from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2872, base_ring=CyclotomicField(358))
M = H._module
chi = DirichletCharacter(H, M([179,0,304]))
chi.galois_orbit()
[g,chi] = znchar(Mod(15,2872))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(2872\) | |
| Conductor: | \(1436\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(358\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 1436.h | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | 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 358 polynomial (not computed) |
First 31 of 178 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{2872}(15,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{159}{358}\right)\) | \(e\left(\frac{36}{179}\right)\) | \(e\left(\frac{125}{358}\right)\) | \(e\left(\frac{159}{179}\right)\) | \(e\left(\frac{59}{358}\right)\) | \(e\left(\frac{14}{179}\right)\) | \(e\left(\frac{231}{358}\right)\) | \(e\left(\frac{151}{179}\right)\) | \(e\left(\frac{293}{358}\right)\) | \(e\left(\frac{142}{179}\right)\) |
| \(\chi_{2872}(23,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{77}{358}\right)\) | \(e\left(\frac{112}{179}\right)\) | \(e\left(\frac{11}{358}\right)\) | \(e\left(\frac{77}{179}\right)\) | \(e\left(\frac{283}{358}\right)\) | \(e\left(\frac{143}{179}\right)\) | \(e\left(\frac{301}{358}\right)\) | \(e\left(\frac{72}{179}\right)\) | \(e\left(\frac{295}{358}\right)\) | \(e\left(\frac{44}{179}\right)\) |
| \(\chi_{2872}(47,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{358}\right)\) | \(e\left(\frac{66}{179}\right)\) | \(e\left(\frac{259}{358}\right)\) | \(e\left(\frac{23}{179}\right)\) | \(e\left(\frac{317}{358}\right)\) | \(e\left(\frac{145}{179}\right)\) | \(e\left(\frac{155}{358}\right)\) | \(e\left(\frac{68}{179}\right)\) | \(e\left(\frac{209}{358}\right)\) | \(e\left(\frac{141}{179}\right)\) |
| \(\chi_{2872}(55,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{225}{358}\right)\) | \(e\left(\frac{132}{179}\right)\) | \(e\left(\frac{339}{358}\right)\) | \(e\left(\frac{46}{179}\right)\) | \(e\left(\frac{97}{358}\right)\) | \(e\left(\frac{111}{179}\right)\) | \(e\left(\frac{131}{358}\right)\) | \(e\left(\frac{136}{179}\right)\) | \(e\left(\frac{239}{358}\right)\) | \(e\left(\frac{103}{179}\right)\) |
| \(\chi_{2872}(79,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{267}{358}\right)\) | \(e\left(\frac{128}{179}\right)\) | \(e\left(\frac{345}{358}\right)\) | \(e\left(\frac{88}{179}\right)\) | \(e\left(\frac{349}{358}\right)\) | \(e\left(\frac{10}{179}\right)\) | \(e\left(\frac{165}{358}\right)\) | \(e\left(\frac{159}{179}\right)\) | \(e\left(\frac{107}{358}\right)\) | \(e\left(\frac{127}{179}\right)\) |
| \(\chi_{2872}(111,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{183}{358}\right)\) | \(e\left(\frac{136}{179}\right)\) | \(e\left(\frac{333}{358}\right)\) | \(e\left(\frac{4}{179}\right)\) | \(e\left(\frac{203}{358}\right)\) | \(e\left(\frac{33}{179}\right)\) | \(e\left(\frac{97}{358}\right)\) | \(e\left(\frac{113}{179}\right)\) | \(e\left(\frac{13}{358}\right)\) | \(e\left(\frac{79}{179}\right)\) |
| \(\chi_{2872}(127,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{295}{358}\right)\) | \(e\left(\frac{6}{179}\right)\) | \(e\left(\frac{349}{358}\right)\) | \(e\left(\frac{116}{179}\right)\) | \(e\left(\frac{159}{358}\right)\) | \(e\left(\frac{62}{179}\right)\) | \(e\left(\frac{307}{358}\right)\) | \(e\left(\frac{55}{179}\right)\) | \(e\left(\frac{19}{358}\right)\) | \(e\left(\frac{143}{179}\right)\) |
| \(\chi_{2872}(135,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{257}{358}\right)\) | \(e\left(\frac{146}{179}\right)\) | \(e\left(\frac{139}{358}\right)\) | \(e\left(\frac{78}{179}\right)\) | \(e\left(\frac{289}{358}\right)\) | \(e\left(\frac{17}{179}\right)\) | \(e\left(\frac{191}{358}\right)\) | \(e\left(\frac{145}{179}\right)\) | \(e\left(\frac{343}{358}\right)\) | \(e\left(\frac{19}{179}\right)\) |
| \(\chi_{2872}(151,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{203}{358}\right)\) | \(e\left(\frac{100}{179}\right)\) | \(e\left(\frac{29}{358}\right)\) | \(e\left(\frac{24}{179}\right)\) | \(e\left(\frac{323}{358}\right)\) | \(e\left(\frac{19}{179}\right)\) | \(e\left(\frac{45}{358}\right)\) | \(e\left(\frac{141}{179}\right)\) | \(e\left(\frac{257}{358}\right)\) | \(e\left(\frac{116}{179}\right)\) |
| \(\chi_{2872}(191,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{27}{358}\right)\) | \(e\left(\frac{23}{179}\right)\) | \(e\left(\frac{55}{358}\right)\) | \(e\left(\frac{27}{179}\right)\) | \(e\left(\frac{341}{358}\right)\) | \(e\left(\frac{178}{179}\right)\) | \(e\left(\frac{73}{358}\right)\) | \(e\left(\frac{2}{179}\right)\) | \(e\left(\frac{43}{358}\right)\) | \(e\left(\frac{41}{179}\right)\) |
| \(\chi_{2872}(207,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{175}{358}\right)\) | \(e\left(\frac{43}{179}\right)\) | \(e\left(\frac{25}{358}\right)\) | \(e\left(\frac{175}{179}\right)\) | \(e\left(\frac{155}{358}\right)\) | \(e\left(\frac{146}{179}\right)\) | \(e\left(\frac{261}{358}\right)\) | \(e\left(\frac{66}{179}\right)\) | \(e\left(\frac{345}{358}\right)\) | \(e\left(\frac{100}{179}\right)\) |
| \(\chi_{2872}(247,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{207}{358}\right)\) | \(e\left(\frac{57}{179}\right)\) | \(e\left(\frac{183}{358}\right)\) | \(e\left(\frac{28}{179}\right)\) | \(e\left(\frac{347}{358}\right)\) | \(e\left(\frac{52}{179}\right)\) | \(e\left(\frac{321}{358}\right)\) | \(e\left(\frac{75}{179}\right)\) | \(e\left(\frac{91}{358}\right)\) | \(e\left(\frac{16}{179}\right)\) |
| \(\chi_{2872}(255,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{153}{358}\right)\) | \(e\left(\frac{11}{179}\right)\) | \(e\left(\frac{73}{358}\right)\) | \(e\left(\frac{153}{179}\right)\) | \(e\left(\frac{23}{358}\right)\) | \(e\left(\frac{54}{179}\right)\) | \(e\left(\frac{175}{358}\right)\) | \(e\left(\frac{71}{179}\right)\) | \(e\left(\frac{5}{358}\right)\) | \(e\left(\frac{113}{179}\right)\) |
| \(\chi_{2872}(303,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{157}{358}\right)\) | \(e\left(\frac{147}{179}\right)\) | \(e\left(\frac{227}{358}\right)\) | \(e\left(\frac{157}{179}\right)\) | \(e\left(\frac{47}{358}\right)\) | \(e\left(\frac{87}{179}\right)\) | \(e\left(\frac{93}{358}\right)\) | \(e\left(\frac{5}{179}\right)\) | \(e\left(\frac{197}{358}\right)\) | \(e\left(\frac{13}{179}\right)\) |
| \(\chi_{2872}(367,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{329}{358}\right)\) | \(e\left(\frac{88}{179}\right)\) | \(e\left(\frac{47}{358}\right)\) | \(e\left(\frac{150}{179}\right)\) | \(e\left(\frac{5}{358}\right)\) | \(e\left(\frac{74}{179}\right)\) | \(e\left(\frac{147}{358}\right)\) | \(e\left(\frac{31}{179}\right)\) | \(e\left(\frac{219}{358}\right)\) | \(e\left(\frac{9}{179}\right)\) |
| \(\chi_{2872}(375,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{21}{358}\right)\) | \(e\left(\frac{177}{179}\right)\) | \(e\left(\frac{3}{358}\right)\) | \(e\left(\frac{21}{179}\right)\) | \(e\left(\frac{305}{358}\right)\) | \(e\left(\frac{39}{179}\right)\) | \(e\left(\frac{17}{358}\right)\) | \(e\left(\frac{101}{179}\right)\) | \(e\left(\frac{113}{358}\right)\) | \(e\left(\frac{12}{179}\right)\) |
| \(\chi_{2872}(383,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{199}{358}\right)\) | \(e\left(\frac{143}{179}\right)\) | \(e\left(\frac{233}{358}\right)\) | \(e\left(\frac{20}{179}\right)\) | \(e\left(\frac{299}{358}\right)\) | \(e\left(\frac{165}{179}\right)\) | \(e\left(\frac{127}{358}\right)\) | \(e\left(\frac{28}{179}\right)\) | \(e\left(\frac{65}{358}\right)\) | \(e\left(\frac{37}{179}\right)\) |
| \(\chi_{2872}(391,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{71}{358}\right)\) | \(e\left(\frac{87}{179}\right)\) | \(e\left(\frac{317}{358}\right)\) | \(e\left(\frac{71}{179}\right)\) | \(e\left(\frac{247}{358}\right)\) | \(e\left(\frac{4}{179}\right)\) | \(e\left(\frac{245}{358}\right)\) | \(e\left(\frac{171}{179}\right)\) | \(e\left(\frac{7}{358}\right)\) | \(e\left(\frac{15}{179}\right)\) |
| \(\chi_{2872}(399,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{81}{358}\right)\) | \(e\left(\frac{69}{179}\right)\) | \(e\left(\frac{165}{358}\right)\) | \(e\left(\frac{81}{179}\right)\) | \(e\left(\frac{307}{358}\right)\) | \(e\left(\frac{176}{179}\right)\) | \(e\left(\frac{219}{358}\right)\) | \(e\left(\frac{6}{179}\right)\) | \(e\left(\frac{129}{358}\right)\) | \(e\left(\frac{123}{179}\right)\) |
| \(\chi_{2872}(407,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{249}{358}\right)\) | \(e\left(\frac{53}{179}\right)\) | \(e\left(\frac{189}{358}\right)\) | \(e\left(\frac{70}{179}\right)\) | \(e\left(\frac{241}{358}\right)\) | \(e\left(\frac{130}{179}\right)\) | \(e\left(\frac{355}{358}\right)\) | \(e\left(\frac{98}{179}\right)\) | \(e\left(\frac{317}{358}\right)\) | \(e\left(\frac{40}{179}\right)\) |
| \(\chi_{2872}(423,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{121}{358}\right)\) | \(e\left(\frac{176}{179}\right)\) | \(e\left(\frac{273}{358}\right)\) | \(e\left(\frac{121}{179}\right)\) | \(e\left(\frac{189}{358}\right)\) | \(e\left(\frac{148}{179}\right)\) | \(e\left(\frac{115}{358}\right)\) | \(e\left(\frac{62}{179}\right)\) | \(e\left(\frac{259}{358}\right)\) | \(e\left(\frac{18}{179}\right)\) |
| \(\chi_{2872}(431,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{69}{358}\right)\) | \(e\left(\frac{19}{179}\right)\) | \(e\left(\frac{61}{358}\right)\) | \(e\left(\frac{69}{179}\right)\) | \(e\left(\frac{235}{358}\right)\) | \(e\left(\frac{77}{179}\right)\) | \(e\left(\frac{107}{358}\right)\) | \(e\left(\frac{25}{179}\right)\) | \(e\left(\frac{269}{358}\right)\) | \(e\left(\frac{65}{179}\right)\) |
| \(\chi_{2872}(439,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{131}{358}\right)\) | \(e\left(\frac{158}{179}\right)\) | \(e\left(\frac{121}{358}\right)\) | \(e\left(\frac{131}{179}\right)\) | \(e\left(\frac{249}{358}\right)\) | \(e\left(\frac{141}{179}\right)\) | \(e\left(\frac{89}{358}\right)\) | \(e\left(\frac{76}{179}\right)\) | \(e\left(\frac{23}{358}\right)\) | \(e\left(\frac{126}{179}\right)\) |
| \(\chi_{2872}(447,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{265}{358}\right)\) | \(e\left(\frac{60}{179}\right)\) | \(e\left(\frac{89}{358}\right)\) | \(e\left(\frac{86}{179}\right)\) | \(e\left(\frac{337}{358}\right)\) | \(e\left(\frac{83}{179}\right)\) | \(e\left(\frac{27}{358}\right)\) | \(e\left(\frac{13}{179}\right)\) | \(e\left(\frac{11}{358}\right)\) | \(e\left(\frac{177}{179}\right)\) |
| \(\chi_{2872}(455,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{299}{358}\right)\) | \(e\left(\frac{142}{179}\right)\) | \(e\left(\frac{145}{358}\right)\) | \(e\left(\frac{120}{179}\right)\) | \(e\left(\frac{183}{358}\right)\) | \(e\left(\frac{95}{179}\right)\) | \(e\left(\frac{225}{358}\right)\) | \(e\left(\frac{168}{179}\right)\) | \(e\left(\frac{211}{358}\right)\) | \(e\left(\frac{43}{179}\right)\) |
| \(\chi_{2872}(479,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{309}{358}\right)\) | \(e\left(\frac{124}{179}\right)\) | \(e\left(\frac{351}{358}\right)\) | \(e\left(\frac{130}{179}\right)\) | \(e\left(\frac{243}{358}\right)\) | \(e\left(\frac{88}{179}\right)\) | \(e\left(\frac{199}{358}\right)\) | \(e\left(\frac{3}{179}\right)\) | \(e\left(\frac{333}{358}\right)\) | \(e\left(\frac{151}{179}\right)\) |
| \(\chi_{2872}(487,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{171}{358}\right)\) | \(e\left(\frac{86}{179}\right)\) | \(e\left(\frac{229}{358}\right)\) | \(e\left(\frac{171}{179}\right)\) | \(e\left(\frac{131}{358}\right)\) | \(e\left(\frac{113}{179}\right)\) | \(e\left(\frac{343}{358}\right)\) | \(e\left(\frac{132}{179}\right)\) | \(e\left(\frac{153}{358}\right)\) | \(e\left(\frac{21}{179}\right)\) |
| \(\chi_{2872}(495,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{323}{358}\right)\) | \(e\left(\frac{63}{179}\right)\) | \(e\left(\frac{353}{358}\right)\) | \(e\left(\frac{144}{179}\right)\) | \(e\left(\frac{327}{358}\right)\) | \(e\left(\frac{114}{179}\right)\) | \(e\left(\frac{91}{358}\right)\) | \(e\left(\frac{130}{179}\right)\) | \(e\left(\frac{289}{358}\right)\) | \(e\left(\frac{159}{179}\right)\) |
| \(\chi_{2872}(503,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{119}{358}\right)\) | \(e\left(\frac{108}{179}\right)\) | \(e\left(\frac{17}{358}\right)\) | \(e\left(\frac{119}{179}\right)\) | \(e\left(\frac{177}{358}\right)\) | \(e\left(\frac{42}{179}\right)\) | \(e\left(\frac{335}{358}\right)\) | \(e\left(\frac{95}{179}\right)\) | \(e\left(\frac{163}{358}\right)\) | \(e\left(\frac{68}{179}\right)\) |
| \(\chi_{2872}(519,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{181}{358}\right)\) | \(e\left(\frac{68}{179}\right)\) | \(e\left(\frac{77}{358}\right)\) | \(e\left(\frac{2}{179}\right)\) | \(e\left(\frac{191}{358}\right)\) | \(e\left(\frac{106}{179}\right)\) | \(e\left(\frac{317}{358}\right)\) | \(e\left(\frac{146}{179}\right)\) | \(e\left(\frac{275}{358}\right)\) | \(e\left(\frac{129}{179}\right)\) |
| \(\chi_{2872}(535,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{315}{358}\right)\) | \(e\left(\frac{149}{179}\right)\) | \(e\left(\frac{45}{358}\right)\) | \(e\left(\frac{136}{179}\right)\) | \(e\left(\frac{279}{358}\right)\) | \(e\left(\frac{48}{179}\right)\) | \(e\left(\frac{255}{358}\right)\) | \(e\left(\frac{83}{179}\right)\) | \(e\left(\frac{263}{358}\right)\) | \(e\left(\frac{1}{179}\right)\) |