from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2888, base_ring=CyclotomicField(342))
M = H._module
chi = DirichletCharacter(H, M([0,171,329]))
chi.galois_orbit()
[g,chi] = znchar(Mod(13,2888))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(2888\) | |
| Conductor: | \(2888\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(342\) | 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: | $\Q(\zeta_{171})$ |
| Fixed field: | Number field defined by a degree 342 polynomial (not computed) |
First 31 of 108 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(21\) | \(23\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{2888}(13,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{37}{171}\right)\) | \(e\left(\frac{233}{342}\right)\) | \(e\left(\frac{17}{57}\right)\) | \(e\left(\frac{74}{171}\right)\) | \(e\left(\frac{71}{114}\right)\) | \(e\left(\frac{170}{171}\right)\) | \(e\left(\frac{307}{342}\right)\) | \(e\left(\frac{70}{171}\right)\) | \(e\left(\frac{88}{171}\right)\) | \(e\left(\frac{77}{171}\right)\) |
| \(\chi_{2888}(21,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{164}{171}\right)\) | \(e\left(\frac{187}{342}\right)\) | \(e\left(\frac{43}{57}\right)\) | \(e\left(\frac{157}{171}\right)\) | \(e\left(\frac{79}{114}\right)\) | \(e\left(\frac{88}{171}\right)\) | \(e\left(\frac{173}{342}\right)\) | \(e\left(\frac{167}{171}\right)\) | \(e\left(\frac{122}{171}\right)\) | \(e\left(\frac{64}{171}\right)\) |
| \(\chi_{2888}(29,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{70}{171}\right)\) | \(e\left(\frac{11}{342}\right)\) | \(e\left(\frac{26}{57}\right)\) | \(e\left(\frac{140}{171}\right)\) | \(e\left(\frac{65}{114}\right)\) | \(e\left(\frac{146}{171}\right)\) | \(e\left(\frac{151}{342}\right)\) | \(e\left(\frac{40}{171}\right)\) | \(e\left(\frac{148}{171}\right)\) | \(e\left(\frac{44}{171}\right)\) |
| \(\chi_{2888}(53,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{40}{171}\right)\) | \(e\left(\frac{275}{342}\right)\) | \(e\left(\frac{23}{57}\right)\) | \(e\left(\frac{80}{171}\right)\) | \(e\left(\frac{29}{114}\right)\) | \(e\left(\frac{59}{171}\right)\) | \(e\left(\frac{13}{342}\right)\) | \(e\left(\frac{145}{171}\right)\) | \(e\left(\frac{109}{171}\right)\) | \(e\left(\frac{74}{171}\right)\) |
| \(\chi_{2888}(109,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{131}{171}\right)\) | \(e\left(\frac{67}{342}\right)\) | \(e\left(\frac{34}{57}\right)\) | \(e\left(\frac{91}{171}\right)\) | \(e\left(\frac{85}{114}\right)\) | \(e\left(\frac{112}{171}\right)\) | \(e\left(\frac{329}{342}\right)\) | \(e\left(\frac{26}{171}\right)\) | \(e\left(\frac{62}{171}\right)\) | \(e\left(\frac{97}{171}\right)\) |
| \(\chi_{2888}(117,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{35}{171}\right)\) | \(e\left(\frac{91}{342}\right)\) | \(e\left(\frac{13}{57}\right)\) | \(e\left(\frac{70}{171}\right)\) | \(e\left(\frac{61}{114}\right)\) | \(e\left(\frac{73}{171}\right)\) | \(e\left(\frac{161}{342}\right)\) | \(e\left(\frac{20}{171}\right)\) | \(e\left(\frac{74}{171}\right)\) | \(e\left(\frac{22}{171}\right)\) |
| \(\chi_{2888}(165,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{127}{171}\right)\) | \(e\left(\frac{125}{342}\right)\) | \(e\left(\frac{26}{57}\right)\) | \(e\left(\frac{83}{171}\right)\) | \(e\left(\frac{65}{114}\right)\) | \(e\left(\frac{89}{171}\right)\) | \(e\left(\frac{37}{342}\right)\) | \(e\left(\frac{97}{171}\right)\) | \(e\left(\frac{34}{171}\right)\) | \(e\left(\frac{158}{171}\right)\) |
| \(\chi_{2888}(173,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{65}{171}\right)\) | \(e\left(\frac{169}{342}\right)\) | \(e\left(\frac{16}{57}\right)\) | \(e\left(\frac{130}{171}\right)\) | \(e\left(\frac{97}{114}\right)\) | \(e\left(\frac{160}{171}\right)\) | \(e\left(\frac{299}{342}\right)\) | \(e\left(\frac{86}{171}\right)\) | \(e\left(\frac{113}{171}\right)\) | \(e\left(\frac{163}{171}\right)\) |
| \(\chi_{2888}(181,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{16}{171}\right)\) | \(e\left(\frac{281}{342}\right)\) | \(e\left(\frac{32}{57}\right)\) | \(e\left(\frac{32}{171}\right)\) | \(e\left(\frac{23}{114}\right)\) | \(e\left(\frac{92}{171}\right)\) | \(e\left(\frac{313}{342}\right)\) | \(e\left(\frac{58}{171}\right)\) | \(e\left(\frac{112}{171}\right)\) | \(e\left(\frac{98}{171}\right)\) |
| \(\chi_{2888}(205,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{4}{171}\right)\) | \(e\left(\frac{113}{342}\right)\) | \(e\left(\frac{8}{57}\right)\) | \(e\left(\frac{8}{171}\right)\) | \(e\left(\frac{77}{114}\right)\) | \(e\left(\frac{23}{171}\right)\) | \(e\left(\frac{121}{342}\right)\) | \(e\left(\frac{100}{171}\right)\) | \(e\left(\frac{28}{171}\right)\) | \(e\left(\frac{110}{171}\right)\) |
| \(\chi_{2888}(261,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{68}{171}\right)\) | \(e\left(\frac{211}{342}\right)\) | \(e\left(\frac{22}{57}\right)\) | \(e\left(\frac{136}{171}\right)\) | \(e\left(\frac{55}{114}\right)\) | \(e\left(\frac{49}{171}\right)\) | \(e\left(\frac{5}{342}\right)\) | \(e\left(\frac{161}{171}\right)\) | \(e\left(\frac{134}{171}\right)\) | \(e\left(\frac{160}{171}\right)\) |
| \(\chi_{2888}(269,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{26}{171}\right)\) | \(e\left(\frac{307}{342}\right)\) | \(e\left(\frac{52}{57}\right)\) | \(e\left(\frac{52}{171}\right)\) | \(e\left(\frac{73}{114}\right)\) | \(e\left(\frac{64}{171}\right)\) | \(e\left(\frac{17}{342}\right)\) | \(e\left(\frac{137}{171}\right)\) | \(e\left(\frac{11}{171}\right)\) | \(e\left(\frac{31}{171}\right)\) |
| \(\chi_{2888}(317,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{46}{171}\right)\) | \(e\left(\frac{17}{342}\right)\) | \(e\left(\frac{35}{57}\right)\) | \(e\left(\frac{92}{171}\right)\) | \(e\left(\frac{59}{114}\right)\) | \(e\left(\frac{8}{171}\right)\) | \(e\left(\frac{109}{342}\right)\) | \(e\left(\frac{124}{171}\right)\) | \(e\left(\frac{151}{171}\right)\) | \(e\left(\frac{68}{171}\right)\) |
| \(\chi_{2888}(325,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{137}{171}\right)\) | \(e\left(\frac{151}{342}\right)\) | \(e\left(\frac{46}{57}\right)\) | \(e\left(\frac{103}{171}\right)\) | \(e\left(\frac{1}{114}\right)\) | \(e\left(\frac{61}{171}\right)\) | \(e\left(\frac{83}{342}\right)\) | \(e\left(\frac{5}{171}\right)\) | \(e\left(\frac{104}{171}\right)\) | \(e\left(\frac{91}{171}\right)\) |
| \(\chi_{2888}(357,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{139}{171}\right)\) | \(e\left(\frac{293}{342}\right)\) | \(e\left(\frac{50}{57}\right)\) | \(e\left(\frac{107}{171}\right)\) | \(e\left(\frac{11}{114}\right)\) | \(e\left(\frac{158}{171}\right)\) | \(e\left(\frac{229}{342}\right)\) | \(e\left(\frac{55}{171}\right)\) | \(e\left(\frac{118}{171}\right)\) | \(e\left(\frac{146}{171}\right)\) |
| \(\chi_{2888}(413,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{171}\right)\) | \(e\left(\frac{13}{342}\right)\) | \(e\left(\frac{10}{57}\right)\) | \(e\left(\frac{10}{171}\right)\) | \(e\left(\frac{25}{114}\right)\) | \(e\left(\frac{157}{171}\right)\) | \(e\left(\frac{23}{342}\right)\) | \(e\left(\frac{125}{171}\right)\) | \(e\left(\frac{35}{171}\right)\) | \(e\left(\frac{52}{171}\right)\) |
| \(\chi_{2888}(421,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{171}\right)\) | \(e\left(\frac{181}{342}\right)\) | \(e\left(\frac{34}{57}\right)\) | \(e\left(\frac{34}{171}\right)\) | \(e\left(\frac{85}{114}\right)\) | \(e\left(\frac{55}{171}\right)\) | \(e\left(\frac{215}{342}\right)\) | \(e\left(\frac{83}{171}\right)\) | \(e\left(\frac{119}{171}\right)\) | \(e\left(\frac{40}{171}\right)\) |
| \(\chi_{2888}(469,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{136}{171}\right)\) | \(e\left(\frac{251}{342}\right)\) | \(e\left(\frac{44}{57}\right)\) | \(e\left(\frac{101}{171}\right)\) | \(e\left(\frac{53}{114}\right)\) | \(e\left(\frac{98}{171}\right)\) | \(e\left(\frac{181}{342}\right)\) | \(e\left(\frac{151}{171}\right)\) | \(e\left(\frac{97}{171}\right)\) | \(e\left(\frac{149}{171}\right)\) |
| \(\chi_{2888}(485,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{79}{171}\right)\) | \(e\left(\frac{137}{342}\right)\) | \(e\left(\frac{44}{57}\right)\) | \(e\left(\frac{158}{171}\right)\) | \(e\left(\frac{53}{114}\right)\) | \(e\left(\frac{155}{171}\right)\) | \(e\left(\frac{295}{342}\right)\) | \(e\left(\frac{94}{171}\right)\) | \(e\left(\frac{40}{171}\right)\) | \(e\left(\frac{35}{171}\right)\) |
| \(\chi_{2888}(509,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{103}{171}\right)\) | \(e\left(\frac{131}{342}\right)\) | \(e\left(\frac{35}{57}\right)\) | \(e\left(\frac{35}{171}\right)\) | \(e\left(\frac{59}{114}\right)\) | \(e\left(\frac{122}{171}\right)\) | \(e\left(\frac{337}{342}\right)\) | \(e\left(\frac{10}{171}\right)\) | \(e\left(\frac{37}{171}\right)\) | \(e\left(\frac{11}{171}\right)\) |
| \(\chi_{2888}(565,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{113}{171}\right)\) | \(e\left(\frac{157}{342}\right)\) | \(e\left(\frac{55}{57}\right)\) | \(e\left(\frac{55}{171}\right)\) | \(e\left(\frac{109}{114}\right)\) | \(e\left(\frac{94}{171}\right)\) | \(e\left(\frac{41}{342}\right)\) | \(e\left(\frac{89}{171}\right)\) | \(e\left(\frac{107}{171}\right)\) | \(e\left(\frac{115}{171}\right)\) |
| \(\chi_{2888}(573,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{8}{171}\right)\) | \(e\left(\frac{55}{342}\right)\) | \(e\left(\frac{16}{57}\right)\) | \(e\left(\frac{16}{171}\right)\) | \(e\left(\frac{97}{114}\right)\) | \(e\left(\frac{46}{171}\right)\) | \(e\left(\frac{71}{342}\right)\) | \(e\left(\frac{29}{171}\right)\) | \(e\left(\frac{56}{171}\right)\) | \(e\left(\frac{49}{171}\right)\) |
| \(\chi_{2888}(621,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{55}{171}\right)\) | \(e\left(\frac{143}{342}\right)\) | \(e\left(\frac{53}{57}\right)\) | \(e\left(\frac{110}{171}\right)\) | \(e\left(\frac{47}{114}\right)\) | \(e\left(\frac{17}{171}\right)\) | \(e\left(\frac{253}{342}\right)\) | \(e\left(\frac{7}{171}\right)\) | \(e\left(\frac{43}{171}\right)\) | \(e\left(\frac{59}{171}\right)\) |
| \(\chi_{2888}(629,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{110}{171}\right)\) | \(e\left(\frac{115}{342}\right)\) | \(e\left(\frac{49}{57}\right)\) | \(e\left(\frac{49}{171}\right)\) | \(e\left(\frac{37}{114}\right)\) | \(e\left(\frac{34}{171}\right)\) | \(e\left(\frac{335}{342}\right)\) | \(e\left(\frac{14}{171}\right)\) | \(e\left(\frac{86}{171}\right)\) | \(e\left(\frac{118}{171}\right)\) |
| \(\chi_{2888}(637,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{25}{171}\right)\) | \(e\left(\frac{65}{342}\right)\) | \(e\left(\frac{50}{57}\right)\) | \(e\left(\frac{50}{171}\right)\) | \(e\left(\frac{11}{114}\right)\) | \(e\left(\frac{101}{171}\right)\) | \(e\left(\frac{115}{342}\right)\) | \(e\left(\frac{112}{171}\right)\) | \(e\left(\frac{4}{171}\right)\) | \(e\left(\frac{89}{171}\right)\) |
| \(\chi_{2888}(661,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{67}{171}\right)\) | \(e\left(\frac{311}{342}\right)\) | \(e\left(\frac{20}{57}\right)\) | \(e\left(\frac{134}{171}\right)\) | \(e\left(\frac{107}{114}\right)\) | \(e\left(\frac{86}{171}\right)\) | \(e\left(\frac{103}{342}\right)\) | \(e\left(\frac{136}{171}\right)\) | \(e\left(\frac{127}{171}\right)\) | \(e\left(\frac{47}{171}\right)\) |
| \(\chi_{2888}(717,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{50}{171}\right)\) | \(e\left(\frac{301}{342}\right)\) | \(e\left(\frac{43}{57}\right)\) | \(e\left(\frac{100}{171}\right)\) | \(e\left(\frac{79}{114}\right)\) | \(e\left(\frac{31}{171}\right)\) | \(e\left(\frac{59}{342}\right)\) | \(e\left(\frac{53}{171}\right)\) | \(e\left(\frac{8}{171}\right)\) | \(e\left(\frac{7}{171}\right)\) |
| \(\chi_{2888}(725,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{170}{171}\right)\) | \(e\left(\frac{271}{342}\right)\) | \(e\left(\frac{55}{57}\right)\) | \(e\left(\frac{169}{171}\right)\) | \(e\left(\frac{109}{114}\right)\) | \(e\left(\frac{37}{171}\right)\) | \(e\left(\frac{269}{342}\right)\) | \(e\left(\frac{146}{171}\right)\) | \(e\left(\frac{164}{171}\right)\) | \(e\left(\frac{58}{171}\right)\) |
| \(\chi_{2888}(773,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{145}{171}\right)\) | \(e\left(\frac{35}{342}\right)\) | \(e\left(\frac{5}{57}\right)\) | \(e\left(\frac{119}{171}\right)\) | \(e\left(\frac{41}{114}\right)\) | \(e\left(\frac{107}{171}\right)\) | \(e\left(\frac{325}{342}\right)\) | \(e\left(\frac{34}{171}\right)\) | \(e\left(\frac{160}{171}\right)\) | \(e\left(\frac{140}{171}\right)\) |
| \(\chi_{2888}(781,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{171}\right)\) | \(e\left(\frac{97}{342}\right)\) | \(e\left(\frac{22}{57}\right)\) | \(e\left(\frac{22}{171}\right)\) | \(e\left(\frac{55}{114}\right)\) | \(e\left(\frac{106}{171}\right)\) | \(e\left(\frac{119}{342}\right)\) | \(e\left(\frac{104}{171}\right)\) | \(e\left(\frac{77}{171}\right)\) | \(e\left(\frac{46}{171}\right)\) |
| \(\chi_{2888}(789,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{142}{171}\right)\) | \(e\left(\frac{335}{342}\right)\) | \(e\left(\frac{56}{57}\right)\) | \(e\left(\frac{113}{171}\right)\) | \(e\left(\frac{83}{114}\right)\) | \(e\left(\frac{47}{171}\right)\) | \(e\left(\frac{277}{342}\right)\) | \(e\left(\frac{130}{171}\right)\) | \(e\left(\frac{139}{171}\right)\) | \(e\left(\frac{143}{171}\right)\) |