from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8281, base_ring=CyclotomicField(364))
M = H._module
chi = DirichletCharacter(H, M([312,7]))
chi.galois_orbit()
[g,chi] = znchar(Mod(8,8281))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(8281\) | |
| Conductor: | \(8281\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(364\) | 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: | $\Q(\zeta_{364})$ |
| Fixed field: | Number field defined by a degree 364 polynomial (not computed) |
First 31 of 144 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{8281}(8,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{111}{364}\right)\) | \(e\left(\frac{22}{91}\right)\) | \(e\left(\frac{111}{182}\right)\) | \(e\left(\frac{11}{364}\right)\) | \(e\left(\frac{199}{364}\right)\) | \(e\left(\frac{333}{364}\right)\) | \(e\left(\frac{44}{91}\right)\) | \(e\left(\frac{61}{182}\right)\) | \(e\left(\frac{97}{364}\right)\) | \(e\left(\frac{155}{182}\right)\) |
| \(\chi_{8281}(57,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{181}{364}\right)\) | \(e\left(\frac{8}{91}\right)\) | \(e\left(\frac{181}{182}\right)\) | \(e\left(\frac{277}{364}\right)\) | \(e\left(\frac{213}{364}\right)\) | \(e\left(\frac{179}{364}\right)\) | \(e\left(\frac{16}{91}\right)\) | \(e\left(\frac{47}{182}\right)\) | \(e\left(\frac{27}{364}\right)\) | \(e\left(\frac{15}{182}\right)\) |
| \(\chi_{8281}(190,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{71}{364}\right)\) | \(e\left(\frac{69}{91}\right)\) | \(e\left(\frac{71}{182}\right)\) | \(e\left(\frac{171}{364}\right)\) | \(e\left(\frac{347}{364}\right)\) | \(e\left(\frac{213}{364}\right)\) | \(e\left(\frac{47}{91}\right)\) | \(e\left(\frac{121}{182}\right)\) | \(e\left(\frac{85}{364}\right)\) | \(e\left(\frac{27}{182}\right)\) |
| \(\chi_{8281}(281,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{51}{364}\right)\) | \(e\left(\frac{47}{91}\right)\) | \(e\left(\frac{51}{182}\right)\) | \(e\left(\frac{251}{364}\right)\) | \(e\left(\frac{239}{364}\right)\) | \(e\left(\frac{153}{364}\right)\) | \(e\left(\frac{3}{91}\right)\) | \(e\left(\frac{151}{182}\right)\) | \(e\left(\frac{261}{364}\right)\) | \(e\left(\frac{145}{182}\right)\) |
| \(\chi_{8281}(330,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{345}{364}\right)\) | \(e\left(\frac{61}{91}\right)\) | \(e\left(\frac{163}{182}\right)\) | \(e\left(\frac{349}{364}\right)\) | \(e\left(\frac{225}{364}\right)\) | \(e\left(\frac{307}{364}\right)\) | \(e\left(\frac{31}{91}\right)\) | \(e\left(\frac{165}{182}\right)\) | \(e\left(\frac{331}{364}\right)\) | \(e\left(\frac{103}{182}\right)\) |
| \(\chi_{8281}(372,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{364}\right)\) | \(e\left(\frac{25}{91}\right)\) | \(e\left(\frac{31}{182}\right)\) | \(e\left(\frac{331}{364}\right)\) | \(e\left(\frac{131}{364}\right)\) | \(e\left(\frac{93}{364}\right)\) | \(e\left(\frac{50}{91}\right)\) | \(e\left(\frac{181}{182}\right)\) | \(e\left(\frac{73}{364}\right)\) | \(e\left(\frac{81}{182}\right)\) |
| \(\chi_{8281}(421,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{157}{364}\right)\) | \(e\left(\frac{18}{91}\right)\) | \(e\left(\frac{157}{182}\right)\) | \(e\left(\frac{9}{364}\right)\) | \(e\left(\frac{229}{364}\right)\) | \(e\left(\frac{107}{364}\right)\) | \(e\left(\frac{36}{91}\right)\) | \(e\left(\frac{83}{182}\right)\) | \(e\left(\frac{311}{364}\right)\) | \(e\left(\frac{11}{182}\right)\) |
| \(\chi_{8281}(463,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{364}\right)\) | \(e\left(\frac{3}{91}\right)\) | \(e\left(\frac{11}{182}\right)\) | \(e\left(\frac{47}{364}\right)\) | \(e\left(\frac{23}{364}\right)\) | \(e\left(\frac{33}{364}\right)\) | \(e\left(\frac{6}{91}\right)\) | \(e\left(\frac{29}{182}\right)\) | \(e\left(\frac{249}{364}\right)\) | \(e\left(\frac{17}{182}\right)\) |
| \(\chi_{8281}(512,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{333}{364}\right)\) | \(e\left(\frac{66}{91}\right)\) | \(e\left(\frac{151}{182}\right)\) | \(e\left(\frac{33}{364}\right)\) | \(e\left(\frac{233}{364}\right)\) | \(e\left(\frac{271}{364}\right)\) | \(e\left(\frac{41}{91}\right)\) | \(e\left(\frac{1}{182}\right)\) | \(e\left(\frac{291}{364}\right)\) | \(e\left(\frac{101}{182}\right)\) |
| \(\chi_{8281}(554,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{355}{364}\right)\) | \(e\left(\frac{72}{91}\right)\) | \(e\left(\frac{173}{182}\right)\) | \(e\left(\frac{127}{364}\right)\) | \(e\left(\frac{279}{364}\right)\) | \(e\left(\frac{337}{364}\right)\) | \(e\left(\frac{53}{91}\right)\) | \(e\left(\frac{59}{182}\right)\) | \(e\left(\frac{61}{364}\right)\) | \(e\left(\frac{135}{182}\right)\) |
| \(\chi_{8281}(603,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{145}{364}\right)\) | \(e\left(\frac{23}{91}\right)\) | \(e\left(\frac{145}{182}\right)\) | \(e\left(\frac{57}{364}\right)\) | \(e\left(\frac{237}{364}\right)\) | \(e\left(\frac{71}{364}\right)\) | \(e\left(\frac{46}{91}\right)\) | \(e\left(\frac{101}{182}\right)\) | \(e\left(\frac{271}{364}\right)\) | \(e\left(\frac{9}{182}\right)\) |
| \(\chi_{8281}(645,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{335}{364}\right)\) | \(e\left(\frac{50}{91}\right)\) | \(e\left(\frac{153}{182}\right)\) | \(e\left(\frac{207}{364}\right)\) | \(e\left(\frac{171}{364}\right)\) | \(e\left(\frac{277}{364}\right)\) | \(e\left(\frac{9}{91}\right)\) | \(e\left(\frac{89}{182}\right)\) | \(e\left(\frac{237}{364}\right)\) | \(e\left(\frac{71}{182}\right)\) |
| \(\chi_{8281}(694,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{321}{364}\right)\) | \(e\left(\frac{71}{91}\right)\) | \(e\left(\frac{139}{182}\right)\) | \(e\left(\frac{81}{364}\right)\) | \(e\left(\frac{241}{364}\right)\) | \(e\left(\frac{235}{364}\right)\) | \(e\left(\frac{51}{91}\right)\) | \(e\left(\frac{19}{182}\right)\) | \(e\left(\frac{251}{364}\right)\) | \(e\left(\frac{99}{182}\right)\) |
| \(\chi_{8281}(827,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{295}{364}\right)\) | \(e\left(\frac{6}{91}\right)\) | \(e\left(\frac{113}{182}\right)\) | \(e\left(\frac{3}{364}\right)\) | \(e\left(\frac{319}{364}\right)\) | \(e\left(\frac{157}{364}\right)\) | \(e\left(\frac{12}{91}\right)\) | \(e\left(\frac{149}{182}\right)\) | \(e\left(\frac{225}{364}\right)\) | \(e\left(\frac{125}{182}\right)\) |
| \(\chi_{8281}(876,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{309}{364}\right)\) | \(e\left(\frac{76}{91}\right)\) | \(e\left(\frac{127}{182}\right)\) | \(e\left(\frac{129}{364}\right)\) | \(e\left(\frac{249}{364}\right)\) | \(e\left(\frac{199}{364}\right)\) | \(e\left(\frac{61}{91}\right)\) | \(e\left(\frac{37}{182}\right)\) | \(e\left(\frac{211}{364}\right)\) | \(e\left(\frac{97}{182}\right)\) |
| \(\chi_{8281}(918,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{275}{364}\right)\) | \(e\left(\frac{75}{91}\right)\) | \(e\left(\frac{93}{182}\right)\) | \(e\left(\frac{83}{364}\right)\) | \(e\left(\frac{211}{364}\right)\) | \(e\left(\frac{97}{364}\right)\) | \(e\left(\frac{59}{91}\right)\) | \(e\left(\frac{179}{182}\right)\) | \(e\left(\frac{37}{364}\right)\) | \(e\left(\frac{61}{182}\right)\) |
| \(\chi_{8281}(967,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{121}{364}\right)\) | \(e\left(\frac{33}{91}\right)\) | \(e\left(\frac{121}{182}\right)\) | \(e\left(\frac{153}{364}\right)\) | \(e\left(\frac{253}{364}\right)\) | \(e\left(\frac{363}{364}\right)\) | \(e\left(\frac{66}{91}\right)\) | \(e\left(\frac{137}{182}\right)\) | \(e\left(\frac{191}{364}\right)\) | \(e\left(\frac{5}{182}\right)\) |
| \(\chi_{8281}(1009,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{255}{364}\right)\) | \(e\left(\frac{53}{91}\right)\) | \(e\left(\frac{73}{182}\right)\) | \(e\left(\frac{163}{364}\right)\) | \(e\left(\frac{103}{364}\right)\) | \(e\left(\frac{37}{364}\right)\) | \(e\left(\frac{15}{91}\right)\) | \(e\left(\frac{27}{182}\right)\) | \(e\left(\frac{213}{364}\right)\) | \(e\left(\frac{179}{182}\right)\) |
| \(\chi_{8281}(1058,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{297}{364}\right)\) | \(e\left(\frac{81}{91}\right)\) | \(e\left(\frac{115}{182}\right)\) | \(e\left(\frac{177}{364}\right)\) | \(e\left(\frac{257}{364}\right)\) | \(e\left(\frac{163}{364}\right)\) | \(e\left(\frac{71}{91}\right)\) | \(e\left(\frac{55}{182}\right)\) | \(e\left(\frac{171}{364}\right)\) | \(e\left(\frac{95}{182}\right)\) |
| \(\chi_{8281}(1100,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{235}{364}\right)\) | \(e\left(\frac{31}{91}\right)\) | \(e\left(\frac{53}{182}\right)\) | \(e\left(\frac{243}{364}\right)\) | \(e\left(\frac{359}{364}\right)\) | \(e\left(\frac{341}{364}\right)\) | \(e\left(\frac{62}{91}\right)\) | \(e\left(\frac{57}{182}\right)\) | \(e\left(\frac{25}{364}\right)\) | \(e\left(\frac{115}{182}\right)\) |
| \(\chi_{8281}(1149,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{109}{364}\right)\) | \(e\left(\frac{38}{91}\right)\) | \(e\left(\frac{109}{182}\right)\) | \(e\left(\frac{201}{364}\right)\) | \(e\left(\frac{261}{364}\right)\) | \(e\left(\frac{327}{364}\right)\) | \(e\left(\frac{76}{91}\right)\) | \(e\left(\frac{155}{182}\right)\) | \(e\left(\frac{151}{364}\right)\) | \(e\left(\frac{3}{182}\right)\) |
| \(\chi_{8281}(1191,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{215}{364}\right)\) | \(e\left(\frac{9}{91}\right)\) | \(e\left(\frac{33}{182}\right)\) | \(e\left(\frac{323}{364}\right)\) | \(e\left(\frac{251}{364}\right)\) | \(e\left(\frac{281}{364}\right)\) | \(e\left(\frac{18}{91}\right)\) | \(e\left(\frac{87}{182}\right)\) | \(e\left(\frac{201}{364}\right)\) | \(e\left(\frac{51}{182}\right)\) |
| \(\chi_{8281}(1240,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{285}{364}\right)\) | \(e\left(\frac{86}{91}\right)\) | \(e\left(\frac{103}{182}\right)\) | \(e\left(\frac{225}{364}\right)\) | \(e\left(\frac{265}{364}\right)\) | \(e\left(\frac{127}{364}\right)\) | \(e\left(\frac{81}{91}\right)\) | \(e\left(\frac{73}{182}\right)\) | \(e\left(\frac{131}{364}\right)\) | \(e\left(\frac{93}{182}\right)\) |
| \(\chi_{8281}(1331,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{97}{364}\right)\) | \(e\left(\frac{43}{91}\right)\) | \(e\left(\frac{97}{182}\right)\) | \(e\left(\frac{249}{364}\right)\) | \(e\left(\frac{269}{364}\right)\) | \(e\left(\frac{291}{364}\right)\) | \(e\left(\frac{86}{91}\right)\) | \(e\left(\frac{173}{182}\right)\) | \(e\left(\frac{111}{364}\right)\) | \(e\left(\frac{1}{182}\right)\) |
| \(\chi_{8281}(1464,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{155}{364}\right)\) | \(e\left(\frac{34}{91}\right)\) | \(e\left(\frac{155}{182}\right)\) | \(e\left(\frac{199}{364}\right)\) | \(e\left(\frac{291}{364}\right)\) | \(e\left(\frac{101}{364}\right)\) | \(e\left(\frac{68}{91}\right)\) | \(e\left(\frac{177}{182}\right)\) | \(e\left(\frac{1}{364}\right)\) | \(e\left(\frac{41}{182}\right)\) |
| \(\chi_{8281}(1513,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{85}{364}\right)\) | \(e\left(\frac{48}{91}\right)\) | \(e\left(\frac{85}{182}\right)\) | \(e\left(\frac{297}{364}\right)\) | \(e\left(\frac{277}{364}\right)\) | \(e\left(\frac{255}{364}\right)\) | \(e\left(\frac{5}{91}\right)\) | \(e\left(\frac{9}{182}\right)\) | \(e\left(\frac{71}{364}\right)\) | \(e\left(\frac{181}{182}\right)\) |
| \(\chi_{8281}(1555,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{135}{364}\right)\) | \(e\left(\frac{12}{91}\right)\) | \(e\left(\frac{135}{182}\right)\) | \(e\left(\frac{279}{364}\right)\) | \(e\left(\frac{183}{364}\right)\) | \(e\left(\frac{41}{364}\right)\) | \(e\left(\frac{24}{91}\right)\) | \(e\left(\frac{25}{182}\right)\) | \(e\left(\frac{177}{364}\right)\) | \(e\left(\frac{159}{182}\right)\) |
| \(\chi_{8281}(1604,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{261}{364}\right)\) | \(e\left(\frac{5}{91}\right)\) | \(e\left(\frac{79}{182}\right)\) | \(e\left(\frac{321}{364}\right)\) | \(e\left(\frac{281}{364}\right)\) | \(e\left(\frac{55}{364}\right)\) | \(e\left(\frac{10}{91}\right)\) | \(e\left(\frac{109}{182}\right)\) | \(e\left(\frac{51}{364}\right)\) | \(e\left(\frac{89}{182}\right)\) |
| \(\chi_{8281}(1646,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{115}{364}\right)\) | \(e\left(\frac{81}{91}\right)\) | \(e\left(\frac{115}{182}\right)\) | \(e\left(\frac{359}{364}\right)\) | \(e\left(\frac{75}{364}\right)\) | \(e\left(\frac{345}{364}\right)\) | \(e\left(\frac{71}{91}\right)\) | \(e\left(\frac{55}{182}\right)\) | \(e\left(\frac{353}{364}\right)\) | \(e\left(\frac{95}{182}\right)\) |
| \(\chi_{8281}(1695,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{73}{364}\right)\) | \(e\left(\frac{53}{91}\right)\) | \(e\left(\frac{73}{182}\right)\) | \(e\left(\frac{345}{364}\right)\) | \(e\left(\frac{285}{364}\right)\) | \(e\left(\frac{219}{364}\right)\) | \(e\left(\frac{15}{91}\right)\) | \(e\left(\frac{27}{182}\right)\) | \(e\left(\frac{31}{364}\right)\) | \(e\left(\frac{179}{182}\right)\) |
| \(\chi_{8281}(1737,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{95}{364}\right)\) | \(e\left(\frac{59}{91}\right)\) | \(e\left(\frac{95}{182}\right)\) | \(e\left(\frac{75}{364}\right)\) | \(e\left(\frac{331}{364}\right)\) | \(e\left(\frac{285}{364}\right)\) | \(e\left(\frac{27}{91}\right)\) | \(e\left(\frac{85}{182}\right)\) | \(e\left(\frac{165}{364}\right)\) | \(e\left(\frac{31}{182}\right)\) |