from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(361, base_ring=CyclotomicField(342))
M = H._module
chi = DirichletCharacter(H, M([2]))
chi.galois_orbit()
[g,chi] = znchar(Mod(4,361))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(361\) | |
| Conductor: | \(361\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(171\) | 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: | $\Q(\zeta_{171})$ |
| Fixed field: | Number field defined by a degree 171 polynomial (not computed) |
First 31 of 108 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{361}(4,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{171}\right)\) | \(e\left(\frac{139}{171}\right)\) | \(e\left(\frac{2}{171}\right)\) | \(e\left(\frac{61}{171}\right)\) | \(e\left(\frac{140}{171}\right)\) | \(e\left(\frac{50}{57}\right)\) | \(e\left(\frac{1}{57}\right)\) | \(e\left(\frac{107}{171}\right)\) | \(e\left(\frac{62}{171}\right)\) | \(e\left(\frac{34}{57}\right)\) |
| \(\chi_{361}(5,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{116}{171}\right)\) | \(e\left(\frac{50}{171}\right)\) | \(e\left(\frac{61}{171}\right)\) | \(e\left(\frac{65}{171}\right)\) | \(e\left(\frac{166}{171}\right)\) | \(e\left(\frac{43}{57}\right)\) | \(e\left(\frac{2}{57}\right)\) | \(e\left(\frac{100}{171}\right)\) | \(e\left(\frac{10}{171}\right)\) | \(e\left(\frac{11}{57}\right)\) |
| \(\chi_{361}(6,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{70}{171}\right)\) | \(e\left(\frac{154}{171}\right)\) | \(e\left(\frac{140}{171}\right)\) | \(e\left(\frac{166}{171}\right)\) | \(e\left(\frac{53}{171}\right)\) | \(e\left(\frac{23}{57}\right)\) | \(e\left(\frac{13}{57}\right)\) | \(e\left(\frac{137}{171}\right)\) | \(e\left(\frac{65}{171}\right)\) | \(e\left(\frac{43}{57}\right)\) |
| \(\chi_{361}(9,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{139}{171}\right)\) | \(e\left(\frac{169}{171}\right)\) | \(e\left(\frac{107}{171}\right)\) | \(e\left(\frac{100}{171}\right)\) | \(e\left(\frac{137}{171}\right)\) | \(e\left(\frac{53}{57}\right)\) | \(e\left(\frac{25}{57}\right)\) | \(e\left(\frac{167}{171}\right)\) | \(e\left(\frac{68}{171}\right)\) | \(e\left(\frac{52}{57}\right)\) |
| \(\chi_{361}(16,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{171}\right)\) | \(e\left(\frac{107}{171}\right)\) | \(e\left(\frac{4}{171}\right)\) | \(e\left(\frac{122}{171}\right)\) | \(e\left(\frac{109}{171}\right)\) | \(e\left(\frac{43}{57}\right)\) | \(e\left(\frac{2}{57}\right)\) | \(e\left(\frac{43}{171}\right)\) | \(e\left(\frac{124}{171}\right)\) | \(e\left(\frac{11}{57}\right)\) |
| \(\chi_{361}(17,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{113}{171}\right)\) | \(e\left(\frac{146}{171}\right)\) | \(e\left(\frac{55}{171}\right)\) | \(e\left(\frac{53}{171}\right)\) | \(e\left(\frac{88}{171}\right)\) | \(e\left(\frac{7}{57}\right)\) | \(e\left(\frac{56}{57}\right)\) | \(e\left(\frac{121}{171}\right)\) | \(e\left(\frac{166}{171}\right)\) | \(e\left(\frac{23}{57}\right)\) |
| \(\chi_{361}(23,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{73}{171}\right)\) | \(e\left(\frac{58}{171}\right)\) | \(e\left(\frac{146}{171}\right)\) | \(e\left(\frac{7}{171}\right)\) | \(e\left(\frac{131}{171}\right)\) | \(e\left(\frac{2}{57}\right)\) | \(e\left(\frac{16}{57}\right)\) | \(e\left(\frac{116}{171}\right)\) | \(e\left(\frac{80}{171}\right)\) | \(e\left(\frac{31}{57}\right)\) |
| \(\chi_{361}(24,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{71}{171}\right)\) | \(e\left(\frac{122}{171}\right)\) | \(e\left(\frac{142}{171}\right)\) | \(e\left(\frac{56}{171}\right)\) | \(e\left(\frac{22}{171}\right)\) | \(e\left(\frac{16}{57}\right)\) | \(e\left(\frac{14}{57}\right)\) | \(e\left(\frac{73}{171}\right)\) | \(e\left(\frac{127}{171}\right)\) | \(e\left(\frac{20}{57}\right)\) |
| \(\chi_{361}(25,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{61}{171}\right)\) | \(e\left(\frac{100}{171}\right)\) | \(e\left(\frac{122}{171}\right)\) | \(e\left(\frac{130}{171}\right)\) | \(e\left(\frac{161}{171}\right)\) | \(e\left(\frac{29}{57}\right)\) | \(e\left(\frac{4}{57}\right)\) | \(e\left(\frac{29}{171}\right)\) | \(e\left(\frac{20}{171}\right)\) | \(e\left(\frac{22}{57}\right)\) |
| \(\chi_{361}(35,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{20}{171}\right)\) | \(e\left(\frac{44}{171}\right)\) | \(e\left(\frac{40}{171}\right)\) | \(e\left(\frac{23}{171}\right)\) | \(e\left(\frac{64}{171}\right)\) | \(e\left(\frac{31}{57}\right)\) | \(e\left(\frac{20}{57}\right)\) | \(e\left(\frac{88}{171}\right)\) | \(e\left(\frac{43}{171}\right)\) | \(e\left(\frac{53}{57}\right)\) |
| \(\chi_{361}(36,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{140}{171}\right)\) | \(e\left(\frac{137}{171}\right)\) | \(e\left(\frac{109}{171}\right)\) | \(e\left(\frac{161}{171}\right)\) | \(e\left(\frac{106}{171}\right)\) | \(e\left(\frac{46}{57}\right)\) | \(e\left(\frac{26}{57}\right)\) | \(e\left(\frac{103}{171}\right)\) | \(e\left(\frac{130}{171}\right)\) | \(e\left(\frac{29}{57}\right)\) |
| \(\chi_{361}(42,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{145}{171}\right)\) | \(e\left(\frac{148}{171}\right)\) | \(e\left(\frac{119}{171}\right)\) | \(e\left(\frac{124}{171}\right)\) | \(e\left(\frac{122}{171}\right)\) | \(e\left(\frac{11}{57}\right)\) | \(e\left(\frac{31}{57}\right)\) | \(e\left(\frac{125}{171}\right)\) | \(e\left(\frac{98}{171}\right)\) | \(e\left(\frac{28}{57}\right)\) |
| \(\chi_{361}(43,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{26}{171}\right)\) | \(e\left(\frac{23}{171}\right)\) | \(e\left(\frac{52}{171}\right)\) | \(e\left(\frac{47}{171}\right)\) | \(e\left(\frac{49}{171}\right)\) | \(e\left(\frac{46}{57}\right)\) | \(e\left(\frac{26}{57}\right)\) | \(e\left(\frac{46}{171}\right)\) | \(e\left(\frac{73}{171}\right)\) | \(e\left(\frac{29}{57}\right)\) |
| \(\chi_{361}(44,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{52}{171}\right)\) | \(e\left(\frac{46}{171}\right)\) | \(e\left(\frac{104}{171}\right)\) | \(e\left(\frac{94}{171}\right)\) | \(e\left(\frac{98}{171}\right)\) | \(e\left(\frac{35}{57}\right)\) | \(e\left(\frac{52}{57}\right)\) | \(e\left(\frac{92}{171}\right)\) | \(e\left(\frac{146}{171}\right)\) | \(e\left(\frac{1}{57}\right)\) |
| \(\chi_{361}(47,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{171}\right)\) | \(e\left(\frac{97}{171}\right)\) | \(e\left(\frac{26}{171}\right)\) | \(e\left(\frac{109}{171}\right)\) | \(e\left(\frac{110}{171}\right)\) | \(e\left(\frac{23}{57}\right)\) | \(e\left(\frac{13}{57}\right)\) | \(e\left(\frac{23}{171}\right)\) | \(e\left(\frac{122}{171}\right)\) | \(e\left(\frac{43}{57}\right)\) |
| \(\chi_{361}(55,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{167}{171}\right)\) | \(e\left(\frac{128}{171}\right)\) | \(e\left(\frac{163}{171}\right)\) | \(e\left(\frac{98}{171}\right)\) | \(e\left(\frac{124}{171}\right)\) | \(e\left(\frac{28}{57}\right)\) | \(e\left(\frac{53}{57}\right)\) | \(e\left(\frac{85}{171}\right)\) | \(e\left(\frac{94}{171}\right)\) | \(e\left(\frac{35}{57}\right)\) |
| \(\chi_{361}(61,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{46}{171}\right)\) | \(e\left(\frac{67}{171}\right)\) | \(e\left(\frac{92}{171}\right)\) | \(e\left(\frac{70}{171}\right)\) | \(e\left(\frac{113}{171}\right)\) | \(e\left(\frac{20}{57}\right)\) | \(e\left(\frac{46}{57}\right)\) | \(e\left(\frac{134}{171}\right)\) | \(e\left(\frac{116}{171}\right)\) | \(e\left(\frac{25}{57}\right)\) |
| \(\chi_{361}(63,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{171}\right)\) | \(e\left(\frac{163}{171}\right)\) | \(e\left(\frac{86}{171}\right)\) | \(e\left(\frac{58}{171}\right)\) | \(e\left(\frac{35}{171}\right)\) | \(e\left(\frac{41}{57}\right)\) | \(e\left(\frac{43}{57}\right)\) | \(e\left(\frac{155}{171}\right)\) | \(e\left(\frac{101}{171}\right)\) | \(e\left(\frac{37}{57}\right)\) |
| \(\chi_{361}(66,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{121}{171}\right)\) | \(e\left(\frac{61}{171}\right)\) | \(e\left(\frac{71}{171}\right)\) | \(e\left(\frac{28}{171}\right)\) | \(e\left(\frac{11}{171}\right)\) | \(e\left(\frac{8}{57}\right)\) | \(e\left(\frac{7}{57}\right)\) | \(e\left(\frac{122}{171}\right)\) | \(e\left(\frac{149}{171}\right)\) | \(e\left(\frac{10}{57}\right)\) |
| \(\chi_{361}(73,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{56}{171}\right)\) | \(e\left(\frac{89}{171}\right)\) | \(e\left(\frac{112}{171}\right)\) | \(e\left(\frac{167}{171}\right)\) | \(e\left(\frac{145}{171}\right)\) | \(e\left(\frac{7}{57}\right)\) | \(e\left(\frac{56}{57}\right)\) | \(e\left(\frac{7}{171}\right)\) | \(e\left(\frac{52}{171}\right)\) | \(e\left(\frac{23}{57}\right)\) |
| \(\chi_{361}(74,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{171}\right)\) | \(e\left(\frac{119}{171}\right)\) | \(e\left(\frac{46}{171}\right)\) | \(e\left(\frac{35}{171}\right)\) | \(e\left(\frac{142}{171}\right)\) | \(e\left(\frac{10}{57}\right)\) | \(e\left(\frac{23}{57}\right)\) | \(e\left(\frac{67}{171}\right)\) | \(e\left(\frac{58}{171}\right)\) | \(e\left(\frac{41}{57}\right)\) |
| \(\chi_{361}(80,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{118}{171}\right)\) | \(e\left(\frac{157}{171}\right)\) | \(e\left(\frac{65}{171}\right)\) | \(e\left(\frac{16}{171}\right)\) | \(e\left(\frac{104}{171}\right)\) | \(e\left(\frac{29}{57}\right)\) | \(e\left(\frac{4}{57}\right)\) | \(e\left(\frac{143}{171}\right)\) | \(e\left(\frac{134}{171}\right)\) | \(e\left(\frac{22}{57}\right)\) |
| \(\chi_{361}(81,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{107}{171}\right)\) | \(e\left(\frac{167}{171}\right)\) | \(e\left(\frac{43}{171}\right)\) | \(e\left(\frac{29}{171}\right)\) | \(e\left(\frac{103}{171}\right)\) | \(e\left(\frac{49}{57}\right)\) | \(e\left(\frac{50}{57}\right)\) | \(e\left(\frac{163}{171}\right)\) | \(e\left(\frac{136}{171}\right)\) | \(e\left(\frac{47}{57}\right)\) |
| \(\chi_{361}(82,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{34}{171}\right)\) | \(e\left(\frac{109}{171}\right)\) | \(e\left(\frac{68}{171}\right)\) | \(e\left(\frac{22}{171}\right)\) | \(e\left(\frac{143}{171}\right)\) | \(e\left(\frac{47}{57}\right)\) | \(e\left(\frac{34}{57}\right)\) | \(e\left(\frac{47}{171}\right)\) | \(e\left(\frac{56}{171}\right)\) | \(e\left(\frac{16}{57}\right)\) |
| \(\chi_{361}(85,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{58}{171}\right)\) | \(e\left(\frac{25}{171}\right)\) | \(e\left(\frac{116}{171}\right)\) | \(e\left(\frac{118}{171}\right)\) | \(e\left(\frac{83}{171}\right)\) | \(e\left(\frac{50}{57}\right)\) | \(e\left(\frac{1}{57}\right)\) | \(e\left(\frac{50}{171}\right)\) | \(e\left(\frac{5}{171}\right)\) | \(e\left(\frac{34}{57}\right)\) |
| \(\chi_{361}(92,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{74}{171}\right)\) | \(e\left(\frac{26}{171}\right)\) | \(e\left(\frac{148}{171}\right)\) | \(e\left(\frac{68}{171}\right)\) | \(e\left(\frac{100}{171}\right)\) | \(e\left(\frac{52}{57}\right)\) | \(e\left(\frac{17}{57}\right)\) | \(e\left(\frac{52}{171}\right)\) | \(e\left(\frac{142}{171}\right)\) | \(e\left(\frac{8}{57}\right)\) |
| \(\chi_{361}(93,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{50}{171}\right)\) | \(e\left(\frac{110}{171}\right)\) | \(e\left(\frac{100}{171}\right)\) | \(e\left(\frac{143}{171}\right)\) | \(e\left(\frac{160}{171}\right)\) | \(e\left(\frac{49}{57}\right)\) | \(e\left(\frac{50}{57}\right)\) | \(e\left(\frac{49}{171}\right)\) | \(e\left(\frac{22}{171}\right)\) | \(e\left(\frac{47}{57}\right)\) |
| \(\chi_{361}(100,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{62}{171}\right)\) | \(e\left(\frac{68}{171}\right)\) | \(e\left(\frac{124}{171}\right)\) | \(e\left(\frac{20}{171}\right)\) | \(e\left(\frac{130}{171}\right)\) | \(e\left(\frac{22}{57}\right)\) | \(e\left(\frac{5}{57}\right)\) | \(e\left(\frac{136}{171}\right)\) | \(e\left(\frac{82}{171}\right)\) | \(e\left(\frac{56}{57}\right)\) |
| \(\chi_{361}(101,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{171}\right)\) | \(e\left(\frac{55}{171}\right)\) | \(e\left(\frac{50}{171}\right)\) | \(e\left(\frac{157}{171}\right)\) | \(e\left(\frac{80}{171}\right)\) | \(e\left(\frac{53}{57}\right)\) | \(e\left(\frac{25}{57}\right)\) | \(e\left(\frac{110}{171}\right)\) | \(e\left(\frac{11}{171}\right)\) | \(e\left(\frac{52}{57}\right)\) |
| \(\chi_{361}(104,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{166}{171}\right)\) | \(e\left(\frac{160}{171}\right)\) | \(e\left(\frac{161}{171}\right)\) | \(e\left(\frac{37}{171}\right)\) | \(e\left(\frac{155}{171}\right)\) | \(e\left(\frac{35}{57}\right)\) | \(e\left(\frac{52}{57}\right)\) | \(e\left(\frac{149}{171}\right)\) | \(e\left(\frac{32}{171}\right)\) | \(e\left(\frac{1}{57}\right)\) |
| \(\chi_{361}(111,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{92}{171}\right)\) | \(e\left(\frac{134}{171}\right)\) | \(e\left(\frac{13}{171}\right)\) | \(e\left(\frac{140}{171}\right)\) | \(e\left(\frac{55}{171}\right)\) | \(e\left(\frac{40}{57}\right)\) | \(e\left(\frac{35}{57}\right)\) | \(e\left(\frac{97}{171}\right)\) | \(e\left(\frac{61}{171}\right)\) | \(e\left(\frac{50}{57}\right)\) |