) | \(e\left(\frac{218}{253}\right)\) | \(e\left(\frac{373}{506}\right)\) | \(e\left(\frac{57}{253}\right)\) | \(e\left(\frac{47}{506}\right)\) | \(e\left(\frac{74}{253}\right)\) | \(e\left(\frac{149}{253}\right)\) | \(e\left(\frac{85}{506}\right)\) | \(e\left(\frac{497}{506}\right)\) |
| \(\chi_{529}(38,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{72}{253}\right)\) | \(e\left(\frac{26}{253}\right)\) | \(e\left(\frac{144}{253}\right)\) | \(e\left(\frac{193}{506}\right)\) | \(e\left(\frac{98}{253}\right)\) | \(e\left(\frac{103}{506}\right)\) | \(e\left(\frac{216}{253}\right)\) | \(e\left(\frac{52}{253}\right)\) | \(e\left(\frac{337}{506}\right)\) | \(e\left(\frac{131}{506}\right)\) |
| \(\chi_{529}(40,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{139}{253}\right)\) | \(e\left(\frac{1}{253}\right)\) | \(e\left(\frac{25}{253}\right)\) | \(e\left(\frac{95}{506}\right)\) | \(e\left(\frac{140}{253}\right)\) | \(e\left(\frac{111}{506}\right)\) | \(e\left(\frac{164}{253}\right)\) | \(e\left(\frac{2}{253}\right)\) | \(e\left(\frac{373}{506}\right)\) | \(e\left(\frac{151}{506}\right)\) |
| \(\chi_{529}(43,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{170}{253}\right)\) | \(e\left(\frac{216}{253}\right)\) | \(e\left(\frac{87}{253}\right)\) | \(e\left(\frac{27}{506}\right)\) | \(e\left(\frac{133}{253}\right)\) | \(e\left(\frac{447}{506}\right)\) | \(e\left(\frac{4}{253}\right)\) | \(e\left(\frac{179}{253}\right)\) | \(e\left(\frac{367}{506}\right)\) | \(e\left(\frac{485}{506}\right)\) |
| \(\chi_{529}(44,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{145}{253}\right)\) | \(e\left(\frac{214}{253}\right)\) | \(e\left(\frac{37}{253}\right)\) | \(e\left(\frac{343}{506}\right)\) | \(e\left(\frac{106}{253}\right)\) | \(e\left(\frac{225}{506}\right)\) | \(e\left(\frac{182}{253}\right)\) | \(e\left(\frac{175}{253}\right)\) | \(e\left(\frac{127}{506}\right)\) | \(e\left(\frac{183}{506}\right)\) |
| \(\chi_{529}(51,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{199}{253}\right)\) | \(e\left(\frac{107}{253}\right)\) | \(e\left(\frac{145}{253}\right)\) | \(e\left(\frac{45}{506}\right)\) | \(e\left(\frac{53}{253}\right)\) | \(e\left(\frac{239}{506}\right)\) | \(e\left(\frac{91}{253}\right)\) | \(e\left(\frac{214}{253}\right)\) | \(e\left(\frac{443}{506}\right)\) | \(e\left(\frac{471}{506}\right)\) |
| \(\chi_{529}(53,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{140}{253}\right)\) | \(e\left(\frac{163}{253}\right)\) | \(e\left(\frac{27}{253}\right)\) | \(e\left(\frac{305}{506}\right)\) | \(e\left(\frac{50}{253}\right)\) | \(e\left(\frac{383}{506}\right)\) | \(e\left(\frac{167}{253}\right)\) | \(e\left(\frac{73}{253}\right)\) | \(e\left(\frac{79}{506}\right)\) | \(e\left(\frac{325}{506}\right)\) |
| \(\chi_{529}(56,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{36}{253}\right)\) | \(e\left(\frac{13}{253}\right)\) | \(e\left(\frac{72}{253}\right)\) | \(e\left(\frac{223}{506}\right)\) | \(e\left(\frac{49}{253}\right)\) | \(e\left(\frac{431}{506}\right)\) | \(e\left(\frac{108}{253}\right)\) | \(e\left(\frac{26}{253}\right)\) | \(e\left(\frac{295}{506}\right)\) | \(e\left(\frac{445}{506}\right)\) |
| \(\chi_{529}(57,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{141}{253}\right)\) | \(e\left(\frac{72}{253}\right)\) | \(e\left(\frac{29}{253}\right)\) | \(e\left(\frac{9}{506}\right)\) | \(e\left(\frac{213}{253}\right)\) | \(e\left(\frac{149}{506}\right)\) | \(e\left(\frac{170}{253}\right)\) | \(e\left(\frac{144}{253}\right)\) | \(e\left(\frac{291}{506}\right)\) | \(e\left(\frac{499}{506}\right)\) |
| \(\chi_{529}(60,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{208}{253}\right)\) | \(e\left(\frac{47}{253}\right)\) | \(e\left(\frac{163}{253}\right)\) | \(e\left(\frac{417}{506}\right)\) | \(e\left(\frac{2}{253}\right)\) | \(e\left(\frac{157}{506}\right)\) | \(e\left(\frac{118}{253}\right)\) | \(e\left(\frac{94}{253}\right)\) | \(e\left(\frac{327}{506}\right)\) | \(e\left(\frac{13}{506}\right)\) |
| \(\chi_{529}(61,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{215}{253}\right)\) | \(e\left(\frac{169}{253}\right)\) | \(e\left(\frac{177}{253}\right)\) | \(e\left(\frac{369}{506}\right)\) | \(e\left(\frac{131}{253}\right)\) | \(e\left(\frac{37}{506}\right)\) | \(e\left(\frac{139}{253}\right)\) | \(e\left(\frac{85}{253}\right)\) | \(e\left(\frac{293}{506}\right)\) | \(e\left(\frac{219}{506}\right)\) |
| \(\chi_{529}(65,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{125}{253}\right)\) | \(e\left(\frac{10}{253}\right)\) | \(e\left(\frac{250}{253}\right)\) | \(e\left(\frac{191}{506}\right)\) | \(e\left(\frac{135}{253}\right)\) | \(e\left(\frac{351}{506}\right)\) | \(e\left(\frac{122}{253}\right)\) | \(e\left(\frac{20}{253}\right)\) | \(e\left(\frac{441}{506}\right)\) | \(e\left(\frac{245}{506}\right)\) |
| \(\chi_{529}(66,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{214}{253}\right)\) | \(e\left(\frac{7}{253}\right)\) | \(e\left(\frac{175}{253}\right)\) | \(e\left(\frac{159}{506}\right)\) | \(e\left(\frac{221}{253}\right)\) | \(e\left(\frac{271}{506}\right)\) | \(e\left(\frac{136}{253}\right)\) | \(e\left(\frac{14}{253}\right)\) | \(e\left(\frac{81}{506}\right)\) | \(e\left(\frac{45}{506}\right)\) |
| \(\chi_{529}(67,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{211}{253}\right)\) | \(e\left(\frac{27}{253}\right)\) | \(e\left(\frac{169}{253}\right)\) | \(e\left(\frac{35}{506}\right)\) | \(e\left(\frac{238}{253}\right)\) | \(e\left(\frac{467}{506}\right)\) | \(e\left(\frac{127}{253}\right)\) | \(e\left(\frac{54}{253}\right)\) | \(e\left(\frac{457}{506}\right)\) | \(e\left(\frac{29}{506}\right)\) |
| \(\chi_{529}(74,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{122}{253}\right)\) | \(e\left(\frac{30}{253}\right)\) | \(e\left(\frac{244}{253}\right)\) | \(e\left(\frac{67}{506}\right)\) | \(e\left(\frac{152}{253}\right)\) | \(e\left(\frac{41}{506}\right)\) | \(e\left(\frac{113}{253}\right)\) | \(e\left(\frac{60}{253}\right)\) | \(e\left(\frac{311}{506}\right)\) | \(e\left(\frac{229}{506}\right)\) |
| \(\chi_{529}(76,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{85}{253}\right)\) | \(e\left(\frac{108}{253}\right)\) | \(e\left(\frac{170}{253}\right)\) | \(e\left(\frac{393}{506}\right)\) | \(e\left(\frac{193}{253}\right)\) | \(e\left(\frac{97}{506}\right)\) | \(e\left(\frac{2}{253}\right)\) | \(e\left(\frac{216}{253}\right)\) | \(e\left(\frac{57}{506}\right)\) | \(e\left(\frac{369}{506}\right)\) |
| \(\chi_{529}(79,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{124}{253}\right)\) | \(e\left(\frac{101}{253}\right)\) | \(e\left(\frac{248}{253}\right)\) | \(e\left(\frac{487}{506}\right)\) | \(e\left(\frac{225}{253}\right)\) | \(e\left(\frac{79}{506}\right)\) | \(e\left(\frac{119}{253}\right)\) | \(e\left(\frac{202}{253}\right)\) | \(e\left(\frac{229}{506}\right)\) | \(e\left(\frac{71}{506}\right)\) |
| \(\chi_{529}(80,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{152}{253}\right)\) | \(e\left(\frac{83}{253}\right)\) | \(e\left(\frac{51}{253}\right)\) | \(e\left(\frac{295}{506}\right)\) | \(e\left(\frac{235}{253}\right)\) | \(e\left(\frac{105}{506}\right)\) | \(e\left(\frac{203}{253}\right)\) | \(e\left(\frac{166}{253}\right)\) | \(e\left(\frac{93}{506}\right)\) | \(e\left(\frac{389}{506}\right)\) |