from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(239, base_ring=CyclotomicField(238))
M = H._module
chi = DirichletCharacter(H, M([1]))
chi.galois_orbit()
[g,chi] = znchar(Mod(7,239))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(239\) | |
| Conductor: | \(239\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(238\) | 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: | $\Q(\zeta_{119})$ |
| Fixed field: | Number field defined by a degree 238 polynomial (not computed) |
First 31 of 96 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{239}(7,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{33}{119}\right)\) | \(e\left(\frac{37}{119}\right)\) | \(e\left(\frac{66}{119}\right)\) | \(e\left(\frac{69}{119}\right)\) | \(e\left(\frac{10}{17}\right)\) | \(e\left(\frac{1}{238}\right)\) | \(e\left(\frac{99}{119}\right)\) | \(e\left(\frac{74}{119}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{2}{119}\right)\) |
| \(\chi_{239}(13,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{110}{119}\right)\) | \(e\left(\frac{44}{119}\right)\) | \(e\left(\frac{101}{119}\right)\) | \(e\left(\frac{111}{119}\right)\) | \(e\left(\frac{5}{17}\right)\) | \(e\left(\frac{43}{238}\right)\) | \(e\left(\frac{92}{119}\right)\) | \(e\left(\frac{88}{119}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{86}{119}\right)\) |
| \(\chi_{239}(14,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{69}{119}\right)\) | \(e\left(\frac{99}{119}\right)\) | \(e\left(\frac{19}{119}\right)\) | \(e\left(\frac{101}{119}\right)\) | \(e\left(\frac{7}{17}\right)\) | \(e\left(\frac{67}{238}\right)\) | \(e\left(\frac{88}{119}\right)\) | \(e\left(\frac{79}{119}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{15}{119}\right)\) |
| \(\chi_{239}(19,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{117}{119}\right)\) | \(e\left(\frac{23}{119}\right)\) | \(e\left(\frac{115}{119}\right)\) | \(e\left(\frac{104}{119}\right)\) | \(e\left(\frac{3}{17}\right)\) | \(e\left(\frac{155}{238}\right)\) | \(e\left(\frac{113}{119}\right)\) | \(e\left(\frac{46}{119}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{72}{119}\right)\) |
| \(\chi_{239}(21,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{95}{119}\right)\) | \(e\left(\frac{38}{119}\right)\) | \(e\left(\frac{71}{119}\right)\) | \(e\left(\frac{58}{119}\right)\) | \(e\left(\frac{2}{17}\right)\) | \(e\left(\frac{75}{238}\right)\) | \(e\left(\frac{47}{119}\right)\) | \(e\left(\frac{76}{119}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{31}{119}\right)\) |
| \(\chi_{239}(26,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{27}{119}\right)\) | \(e\left(\frac{106}{119}\right)\) | \(e\left(\frac{54}{119}\right)\) | \(e\left(\frac{24}{119}\right)\) | \(e\left(\frac{2}{17}\right)\) | \(e\left(\frac{109}{238}\right)\) | \(e\left(\frac{81}{119}\right)\) | \(e\left(\frac{93}{119}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{99}{119}\right)\) |
| \(\chi_{239}(35,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{65}{119}\right)\) | \(e\left(\frac{26}{119}\right)\) | \(e\left(\frac{11}{119}\right)\) | \(e\left(\frac{71}{119}\right)\) | \(e\left(\frac{13}{17}\right)\) | \(e\left(\frac{139}{238}\right)\) | \(e\left(\frac{76}{119}\right)\) | \(e\left(\frac{52}{119}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{40}{119}\right)\) |
| \(\chi_{239}(37,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{71}{119}\right)\) | \(e\left(\frac{76}{119}\right)\) | \(e\left(\frac{23}{119}\right)\) | \(e\left(\frac{116}{119}\right)\) | \(e\left(\frac{4}{17}\right)\) | \(e\left(\frac{31}{238}\right)\) | \(e\left(\frac{94}{119}\right)\) | \(e\left(\frac{33}{119}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{62}{119}\right)\) |
| \(\chi_{239}(39,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{53}{119}\right)\) | \(e\left(\frac{45}{119}\right)\) | \(e\left(\frac{106}{119}\right)\) | \(e\left(\frac{100}{119}\right)\) | \(e\left(\frac{14}{17}\right)\) | \(e\left(\frac{117}{238}\right)\) | \(e\left(\frac{40}{119}\right)\) | \(e\left(\frac{90}{119}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{115}{119}\right)\) |
| \(\chi_{239}(41,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{54}{119}\right)\) | \(e\left(\frac{93}{119}\right)\) | \(e\left(\frac{108}{119}\right)\) | \(e\left(\frac{48}{119}\right)\) | \(e\left(\frac{4}{17}\right)\) | \(e\left(\frac{99}{238}\right)\) | \(e\left(\frac{43}{119}\right)\) | \(e\left(\frac{67}{119}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{79}{119}\right)\) |
| \(\chi_{239}(42,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{12}{119}\right)\) | \(e\left(\frac{100}{119}\right)\) | \(e\left(\frac{24}{119}\right)\) | \(e\left(\frac{90}{119}\right)\) | \(e\left(\frac{16}{17}\right)\) | \(e\left(\frac{141}{238}\right)\) | \(e\left(\frac{36}{119}\right)\) | \(e\left(\frac{81}{119}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{44}{119}\right)\) |
| \(\chi_{239}(43,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{119}\right)\) | \(e\left(\frac{79}{119}\right)\) | \(e\left(\frac{38}{119}\right)\) | \(e\left(\frac{83}{119}\right)\) | \(e\left(\frac{14}{17}\right)\) | \(e\left(\frac{15}{238}\right)\) | \(e\left(\frac{57}{119}\right)\) | \(e\left(\frac{39}{119}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{30}{119}\right)\) |
| \(\chi_{239}(46,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{92}{119}\right)\) | \(e\left(\frac{13}{119}\right)\) | \(e\left(\frac{65}{119}\right)\) | \(e\left(\frac{95}{119}\right)\) | \(e\left(\frac{15}{17}\right)\) | \(e\left(\frac{129}{238}\right)\) | \(e\left(\frac{38}{119}\right)\) | \(e\left(\frac{26}{119}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{20}{119}\right)\) |
| \(\chi_{239}(47,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{40}{119}\right)\) | \(e\left(\frac{16}{119}\right)\) | \(e\left(\frac{80}{119}\right)\) | \(e\left(\frac{62}{119}\right)\) | \(e\left(\frac{8}{17}\right)\) | \(e\left(\frac{113}{238}\right)\) | \(e\left(\frac{1}{119}\right)\) | \(e\left(\frac{32}{119}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{107}{119}\right)\) |
| \(\chi_{239}(53,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{114}{119}\right)\) | \(e\left(\frac{117}{119}\right)\) | \(e\left(\frac{109}{119}\right)\) | \(e\left(\frac{22}{119}\right)\) | \(e\left(\frac{16}{17}\right)\) | \(e\left(\frac{209}{238}\right)\) | \(e\left(\frac{104}{119}\right)\) | \(e\left(\frac{115}{119}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{61}{119}\right)\) |
| \(\chi_{239}(56,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{22}{119}\right)\) | \(e\left(\frac{104}{119}\right)\) | \(e\left(\frac{44}{119}\right)\) | \(e\left(\frac{46}{119}\right)\) | \(e\left(\frac{1}{17}\right)\) | \(e\left(\frac{199}{238}\right)\) | \(e\left(\frac{66}{119}\right)\) | \(e\left(\frac{89}{119}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{41}{119}\right)\) |
| \(\chi_{239}(57,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{60}{119}\right)\) | \(e\left(\frac{24}{119}\right)\) | \(e\left(\frac{1}{119}\right)\) | \(e\left(\frac{93}{119}\right)\) | \(e\left(\frac{12}{17}\right)\) | \(e\left(\frac{229}{238}\right)\) | \(e\left(\frac{61}{119}\right)\) | \(e\left(\frac{48}{119}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{101}{119}\right)\) |
| \(\chi_{239}(59,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{109}{119}\right)\) | \(e\left(\frac{115}{119}\right)\) | \(e\left(\frac{99}{119}\right)\) | \(e\left(\frac{44}{119}\right)\) | \(e\left(\frac{15}{17}\right)\) | \(e\left(\frac{61}{238}\right)\) | \(e\left(\frac{89}{119}\right)\) | \(e\left(\frac{111}{119}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{3}{119}\right)\) |
| \(\chi_{239}(63,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{38}{119}\right)\) | \(e\left(\frac{39}{119}\right)\) | \(e\left(\frac{76}{119}\right)\) | \(e\left(\frac{47}{119}\right)\) | \(e\left(\frac{11}{17}\right)\) | \(e\left(\frac{149}{238}\right)\) | \(e\left(\frac{114}{119}\right)\) | \(e\left(\frac{78}{119}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{60}{119}\right)\) |
| \(\chi_{239}(65,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{119}\right)\) | \(e\left(\frac{33}{119}\right)\) | \(e\left(\frac{46}{119}\right)\) | \(e\left(\frac{113}{119}\right)\) | \(e\left(\frac{8}{17}\right)\) | \(e\left(\frac{181}{238}\right)\) | \(e\left(\frac{69}{119}\right)\) | \(e\left(\frac{66}{119}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{5}{119}\right)\) |
| \(\chi_{239}(69,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{118}{119}\right)\) | \(e\left(\frac{71}{119}\right)\) | \(e\left(\frac{117}{119}\right)\) | \(e\left(\frac{52}{119}\right)\) | \(e\left(\frac{10}{17}\right)\) | \(e\left(\frac{137}{238}\right)\) | \(e\left(\frac{116}{119}\right)\) | \(e\left(\frac{23}{119}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{36}{119}\right)\) |
| \(\chi_{239}(70,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{101}{119}\right)\) | \(e\left(\frac{88}{119}\right)\) | \(e\left(\frac{83}{119}\right)\) | \(e\left(\frac{103}{119}\right)\) | \(e\left(\frac{10}{17}\right)\) | \(e\left(\frac{205}{238}\right)\) | \(e\left(\frac{65}{119}\right)\) | \(e\left(\frac{57}{119}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{53}{119}\right)\) |
| \(\chi_{239}(74,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{107}{119}\right)\) | \(e\left(\frac{19}{119}\right)\) | \(e\left(\frac{95}{119}\right)\) | \(e\left(\frac{29}{119}\right)\) | \(e\left(\frac{1}{17}\right)\) | \(e\left(\frac{97}{238}\right)\) | \(e\left(\frac{83}{119}\right)\) | \(e\left(\frac{38}{119}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{75}{119}\right)\) |
| \(\chi_{239}(77,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{46}{119}\right)\) | \(e\left(\frac{66}{119}\right)\) | \(e\left(\frac{92}{119}\right)\) | \(e\left(\frac{107}{119}\right)\) | \(e\left(\frac{16}{17}\right)\) | \(e\left(\frac{5}{238}\right)\) | \(e\left(\frac{19}{119}\right)\) | \(e\left(\frac{13}{119}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{10}{119}\right)\) |
| \(\chi_{239}(78,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{89}{119}\right)\) | \(e\left(\frac{107}{119}\right)\) | \(e\left(\frac{59}{119}\right)\) | \(e\left(\frac{13}{119}\right)\) | \(e\left(\frac{11}{17}\right)\) | \(e\left(\frac{183}{238}\right)\) | \(e\left(\frac{29}{119}\right)\) | \(e\left(\frac{95}{119}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{9}{119}\right)\) |
| \(\chi_{239}(79,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{93}{119}\right)\) | \(e\left(\frac{61}{119}\right)\) | \(e\left(\frac{67}{119}\right)\) | \(e\left(\frac{43}{119}\right)\) | \(e\left(\frac{5}{17}\right)\) | \(e\left(\frac{111}{238}\right)\) | \(e\left(\frac{41}{119}\right)\) | \(e\left(\frac{3}{119}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{103}{119}\right)\) |
| \(\chi_{239}(82,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{90}{119}\right)\) | \(e\left(\frac{36}{119}\right)\) | \(e\left(\frac{61}{119}\right)\) | \(e\left(\frac{80}{119}\right)\) | \(e\left(\frac{1}{17}\right)\) | \(e\left(\frac{165}{238}\right)\) | \(e\left(\frac{32}{119}\right)\) | \(e\left(\frac{72}{119}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{92}{119}\right)\) |
| \(\chi_{239}(84,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{48}{119}\right)\) | \(e\left(\frac{43}{119}\right)\) | \(e\left(\frac{96}{119}\right)\) | \(e\left(\frac{3}{119}\right)\) | \(e\left(\frac{13}{17}\right)\) | \(e\left(\frac{207}{238}\right)\) | \(e\left(\frac{25}{119}\right)\) | \(e\left(\frac{86}{119}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{57}{119}\right)\) |
| \(\chi_{239}(86,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{55}{119}\right)\) | \(e\left(\frac{22}{119}\right)\) | \(e\left(\frac{110}{119}\right)\) | \(e\left(\frac{115}{119}\right)\) | \(e\left(\frac{11}{17}\right)\) | \(e\left(\frac{81}{238}\right)\) | \(e\left(\frac{46}{119}\right)\) | \(e\left(\frac{44}{119}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{43}{119}\right)\) |
| \(\chi_{239}(89,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{119}\right)\) | \(e\left(\frac{41}{119}\right)\) | \(e\left(\frac{86}{119}\right)\) | \(e\left(\frac{25}{119}\right)\) | \(e\left(\frac{12}{17}\right)\) | \(e\left(\frac{59}{238}\right)\) | \(e\left(\frac{10}{119}\right)\) | \(e\left(\frac{82}{119}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{118}{119}\right)\) |
| \(\chi_{239}(92,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{9}{119}\right)\) | \(e\left(\frac{75}{119}\right)\) | \(e\left(\frac{18}{119}\right)\) | \(e\left(\frac{8}{119}\right)\) | \(e\left(\frac{12}{17}\right)\) | \(e\left(\frac{195}{238}\right)\) | \(e\left(\frac{27}{119}\right)\) | \(e\left(\frac{31}{119}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{33}{119}\right)\) |