from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2153, base_ring=CyclotomicField(538))
M = H._module
chi = DirichletCharacter(H, M([488]))
chi.galois_orbit()
[g,chi] = znchar(Mod(7,2153))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(2153\) | |
| Conductor: | \(2153\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(269\) | 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: | $\Q(\zeta_{269})$ |
| Fixed field: | Number field defined by a degree 269 polynomial (not computed) |
First 31 of 268 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{2153}(7,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{132}{269}\right)\) | \(e\left(\frac{244}{269}\right)\) | \(e\left(\frac{264}{269}\right)\) | \(e\left(\frac{243}{269}\right)\) | \(e\left(\frac{107}{269}\right)\) | \(e\left(\frac{158}{269}\right)\) | \(e\left(\frac{127}{269}\right)\) | \(e\left(\frac{219}{269}\right)\) | \(e\left(\frac{106}{269}\right)\) | \(e\left(\frac{202}{269}\right)\) |
| \(\chi_{2153}(16,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{230}{269}\right)\) | \(e\left(\frac{148}{269}\right)\) | \(e\left(\frac{191}{269}\right)\) | \(e\left(\frac{240}{269}\right)\) | \(e\left(\frac{109}{269}\right)\) | \(e\left(\frac{259}{269}\right)\) | \(e\left(\frac{152}{269}\right)\) | \(e\left(\frac{27}{269}\right)\) | \(e\left(\frac{201}{269}\right)\) | \(e\left(\frac{246}{269}\right)\) |
| \(\chi_{2153}(18,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{266}{269}\right)\) | \(e\left(\frac{239}{269}\right)\) | \(e\left(\frac{263}{269}\right)\) | \(e\left(\frac{184}{269}\right)\) | \(e\left(\frac{236}{269}\right)\) | \(e\left(\frac{82}{269}\right)\) | \(e\left(\frac{260}{269}\right)\) | \(e\left(\frac{209}{269}\right)\) | \(e\left(\frac{181}{269}\right)\) | \(e\left(\frac{81}{269}\right)\) |
| \(\chi_{2153}(29,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{231}{269}\right)\) | \(e\left(\frac{158}{269}\right)\) | \(e\left(\frac{193}{269}\right)\) | \(e\left(\frac{89}{269}\right)\) | \(e\left(\frac{120}{269}\right)\) | \(e\left(\frac{142}{269}\right)\) | \(e\left(\frac{155}{269}\right)\) | \(e\left(\frac{47}{269}\right)\) | \(e\left(\frac{51}{269}\right)\) | \(e\left(\frac{219}{269}\right)\) |
| \(\chi_{2153}(49,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{264}{269}\right)\) | \(e\left(\frac{219}{269}\right)\) | \(e\left(\frac{259}{269}\right)\) | \(e\left(\frac{217}{269}\right)\) | \(e\left(\frac{214}{269}\right)\) | \(e\left(\frac{47}{269}\right)\) | \(e\left(\frac{254}{269}\right)\) | \(e\left(\frac{169}{269}\right)\) | \(e\left(\frac{212}{269}\right)\) | \(e\left(\frac{135}{269}\right)\) |
| \(\chi_{2153}(50,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{269}\right)\) | \(e\left(\frac{161}{269}\right)\) | \(e\left(\frac{86}{269}\right)\) | \(e\left(\frac{232}{269}\right)\) | \(e\left(\frac{204}{269}\right)\) | \(e\left(\frac{80}{269}\right)\) | \(e\left(\frac{129}{269}\right)\) | \(e\left(\frac{53}{269}\right)\) | \(e\left(\frac{6}{269}\right)\) | \(e\left(\frac{184}{269}\right)\) |
| \(\chi_{2153}(65,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{74}{269}\right)\) | \(e\left(\frac{202}{269}\right)\) | \(e\left(\frac{148}{269}\right)\) | \(e\left(\frac{124}{269}\right)\) | \(e\left(\frac{7}{269}\right)\) | \(e\left(\frac{219}{269}\right)\) | \(e\left(\frac{222}{269}\right)\) | \(e\left(\frac{135}{269}\right)\) | \(e\left(\frac{198}{269}\right)\) | \(e\left(\frac{154}{269}\right)\) |
| \(\chi_{2153}(67,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{190}{269}\right)\) | \(e\left(\frac{17}{269}\right)\) | \(e\left(\frac{111}{269}\right)\) | \(e\left(\frac{93}{269}\right)\) | \(e\left(\frac{207}{269}\right)\) | \(e\left(\frac{97}{269}\right)\) | \(e\left(\frac{32}{269}\right)\) | \(e\left(\frac{34}{269}\right)\) | \(e\left(\frac{14}{269}\right)\) | \(e\left(\frac{250}{269}\right)\) |
| \(\chi_{2153}(102,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{27}{269}\right)\) | \(e\left(\frac{1}{269}\right)\) | \(e\left(\frac{54}{269}\right)\) | \(e\left(\frac{227}{269}\right)\) | \(e\left(\frac{28}{269}\right)\) | \(e\left(\frac{69}{269}\right)\) | \(e\left(\frac{81}{269}\right)\) | \(e\left(\frac{2}{269}\right)\) | \(e\left(\frac{254}{269}\right)\) | \(e\left(\frac{78}{269}\right)\) |
| \(\chi_{2153}(110,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{179}{269}\right)\) | \(e\left(\frac{176}{269}\right)\) | \(e\left(\frac{89}{269}\right)\) | \(e\left(\frac{140}{269}\right)\) | \(e\left(\frac{86}{269}\right)\) | \(e\left(\frac{39}{269}\right)\) | \(e\left(\frac{268}{269}\right)\) | \(e\left(\frac{83}{269}\right)\) | \(e\left(\frac{50}{269}\right)\) | \(e\left(\frac{9}{269}\right)\) |
| \(\chi_{2153}(112,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{93}{269}\right)\) | \(e\left(\frac{123}{269}\right)\) | \(e\left(\frac{186}{269}\right)\) | \(e\left(\frac{214}{269}\right)\) | \(e\left(\frac{216}{269}\right)\) | \(e\left(\frac{148}{269}\right)\) | \(e\left(\frac{10}{269}\right)\) | \(e\left(\frac{246}{269}\right)\) | \(e\left(\frac{38}{269}\right)\) | \(e\left(\frac{179}{269}\right)\) |
| \(\chi_{2153}(115,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{126}{269}\right)\) | \(e\left(\frac{184}{269}\right)\) | \(e\left(\frac{252}{269}\right)\) | \(e\left(\frac{73}{269}\right)\) | \(e\left(\frac{41}{269}\right)\) | \(e\left(\frac{53}{269}\right)\) | \(e\left(\frac{109}{269}\right)\) | \(e\left(\frac{99}{269}\right)\) | \(e\left(\frac{199}{269}\right)\) | \(e\left(\frac{95}{269}\right)\) |
| \(\chi_{2153}(120,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{135}{269}\right)\) | \(e\left(\frac{5}{269}\right)\) | \(e\left(\frac{1}{269}\right)\) | \(e\left(\frac{59}{269}\right)\) | \(e\left(\frac{140}{269}\right)\) | \(e\left(\frac{76}{269}\right)\) | \(e\left(\frac{136}{269}\right)\) | \(e\left(\frac{10}{269}\right)\) | \(e\left(\frac{194}{269}\right)\) | \(e\left(\frac{121}{269}\right)\) |
| \(\chi_{2153}(126,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{129}{269}\right)\) | \(e\left(\frac{214}{269}\right)\) | \(e\left(\frac{258}{269}\right)\) | \(e\left(\frac{158}{269}\right)\) | \(e\left(\frac{74}{269}\right)\) | \(e\left(\frac{240}{269}\right)\) | \(e\left(\frac{118}{269}\right)\) | \(e\left(\frac{159}{269}\right)\) | \(e\left(\frac{18}{269}\right)\) | \(e\left(\frac{14}{269}\right)\) |
| \(\chi_{2153}(135,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{171}{269}\right)\) | \(e\left(\frac{96}{269}\right)\) | \(e\left(\frac{73}{269}\right)\) | \(e\left(\frac{3}{269}\right)\) | \(e\left(\frac{267}{269}\right)\) | \(e\left(\frac{168}{269}\right)\) | \(e\left(\frac{244}{269}\right)\) | \(e\left(\frac{192}{269}\right)\) | \(e\left(\frac{174}{269}\right)\) | \(e\left(\frac{225}{269}\right)\) |
| \(\chi_{2153}(143,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{210}{269}\right)\) | \(e\left(\frac{217}{269}\right)\) | \(e\left(\frac{151}{269}\right)\) | \(e\left(\frac{32}{269}\right)\) | \(e\left(\frac{158}{269}\right)\) | \(e\left(\frac{178}{269}\right)\) | \(e\left(\frac{92}{269}\right)\) | \(e\left(\frac{165}{269}\right)\) | \(e\left(\frac{242}{269}\right)\) | \(e\left(\frac{248}{269}\right)\) |
| \(\chi_{2153}(146,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{150}{269}\right)\) | \(e\left(\frac{155}{269}\right)\) | \(e\left(\frac{31}{269}\right)\) | \(e\left(\frac{215}{269}\right)\) | \(e\left(\frac{36}{269}\right)\) | \(e\left(\frac{204}{269}\right)\) | \(e\left(\frac{181}{269}\right)\) | \(e\left(\frac{41}{269}\right)\) | \(e\left(\frac{96}{269}\right)\) | \(e\left(\frac{254}{269}\right)\) |
| \(\chi_{2153}(152,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{65}{269}\right)\) | \(e\left(\frac{112}{269}\right)\) | \(e\left(\frac{130}{269}\right)\) | \(e\left(\frac{138}{269}\right)\) | \(e\left(\frac{177}{269}\right)\) | \(e\left(\frac{196}{269}\right)\) | \(e\left(\frac{195}{269}\right)\) | \(e\left(\frac{224}{269}\right)\) | \(e\left(\frac{203}{269}\right)\) | \(e\left(\frac{128}{269}\right)\) |
| \(\chi_{2153}(156,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{166}{269}\right)\) | \(e\left(\frac{46}{269}\right)\) | \(e\left(\frac{63}{269}\right)\) | \(e\left(\frac{220}{269}\right)\) | \(e\left(\frac{212}{269}\right)\) | \(e\left(\frac{215}{269}\right)\) | \(e\left(\frac{229}{269}\right)\) | \(e\left(\frac{92}{269}\right)\) | \(e\left(\frac{117}{269}\right)\) | \(e\left(\frac{91}{269}\right)\) |
| \(\chi_{2153}(158,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{256}{269}\right)\) | \(e\left(\frac{139}{269}\right)\) | \(e\left(\frac{243}{269}\right)\) | \(e\left(\frac{80}{269}\right)\) | \(e\left(\frac{126}{269}\right)\) | \(e\left(\frac{176}{269}\right)\) | \(e\left(\frac{230}{269}\right)\) | \(e\left(\frac{9}{269}\right)\) | \(e\left(\frac{67}{269}\right)\) | \(e\left(\frac{82}{269}\right)\) |
| \(\chi_{2153}(163,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{245}{269}\right)\) | \(e\left(\frac{29}{269}\right)\) | \(e\left(\frac{221}{269}\right)\) | \(e\left(\frac{127}{269}\right)\) | \(e\left(\frac{5}{269}\right)\) | \(e\left(\frac{118}{269}\right)\) | \(e\left(\frac{197}{269}\right)\) | \(e\left(\frac{58}{269}\right)\) | \(e\left(\frac{103}{269}\right)\) | \(e\left(\frac{110}{269}\right)\) |
| \(\chi_{2153}(171,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{101}{269}\right)\) | \(e\left(\frac{203}{269}\right)\) | \(e\left(\frac{202}{269}\right)\) | \(e\left(\frac{82}{269}\right)\) | \(e\left(\frac{35}{269}\right)\) | \(e\left(\frac{19}{269}\right)\) | \(e\left(\frac{34}{269}\right)\) | \(e\left(\frac{137}{269}\right)\) | \(e\left(\frac{183}{269}\right)\) | \(e\left(\frac{232}{269}\right)\) |
| \(\chi_{2153}(178,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{184}{269}\right)\) | \(e\left(\frac{226}{269}\right)\) | \(e\left(\frac{99}{269}\right)\) | \(e\left(\frac{192}{269}\right)\) | \(e\left(\frac{141}{269}\right)\) | \(e\left(\frac{261}{269}\right)\) | \(e\left(\frac{14}{269}\right)\) | \(e\left(\frac{183}{269}\right)\) | \(e\left(\frac{107}{269}\right)\) | \(e\left(\frac{143}{269}\right)\) |
| \(\chi_{2153}(183,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{111}{269}\right)\) | \(e\left(\frac{34}{269}\right)\) | \(e\left(\frac{222}{269}\right)\) | \(e\left(\frac{186}{269}\right)\) | \(e\left(\frac{145}{269}\right)\) | \(e\left(\frac{194}{269}\right)\) | \(e\left(\frac{64}{269}\right)\) | \(e\left(\frac{68}{269}\right)\) | \(e\left(\frac{28}{269}\right)\) | \(e\left(\frac{231}{269}\right)\) |
| \(\chi_{2153}(203,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{94}{269}\right)\) | \(e\left(\frac{133}{269}\right)\) | \(e\left(\frac{188}{269}\right)\) | \(e\left(\frac{63}{269}\right)\) | \(e\left(\frac{227}{269}\right)\) | \(e\left(\frac{31}{269}\right)\) | \(e\left(\frac{13}{269}\right)\) | \(e\left(\frac{266}{269}\right)\) | \(e\left(\frac{157}{269}\right)\) | \(e\left(\frac{152}{269}\right)\) |
| \(\chi_{2153}(206,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{6}{269}\right)\) | \(e\left(\frac{60}{269}\right)\) | \(e\left(\frac{12}{269}\right)\) | \(e\left(\frac{170}{269}\right)\) | \(e\left(\frac{66}{269}\right)\) | \(e\left(\frac{105}{269}\right)\) | \(e\left(\frac{18}{269}\right)\) | \(e\left(\frac{120}{269}\right)\) | \(e\left(\frac{176}{269}\right)\) | \(e\left(\frac{107}{269}\right)\) |
| \(\chi_{2153}(218,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{128}{269}\right)\) | \(e\left(\frac{204}{269}\right)\) | \(e\left(\frac{256}{269}\right)\) | \(e\left(\frac{40}{269}\right)\) | \(e\left(\frac{63}{269}\right)\) | \(e\left(\frac{88}{269}\right)\) | \(e\left(\frac{115}{269}\right)\) | \(e\left(\frac{139}{269}\right)\) | \(e\left(\frac{168}{269}\right)\) | \(e\left(\frac{41}{269}\right)\) |
| \(\chi_{2153}(239,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{118}{269}\right)\) | \(e\left(\frac{104}{269}\right)\) | \(e\left(\frac{236}{269}\right)\) | \(e\left(\frac{205}{269}\right)\) | \(e\left(\frac{222}{269}\right)\) | \(e\left(\frac{182}{269}\right)\) | \(e\left(\frac{85}{269}\right)\) | \(e\left(\frac{208}{269}\right)\) | \(e\left(\frac{54}{269}\right)\) | \(e\left(\frac{42}{269}\right)\) |
| \(\chi_{2153}(242,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{46}{269}\right)\) | \(e\left(\frac{191}{269}\right)\) | \(e\left(\frac{92}{269}\right)\) | \(e\left(\frac{48}{269}\right)\) | \(e\left(\frac{237}{269}\right)\) | \(e\left(\frac{267}{269}\right)\) | \(e\left(\frac{138}{269}\right)\) | \(e\left(\frac{113}{269}\right)\) | \(e\left(\frac{94}{269}\right)\) | \(e\left(\frac{103}{269}\right)\) |
| \(\chi_{2153}(248,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{259}{269}\right)\) | \(e\left(\frac{169}{269}\right)\) | \(e\left(\frac{249}{269}\right)\) | \(e\left(\frac{165}{269}\right)\) | \(e\left(\frac{159}{269}\right)\) | \(e\left(\frac{94}{269}\right)\) | \(e\left(\frac{239}{269}\right)\) | \(e\left(\frac{69}{269}\right)\) | \(e\left(\frac{155}{269}\right)\) | \(e\left(\frac{1}{269}\right)\) |
| \(\chi_{2153}(253,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{262}{269}\right)\) | \(e\left(\frac{199}{269}\right)\) | \(e\left(\frac{255}{269}\right)\) | \(e\left(\frac{250}{269}\right)\) | \(e\left(\frac{192}{269}\right)\) | \(e\left(\frac{12}{269}\right)\) | \(e\left(\frac{248}{269}\right)\) | \(e\left(\frac{129}{269}\right)\) | \(e\left(\frac{243}{269}\right)\) | \(e\left(\frac{189}{269}\right)\) |