from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1077, base_ring=CyclotomicField(358))
M = H._module
chi = DirichletCharacter(H, M([0,270]))
chi.galois_orbit()
[g,chi] = znchar(Mod(4,1077))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(1077\) | |
| Conductor: | \(359\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(179\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 359.c | 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: | $\Q(\zeta_{179})$ |
| Fixed field: | Number field defined by a degree 179 polynomial (not computed) |
First 31 of 178 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{1077}(4,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{146}{179}\right)\) | \(e\left(\frac{113}{179}\right)\) | \(e\left(\frac{178}{179}\right)\) | \(e\left(\frac{135}{179}\right)\) | \(e\left(\frac{80}{179}\right)\) | \(e\left(\frac{145}{179}\right)\) | \(e\left(\frac{121}{179}\right)\) | \(e\left(\frac{109}{179}\right)\) | \(e\left(\frac{102}{179}\right)\) | \(e\left(\frac{47}{179}\right)\) |
| \(\chi_{1077}(10,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{162}{179}\right)\) | \(e\left(\frac{145}{179}\right)\) | \(e\left(\frac{70}{179}\right)\) | \(e\left(\frac{37}{179}\right)\) | \(e\left(\frac{128}{179}\right)\) | \(e\left(\frac{53}{179}\right)\) | \(e\left(\frac{122}{179}\right)\) | \(e\left(\frac{67}{179}\right)\) | \(e\left(\frac{20}{179}\right)\) | \(e\left(\frac{111}{179}\right)\) |
| \(\chi_{1077}(16,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{113}{179}\right)\) | \(e\left(\frac{47}{179}\right)\) | \(e\left(\frac{177}{179}\right)\) | \(e\left(\frac{91}{179}\right)\) | \(e\left(\frac{160}{179}\right)\) | \(e\left(\frac{111}{179}\right)\) | \(e\left(\frac{63}{179}\right)\) | \(e\left(\frac{39}{179}\right)\) | \(e\left(\frac{25}{179}\right)\) | \(e\left(\frac{94}{179}\right)\) |
| \(\chi_{1077}(22,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{44}{179}\right)\) | \(e\left(\frac{88}{179}\right)\) | \(e\left(\frac{61}{179}\right)\) | \(e\left(\frac{178}{179}\right)\) | \(e\left(\frac{132}{179}\right)\) | \(e\left(\frac{105}{179}\right)\) | \(e\left(\frac{137}{179}\right)\) | \(e\left(\frac{153}{179}\right)\) | \(e\left(\frac{43}{179}\right)\) | \(e\left(\frac{176}{179}\right)\) |
| \(\chi_{1077}(25,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{178}{179}\right)\) | \(e\left(\frac{177}{179}\right)\) | \(e\left(\frac{141}{179}\right)\) | \(e\left(\frac{118}{179}\right)\) | \(e\left(\frac{176}{179}\right)\) | \(e\left(\frac{140}{179}\right)\) | \(e\left(\frac{123}{179}\right)\) | \(e\left(\frac{25}{179}\right)\) | \(e\left(\frac{117}{179}\right)\) | \(e\left(\frac{175}{179}\right)\) |
| \(\chi_{1077}(34,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{143}{179}\right)\) | \(e\left(\frac{107}{179}\right)\) | \(e\left(\frac{64}{179}\right)\) | \(e\left(\frac{131}{179}\right)\) | \(e\left(\frac{71}{179}\right)\) | \(e\left(\frac{28}{179}\right)\) | \(e\left(\frac{132}{179}\right)\) | \(e\left(\frac{5}{179}\right)\) | \(e\left(\frac{95}{179}\right)\) | \(e\left(\frac{35}{179}\right)\) |
| \(\chi_{1077}(37,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{167}{179}\right)\) | \(e\left(\frac{155}{179}\right)\) | \(e\left(\frac{81}{179}\right)\) | \(e\left(\frac{163}{179}\right)\) | \(e\left(\frac{143}{179}\right)\) | \(e\left(\frac{69}{179}\right)\) | \(e\left(\frac{44}{179}\right)\) | \(e\left(\frac{121}{179}\right)\) | \(e\left(\frac{151}{179}\right)\) | \(e\left(\frac{131}{179}\right)\) |
| \(\chi_{1077}(40,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{129}{179}\right)\) | \(e\left(\frac{79}{179}\right)\) | \(e\left(\frac{69}{179}\right)\) | \(e\left(\frac{172}{179}\right)\) | \(e\left(\frac{29}{179}\right)\) | \(e\left(\frac{19}{179}\right)\) | \(e\left(\frac{64}{179}\right)\) | \(e\left(\frac{176}{179}\right)\) | \(e\left(\frac{122}{179}\right)\) | \(e\left(\frac{158}{179}\right)\) |
| \(\chi_{1077}(46,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{10}{179}\right)\) | \(e\left(\frac{20}{179}\right)\) | \(e\left(\frac{22}{179}\right)\) | \(e\left(\frac{73}{179}\right)\) | \(e\left(\frac{30}{179}\right)\) | \(e\left(\frac{32}{179}\right)\) | \(e\left(\frac{23}{179}\right)\) | \(e\left(\frac{108}{179}\right)\) | \(e\left(\frac{83}{179}\right)\) | \(e\left(\frac{40}{179}\right)\) |
| \(\chi_{1077}(49,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{135}{179}\right)\) | \(e\left(\frac{91}{179}\right)\) | \(e\left(\frac{118}{179}\right)\) | \(e\left(\frac{1}{179}\right)\) | \(e\left(\frac{47}{179}\right)\) | \(e\left(\frac{74}{179}\right)\) | \(e\left(\frac{42}{179}\right)\) | \(e\left(\frac{26}{179}\right)\) | \(e\left(\frac{136}{179}\right)\) | \(e\left(\frac{3}{179}\right)\) |
| \(\chi_{1077}(55,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{60}{179}\right)\) | \(e\left(\frac{120}{179}\right)\) | \(e\left(\frac{132}{179}\right)\) | \(e\left(\frac{80}{179}\right)\) | \(e\left(\frac{1}{179}\right)\) | \(e\left(\frac{13}{179}\right)\) | \(e\left(\frac{138}{179}\right)\) | \(e\left(\frac{111}{179}\right)\) | \(e\left(\frac{140}{179}\right)\) | \(e\left(\frac{61}{179}\right)\) |
| \(\chi_{1077}(64,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{80}{179}\right)\) | \(e\left(\frac{160}{179}\right)\) | \(e\left(\frac{176}{179}\right)\) | \(e\left(\frac{47}{179}\right)\) | \(e\left(\frac{61}{179}\right)\) | \(e\left(\frac{77}{179}\right)\) | \(e\left(\frac{5}{179}\right)\) | \(e\left(\frac{148}{179}\right)\) | \(e\left(\frac{127}{179}\right)\) | \(e\left(\frac{141}{179}\right)\) |
| \(\chi_{1077}(73,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{179}\right)\) | \(e\left(\frac{18}{179}\right)\) | \(e\left(\frac{163}{179}\right)\) | \(e\left(\frac{12}{179}\right)\) | \(e\left(\frac{27}{179}\right)\) | \(e\left(\frac{172}{179}\right)\) | \(e\left(\frac{146}{179}\right)\) | \(e\left(\frac{133}{179}\right)\) | \(e\left(\frac{21}{179}\right)\) | \(e\left(\frac{36}{179}\right)\) |
| \(\chi_{1077}(79,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{107}{179}\right)\) | \(e\left(\frac{35}{179}\right)\) | \(e\left(\frac{128}{179}\right)\) | \(e\left(\frac{83}{179}\right)\) | \(e\left(\frac{142}{179}\right)\) | \(e\left(\frac{56}{179}\right)\) | \(e\left(\frac{85}{179}\right)\) | \(e\left(\frac{10}{179}\right)\) | \(e\left(\frac{11}{179}\right)\) | \(e\left(\frac{70}{179}\right)\) |
| \(\chi_{1077}(82,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{6}{179}\right)\) | \(e\left(\frac{12}{179}\right)\) | \(e\left(\frac{49}{179}\right)\) | \(e\left(\frac{8}{179}\right)\) | \(e\left(\frac{18}{179}\right)\) | \(e\left(\frac{55}{179}\right)\) | \(e\left(\frac{157}{179}\right)\) | \(e\left(\frac{29}{179}\right)\) | \(e\left(\frac{14}{179}\right)\) | \(e\left(\frac{24}{179}\right)\) |
| \(\chi_{1077}(85,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{159}{179}\right)\) | \(e\left(\frac{139}{179}\right)\) | \(e\left(\frac{135}{179}\right)\) | \(e\left(\frac{33}{179}\right)\) | \(e\left(\frac{119}{179}\right)\) | \(e\left(\frac{115}{179}\right)\) | \(e\left(\frac{133}{179}\right)\) | \(e\left(\frac{142}{179}\right)\) | \(e\left(\frac{13}{179}\right)\) | \(e\left(\frac{99}{179}\right)\) |
| \(\chi_{1077}(88,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{179}\right)\) | \(e\left(\frac{22}{179}\right)\) | \(e\left(\frac{60}{179}\right)\) | \(e\left(\frac{134}{179}\right)\) | \(e\left(\frac{33}{179}\right)\) | \(e\left(\frac{71}{179}\right)\) | \(e\left(\frac{79}{179}\right)\) | \(e\left(\frac{83}{179}\right)\) | \(e\left(\frac{145}{179}\right)\) | \(e\left(\frac{44}{179}\right)\) |
| \(\chi_{1077}(91,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{122}{179}\right)\) | \(e\left(\frac{65}{179}\right)\) | \(e\left(\frac{161}{179}\right)\) | \(e\left(\frac{103}{179}\right)\) | \(e\left(\frac{8}{179}\right)\) | \(e\left(\frac{104}{179}\right)\) | \(e\left(\frac{30}{179}\right)\) | \(e\left(\frac{172}{179}\right)\) | \(e\left(\frac{46}{179}\right)\) | \(e\left(\frac{130}{179}\right)\) |
| \(\chi_{1077}(94,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{103}{179}\right)\) | \(e\left(\frac{27}{179}\right)\) | \(e\left(\frac{155}{179}\right)\) | \(e\left(\frac{18}{179}\right)\) | \(e\left(\frac{130}{179}\right)\) | \(e\left(\frac{79}{179}\right)\) | \(e\left(\frac{40}{179}\right)\) | \(e\left(\frac{110}{179}\right)\) | \(e\left(\frac{121}{179}\right)\) | \(e\left(\frac{54}{179}\right)\) |
| \(\chi_{1077}(100,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{145}{179}\right)\) | \(e\left(\frac{111}{179}\right)\) | \(e\left(\frac{140}{179}\right)\) | \(e\left(\frac{74}{179}\right)\) | \(e\left(\frac{77}{179}\right)\) | \(e\left(\frac{106}{179}\right)\) | \(e\left(\frac{65}{179}\right)\) | \(e\left(\frac{134}{179}\right)\) | \(e\left(\frac{40}{179}\right)\) | \(e\left(\frac{43}{179}\right)\) |
| \(\chi_{1077}(115,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{26}{179}\right)\) | \(e\left(\frac{52}{179}\right)\) | \(e\left(\frac{93}{179}\right)\) | \(e\left(\frac{154}{179}\right)\) | \(e\left(\frac{78}{179}\right)\) | \(e\left(\frac{119}{179}\right)\) | \(e\left(\frac{24}{179}\right)\) | \(e\left(\frac{66}{179}\right)\) | \(e\left(\frac{1}{179}\right)\) | \(e\left(\frac{104}{179}\right)\) |
| \(\chi_{1077}(121,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{121}{179}\right)\) | \(e\left(\frac{63}{179}\right)\) | \(e\left(\frac{123}{179}\right)\) | \(e\left(\frac{42}{179}\right)\) | \(e\left(\frac{5}{179}\right)\) | \(e\left(\frac{65}{179}\right)\) | \(e\left(\frac{153}{179}\right)\) | \(e\left(\frac{18}{179}\right)\) | \(e\left(\frac{163}{179}\right)\) | \(e\left(\frac{126}{179}\right)\) |
| \(\chi_{1077}(127,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{179}\right)\) | \(e\left(\frac{38}{179}\right)\) | \(e\left(\frac{6}{179}\right)\) | \(e\left(\frac{85}{179}\right)\) | \(e\left(\frac{57}{179}\right)\) | \(e\left(\frac{25}{179}\right)\) | \(e\left(\frac{169}{179}\right)\) | \(e\left(\frac{62}{179}\right)\) | \(e\left(\frac{104}{179}\right)\) | \(e\left(\frac{76}{179}\right)\) |
| \(\chi_{1077}(133,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{104}{179}\right)\) | \(e\left(\frac{29}{179}\right)\) | \(e\left(\frac{14}{179}\right)\) | \(e\left(\frac{79}{179}\right)\) | \(e\left(\frac{133}{179}\right)\) | \(e\left(\frac{118}{179}\right)\) | \(e\left(\frac{96}{179}\right)\) | \(e\left(\frac{85}{179}\right)\) | \(e\left(\frac{4}{179}\right)\) | \(e\left(\frac{58}{179}\right)\) |
| \(\chi_{1077}(136,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{110}{179}\right)\) | \(e\left(\frac{41}{179}\right)\) | \(e\left(\frac{63}{179}\right)\) | \(e\left(\frac{87}{179}\right)\) | \(e\left(\frac{151}{179}\right)\) | \(e\left(\frac{173}{179}\right)\) | \(e\left(\frac{74}{179}\right)\) | \(e\left(\frac{114}{179}\right)\) | \(e\left(\frac{18}{179}\right)\) | \(e\left(\frac{82}{179}\right)\) |
| \(\chi_{1077}(148,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{134}{179}\right)\) | \(e\left(\frac{89}{179}\right)\) | \(e\left(\frac{80}{179}\right)\) | \(e\left(\frac{119}{179}\right)\) | \(e\left(\frac{44}{179}\right)\) | \(e\left(\frac{35}{179}\right)\) | \(e\left(\frac{165}{179}\right)\) | \(e\left(\frac{51}{179}\right)\) | \(e\left(\frac{74}{179}\right)\) | \(e\left(\frac{178}{179}\right)\) |
| \(\chi_{1077}(151,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{78}{179}\right)\) | \(e\left(\frac{156}{179}\right)\) | \(e\left(\frac{100}{179}\right)\) | \(e\left(\frac{104}{179}\right)\) | \(e\left(\frac{55}{179}\right)\) | \(e\left(\frac{178}{179}\right)\) | \(e\left(\frac{72}{179}\right)\) | \(e\left(\frac{19}{179}\right)\) | \(e\left(\frac{3}{179}\right)\) | \(e\left(\frac{133}{179}\right)\) |
| \(\chi_{1077}(160,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{96}{179}\right)\) | \(e\left(\frac{13}{179}\right)\) | \(e\left(\frac{68}{179}\right)\) | \(e\left(\frac{128}{179}\right)\) | \(e\left(\frac{109}{179}\right)\) | \(e\left(\frac{164}{179}\right)\) | \(e\left(\frac{6}{179}\right)\) | \(e\left(\frac{106}{179}\right)\) | \(e\left(\frac{45}{179}\right)\) | \(e\left(\frac{26}{179}\right)\) |
| \(\chi_{1077}(169,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{109}{179}\right)\) | \(e\left(\frac{39}{179}\right)\) | \(e\left(\frac{25}{179}\right)\) | \(e\left(\frac{26}{179}\right)\) | \(e\left(\frac{148}{179}\right)\) | \(e\left(\frac{134}{179}\right)\) | \(e\left(\frac{18}{179}\right)\) | \(e\left(\frac{139}{179}\right)\) | \(e\left(\frac{135}{179}\right)\) | \(e\left(\frac{78}{179}\right)\) |
| \(\chi_{1077}(181,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{131}{179}\right)\) | \(e\left(\frac{83}{179}\right)\) | \(e\left(\frac{145}{179}\right)\) | \(e\left(\frac{115}{179}\right)\) | \(e\left(\frac{35}{179}\right)\) | \(e\left(\frac{97}{179}\right)\) | \(e\left(\frac{176}{179}\right)\) | \(e\left(\frac{126}{179}\right)\) | \(e\left(\frac{67}{179}\right)\) | \(e\left(\frac{166}{179}\right)\) |
| \(\chi_{1077}(184,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{156}{179}\right)\) | \(e\left(\frac{133}{179}\right)\) | \(e\left(\frac{21}{179}\right)\) | \(e\left(\frac{29}{179}\right)\) | \(e\left(\frac{110}{179}\right)\) | \(e\left(\frac{177}{179}\right)\) | \(e\left(\frac{144}{179}\right)\) | \(e\left(\frac{38}{179}\right)\) | \(e\left(\frac{6}{179}\right)\) | \(e\left(\frac{87}{179}\right)\) |