from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1723, base_ring=CyclotomicField(574))
M = H._module
chi = DirichletCharacter(H, M([69]))
chi.galois_orbit()
[g,chi] = znchar(Mod(2,1723))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(1723\) | |
| Conductor: | \(1723\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(574\) | 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: | $\Q(\zeta_{287})$ |
| Fixed field: | Number field defined by a degree 574 polynomial (not computed) |
First 31 of 240 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{1723}(2,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{507}{574}\right)\) | \(e\left(\frac{69}{574}\right)\) | \(e\left(\frac{220}{287}\right)\) | \(e\left(\frac{379}{574}\right)\) | \(e\left(\frac{1}{287}\right)\) | \(e\left(\frac{285}{574}\right)\) | \(e\left(\frac{373}{574}\right)\) | \(e\left(\frac{69}{287}\right)\) | \(e\left(\frac{156}{287}\right)\) | \(e\left(\frac{5}{7}\right)\) |
| \(\chi_{1723}(5,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{379}{574}\right)\) | \(e\left(\frac{415}{574}\right)\) | \(e\left(\frac{92}{287}\right)\) | \(e\left(\frac{75}{574}\right)\) | \(e\left(\frac{110}{287}\right)\) | \(e\left(\frac{67}{574}\right)\) | \(e\left(\frac{563}{574}\right)\) | \(e\left(\frac{128}{287}\right)\) | \(e\left(\frac{227}{287}\right)\) | \(e\left(\frac{4}{7}\right)\) |
| \(\chi_{1723}(7,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{285}{574}\right)\) | \(e\left(\frac{409}{574}\right)\) | \(e\left(\frac{285}{287}\right)\) | \(e\left(\frac{67}{574}\right)\) | \(e\left(\frac{60}{287}\right)\) | \(e\left(\frac{167}{574}\right)\) | \(e\left(\frac{281}{574}\right)\) | \(e\left(\frac{122}{287}\right)\) | \(e\left(\frac{176}{287}\right)\) | \(e\left(\frac{6}{7}\right)\) |
| \(\chi_{1723}(8,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{373}{574}\right)\) | \(e\left(\frac{207}{574}\right)\) | \(e\left(\frac{86}{287}\right)\) | \(e\left(\frac{563}{574}\right)\) | \(e\left(\frac{3}{287}\right)\) | \(e\left(\frac{281}{574}\right)\) | \(e\left(\frac{545}{574}\right)\) | \(e\left(\frac{207}{287}\right)\) | \(e\left(\frac{181}{287}\right)\) | \(e\left(\frac{1}{7}\right)\) |
| \(\chi_{1723}(13,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{227}{574}\right)\) | \(e\left(\frac{503}{574}\right)\) | \(e\left(\frac{227}{287}\right)\) | \(e\left(\frac{1}{574}\right)\) | \(e\left(\frac{78}{287}\right)\) | \(e\left(\frac{131}{574}\right)\) | \(e\left(\frac{107}{574}\right)\) | \(e\left(\frac{216}{287}\right)\) | \(e\left(\frac{114}{287}\right)\) | \(e\left(\frac{5}{7}\right)\) |
| \(\chi_{1723}(27,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{207}{574}\right)\) | \(e\left(\frac{1}{574}\right)\) | \(e\left(\frac{207}{287}\right)\) | \(e\left(\frac{97}{574}\right)\) | \(e\left(\frac{104}{287}\right)\) | \(e\left(\frac{79}{574}\right)\) | \(e\left(\frac{47}{574}\right)\) | \(e\left(\frac{1}{287}\right)\) | \(e\left(\frac{152}{287}\right)\) | \(e\left(\frac{2}{7}\right)\) |
| \(\chi_{1723}(28,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{151}{574}\right)\) | \(e\left(\frac{547}{574}\right)\) | \(e\left(\frac{151}{287}\right)\) | \(e\left(\frac{251}{574}\right)\) | \(e\left(\frac{62}{287}\right)\) | \(e\left(\frac{163}{574}\right)\) | \(e\left(\frac{453}{574}\right)\) | \(e\left(\frac{260}{287}\right)\) | \(e\left(\frac{201}{287}\right)\) | \(e\left(\frac{2}{7}\right)\) |
| \(\chi_{1723}(31,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{143}{574}\right)\) | \(e\left(\frac{461}{574}\right)\) | \(e\left(\frac{143}{287}\right)\) | \(e\left(\frac{519}{574}\right)\) | \(e\left(\frac{15}{287}\right)\) | \(e\left(\frac{257}{574}\right)\) | \(e\left(\frac{429}{574}\right)\) | \(e\left(\frac{174}{287}\right)\) | \(e\left(\frac{44}{287}\right)\) | \(e\left(\frac{5}{7}\right)\) |
| \(\chi_{1723}(32,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{239}{574}\right)\) | \(e\left(\frac{345}{574}\right)\) | \(e\left(\frac{239}{287}\right)\) | \(e\left(\frac{173}{574}\right)\) | \(e\left(\frac{5}{287}\right)\) | \(e\left(\frac{277}{574}\right)\) | \(e\left(\frac{143}{574}\right)\) | \(e\left(\frac{58}{287}\right)\) | \(e\left(\frac{206}{287}\right)\) | \(e\left(\frac{4}{7}\right)\) |
| \(\chi_{1723}(33,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{479}{574}\right)\) | \(e\left(\frac{55}{574}\right)\) | \(e\left(\frac{192}{287}\right)\) | \(e\left(\frac{169}{574}\right)\) | \(e\left(\frac{267}{287}\right)\) | \(e\left(\frac{327}{574}\right)\) | \(e\left(\frac{289}{574}\right)\) | \(e\left(\frac{55}{287}\right)\) | \(e\left(\frac{37}{287}\right)\) | \(e\left(\frac{5}{7}\right)\) |
| \(\chi_{1723}(50,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{117}{574}\right)\) | \(e\left(\frac{325}{574}\right)\) | \(e\left(\frac{117}{287}\right)\) | \(e\left(\frac{529}{574}\right)\) | \(e\left(\frac{221}{287}\right)\) | \(e\left(\frac{419}{574}\right)\) | \(e\left(\frac{351}{574}\right)\) | \(e\left(\frac{38}{287}\right)\) | \(e\left(\frac{36}{287}\right)\) | \(e\left(\frac{6}{7}\right)\) |
| \(\chi_{1723}(52,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{93}{574}\right)\) | \(e\left(\frac{67}{574}\right)\) | \(e\left(\frac{93}{287}\right)\) | \(e\left(\frac{185}{574}\right)\) | \(e\left(\frac{80}{287}\right)\) | \(e\left(\frac{127}{574}\right)\) | \(e\left(\frac{279}{574}\right)\) | \(e\left(\frac{67}{287}\right)\) | \(e\left(\frac{139}{287}\right)\) | \(e\left(\frac{1}{7}\right)\) |
| \(\chi_{1723}(53,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{255}{574}\right)\) | \(e\left(\frac{517}{574}\right)\) | \(e\left(\frac{255}{287}\right)\) | \(e\left(\frac{211}{574}\right)\) | \(e\left(\frac{99}{287}\right)\) | \(e\left(\frac{89}{574}\right)\) | \(e\left(\frac{191}{574}\right)\) | \(e\left(\frac{230}{287}\right)\) | \(e\left(\frac{233}{287}\right)\) | \(e\left(\frac{5}{7}\right)\) |
| \(\chi_{1723}(59,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{45}{574}\right)\) | \(e\left(\frac{125}{574}\right)\) | \(e\left(\frac{45}{287}\right)\) | \(e\left(\frac{71}{574}\right)\) | \(e\left(\frac{85}{287}\right)\) | \(e\left(\frac{117}{574}\right)\) | \(e\left(\frac{135}{574}\right)\) | \(e\left(\frac{125}{287}\right)\) | \(e\left(\frac{58}{287}\right)\) | \(e\left(\frac{5}{7}\right)\) |
| \(\chi_{1723}(69,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{323}{574}\right)\) | \(e\left(\frac{387}{574}\right)\) | \(e\left(\frac{36}{287}\right)\) | \(e\left(\frac{229}{574}\right)\) | \(e\left(\frac{68}{287}\right)\) | \(e\left(\frac{151}{574}\right)\) | \(e\left(\frac{395}{574}\right)\) | \(e\left(\frac{100}{287}\right)\) | \(e\left(\frac{276}{287}\right)\) | \(e\left(\frac{4}{7}\right)\) |
| \(\chi_{1723}(70,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{574}\right)\) | \(e\left(\frac{319}{574}\right)\) | \(e\left(\frac{23}{287}\right)\) | \(e\left(\frac{521}{574}\right)\) | \(e\left(\frac{171}{287}\right)\) | \(e\left(\frac{519}{574}\right)\) | \(e\left(\frac{69}{574}\right)\) | \(e\left(\frac{32}{287}\right)\) | \(e\left(\frac{272}{287}\right)\) | \(e\left(\frac{1}{7}\right)\) |
| \(\chi_{1723}(73,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{309}{574}\right)\) | \(e\left(\frac{93}{574}\right)\) | \(e\left(\frac{22}{287}\right)\) | \(e\left(\frac{411}{574}\right)\) | \(e\left(\frac{201}{287}\right)\) | \(e\left(\frac{459}{574}\right)\) | \(e\left(\frac{353}{574}\right)\) | \(e\left(\frac{93}{287}\right)\) | \(e\left(\frac{73}{287}\right)\) | \(e\left(\frac{4}{7}\right)\) |
| \(\chi_{1723}(80,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{111}{574}\right)\) | \(e\left(\frac{117}{574}\right)\) | \(e\left(\frac{111}{287}\right)\) | \(e\left(\frac{443}{574}\right)\) | \(e\left(\frac{114}{287}\right)\) | \(e\left(\frac{59}{574}\right)\) | \(e\left(\frac{333}{574}\right)\) | \(e\left(\frac{117}{287}\right)\) | \(e\left(\frac{277}{287}\right)\) | \(e\left(\frac{3}{7}\right)\) |
| \(\chi_{1723}(97,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{183}{574}\right)\) | \(e\left(\frac{317}{574}\right)\) | \(e\left(\frac{183}{287}\right)\) | \(e\left(\frac{327}{574}\right)\) | \(e\left(\frac{250}{287}\right)\) | \(e\left(\frac{361}{574}\right)\) | \(e\left(\frac{549}{574}\right)\) | \(e\left(\frac{30}{287}\right)\) | \(e\left(\frac{255}{287}\right)\) | \(e\left(\frac{4}{7}\right)\) |
| \(\chi_{1723}(98,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{503}{574}\right)\) | \(e\left(\frac{313}{574}\right)\) | \(e\left(\frac{216}{287}\right)\) | \(e\left(\frac{513}{574}\right)\) | \(e\left(\frac{121}{287}\right)\) | \(e\left(\frac{45}{574}\right)\) | \(e\left(\frac{361}{574}\right)\) | \(e\left(\frac{26}{287}\right)\) | \(e\left(\frac{221}{287}\right)\) | \(e\left(\frac{3}{7}\right)\) |
| \(\chi_{1723}(102,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{493}{574}\right)\) | \(e\left(\frac{349}{574}\right)\) | \(e\left(\frac{206}{287}\right)\) | \(e\left(\frac{561}{574}\right)\) | \(e\left(\frac{134}{287}\right)\) | \(e\left(\frac{19}{574}\right)\) | \(e\left(\frac{331}{574}\right)\) | \(e\left(\frac{62}{287}\right)\) | \(e\left(\frac{240}{287}\right)\) | \(e\left(\frac{5}{7}\right)\) |
| \(\chi_{1723}(108,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{73}{574}\right)\) | \(e\left(\frac{139}{574}\right)\) | \(e\left(\frac{73}{287}\right)\) | \(e\left(\frac{281}{574}\right)\) | \(e\left(\frac{106}{287}\right)\) | \(e\left(\frac{75}{574}\right)\) | \(e\left(\frac{219}{574}\right)\) | \(e\left(\frac{139}{287}\right)\) | \(e\left(\frac{177}{287}\right)\) | \(e\left(\frac{5}{7}\right)\) |
| \(\chi_{1723}(111,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{523}{574}\right)\) | \(e\left(\frac{241}{574}\right)\) | \(e\left(\frac{236}{287}\right)\) | \(e\left(\frac{417}{574}\right)\) | \(e\left(\frac{95}{287}\right)\) | \(e\left(\frac{97}{574}\right)\) | \(e\left(\frac{421}{574}\right)\) | \(e\left(\frac{241}{287}\right)\) | \(e\left(\frac{183}{287}\right)\) | \(e\left(\frac{6}{7}\right)\) |
| \(\chi_{1723}(112,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{574}\right)\) | \(e\left(\frac{111}{574}\right)\) | \(e\left(\frac{17}{287}\right)\) | \(e\left(\frac{435}{574}\right)\) | \(e\left(\frac{64}{287}\right)\) | \(e\left(\frac{159}{574}\right)\) | \(e\left(\frac{51}{574}\right)\) | \(e\left(\frac{111}{287}\right)\) | \(e\left(\frac{226}{287}\right)\) | \(e\left(\frac{5}{7}\right)\) |
| \(\chi_{1723}(123,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{69}{574}\right)\) | \(e\left(\frac{383}{574}\right)\) | \(e\left(\frac{69}{287}\right)\) | \(e\left(\frac{415}{574}\right)\) | \(e\left(\frac{226}{287}\right)\) | \(e\left(\frac{409}{574}\right)\) | \(e\left(\frac{207}{574}\right)\) | \(e\left(\frac{96}{287}\right)\) | \(e\left(\frac{242}{287}\right)\) | \(e\left(\frac{3}{7}\right)\) |
| \(\chi_{1723}(124,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{9}{574}\right)\) | \(e\left(\frac{25}{574}\right)\) | \(e\left(\frac{9}{287}\right)\) | \(e\left(\frac{129}{574}\right)\) | \(e\left(\frac{17}{287}\right)\) | \(e\left(\frac{253}{574}\right)\) | \(e\left(\frac{27}{574}\right)\) | \(e\left(\frac{25}{287}\right)\) | \(e\left(\frac{69}{287}\right)\) | \(e\left(\frac{1}{7}\right)\) |
| \(\chi_{1723}(125,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{563}{574}\right)\) | \(e\left(\frac{97}{574}\right)\) | \(e\left(\frac{276}{287}\right)\) | \(e\left(\frac{225}{574}\right)\) | \(e\left(\frac{43}{287}\right)\) | \(e\left(\frac{201}{574}\right)\) | \(e\left(\frac{541}{574}\right)\) | \(e\left(\frac{97}{287}\right)\) | \(e\left(\frac{107}{287}\right)\) | \(e\left(\frac{5}{7}\right)\) |
| \(\chi_{1723}(131,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{305}{574}\right)\) | \(e\left(\frac{337}{574}\right)\) | \(e\left(\frac{18}{287}\right)\) | \(e\left(\frac{545}{574}\right)\) | \(e\left(\frac{34}{287}\right)\) | \(e\left(\frac{219}{574}\right)\) | \(e\left(\frac{341}{574}\right)\) | \(e\left(\frac{50}{287}\right)\) | \(e\left(\frac{138}{287}\right)\) | \(e\left(\frac{2}{7}\right)\) |
| \(\chi_{1723}(132,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{345}{574}\right)\) | \(e\left(\frac{193}{574}\right)\) | \(e\left(\frac{58}{287}\right)\) | \(e\left(\frac{353}{574}\right)\) | \(e\left(\frac{269}{287}\right)\) | \(e\left(\frac{323}{574}\right)\) | \(e\left(\frac{461}{574}\right)\) | \(e\left(\frac{193}{287}\right)\) | \(e\left(\frac{62}{287}\right)\) | \(e\left(\frac{1}{7}\right)\) |
| \(\chi_{1723}(173,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{547}{574}\right)\) | \(e\left(\frac{499}{574}\right)\) | \(e\left(\frac{260}{287}\right)\) | \(e\left(\frac{187}{574}\right)\) | \(e\left(\frac{236}{287}\right)\) | \(e\left(\frac{389}{574}\right)\) | \(e\left(\frac{493}{574}\right)\) | \(e\left(\frac{212}{287}\right)\) | \(e\left(\frac{80}{287}\right)\) | \(e\left(\frac{4}{7}\right)\) |
| \(\chi_{1723}(182,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{445}{574}\right)\) | \(e\left(\frac{407}{574}\right)\) | \(e\left(\frac{158}{287}\right)\) | \(e\left(\frac{447}{574}\right)\) | \(e\left(\frac{139}{287}\right)\) | \(e\left(\frac{9}{574}\right)\) | \(e\left(\frac{187}{574}\right)\) | \(e\left(\frac{120}{287}\right)\) | \(e\left(\frac{159}{287}\right)\) | \(e\left(\frac{2}{7}\right)\) |