from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1859, base_ring=CyclotomicField(390))
M = H._module
chi = DirichletCharacter(H, M([273,100]))
chi.galois_orbit()
[g,chi] = znchar(Mod(29,1859))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(1859\) | |
| Conductor: | \(1859\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(390\) | 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: | $\Q(\zeta_{195})$ |
| Fixed field: | Number field defined by a degree 390 polynomial (not computed) |
First 31 of 96 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{1859}(29,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{373}{390}\right)\) | \(e\left(\frac{77}{195}\right)\) | \(e\left(\frac{178}{195}\right)\) | \(e\left(\frac{7}{65}\right)\) | \(e\left(\frac{137}{390}\right)\) | \(e\left(\frac{131}{390}\right)\) | \(e\left(\frac{113}{130}\right)\) | \(e\left(\frac{154}{195}\right)\) | \(e\left(\frac{5}{78}\right)\) | \(e\left(\frac{4}{13}\right)\) |
| \(\chi_{1859}(35,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{329}{390}\right)\) | \(e\left(\frac{1}{195}\right)\) | \(e\left(\frac{134}{195}\right)\) | \(e\left(\frac{6}{65}\right)\) | \(e\left(\frac{331}{390}\right)\) | \(e\left(\frac{103}{390}\right)\) | \(e\left(\frac{69}{130}\right)\) | \(e\left(\frac{2}{195}\right)\) | \(e\left(\frac{73}{78}\right)\) | \(e\left(\frac{9}{13}\right)\) |
| \(\chi_{1859}(61,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{311}{390}\right)\) | \(e\left(\frac{94}{195}\right)\) | \(e\left(\frac{116}{195}\right)\) | \(e\left(\frac{44}{65}\right)\) | \(e\left(\frac{109}{390}\right)\) | \(e\left(\frac{127}{390}\right)\) | \(e\left(\frac{51}{130}\right)\) | \(e\left(\frac{188}{195}\right)\) | \(e\left(\frac{37}{78}\right)\) | \(e\left(\frac{1}{13}\right)\) |
| \(\chi_{1859}(68,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{390}\right)\) | \(e\left(\frac{86}{195}\right)\) | \(e\left(\frac{19}{195}\right)\) | \(e\left(\frac{61}{65}\right)\) | \(e\left(\frac{191}{390}\right)\) | \(e\left(\frac{83}{390}\right)\) | \(e\left(\frac{19}{130}\right)\) | \(e\left(\frac{172}{195}\right)\) | \(e\left(\frac{77}{78}\right)\) | \(e\left(\frac{7}{13}\right)\) |
| \(\chi_{1859}(74,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{107}{390}\right)\) | \(e\left(\frac{43}{195}\right)\) | \(e\left(\frac{107}{195}\right)\) | \(e\left(\frac{63}{65}\right)\) | \(e\left(\frac{193}{390}\right)\) | \(e\left(\frac{139}{390}\right)\) | \(e\left(\frac{107}{130}\right)\) | \(e\left(\frac{86}{195}\right)\) | \(e\left(\frac{19}{78}\right)\) | \(e\left(\frac{10}{13}\right)\) |
| \(\chi_{1859}(94,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{121}{390}\right)\) | \(e\left(\frac{14}{195}\right)\) | \(e\left(\frac{121}{195}\right)\) | \(e\left(\frac{19}{65}\right)\) | \(e\left(\frac{149}{390}\right)\) | \(e\left(\frac{77}{390}\right)\) | \(e\left(\frac{121}{130}\right)\) | \(e\left(\frac{28}{195}\right)\) | \(e\left(\frac{47}{78}\right)\) | \(e\left(\frac{9}{13}\right)\) |
| \(\chi_{1859}(107,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{367}{390}\right)\) | \(e\left(\frac{173}{195}\right)\) | \(e\left(\frac{172}{195}\right)\) | \(e\left(\frac{63}{65}\right)\) | \(e\left(\frac{323}{390}\right)\) | \(e\left(\frac{269}{390}\right)\) | \(e\left(\frac{107}{130}\right)\) | \(e\left(\frac{151}{195}\right)\) | \(e\left(\frac{71}{78}\right)\) | \(e\left(\frac{10}{13}\right)\) |
| \(\chi_{1859}(139,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{390}\right)\) | \(e\left(\frac{22}{195}\right)\) | \(e\left(\frac{23}{195}\right)\) | \(e\left(\frac{2}{65}\right)\) | \(e\left(\frac{67}{390}\right)\) | \(e\left(\frac{121}{390}\right)\) | \(e\left(\frac{23}{130}\right)\) | \(e\left(\frac{44}{195}\right)\) | \(e\left(\frac{7}{78}\right)\) | \(e\left(\frac{3}{13}\right)\) |
| \(\chi_{1859}(172,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{193}{390}\right)\) | \(e\left(\frac{32}{195}\right)\) | \(e\left(\frac{193}{195}\right)\) | \(e\left(\frac{62}{65}\right)\) | \(e\left(\frac{257}{390}\right)\) | \(e\left(\frac{371}{390}\right)\) | \(e\left(\frac{63}{130}\right)\) | \(e\left(\frac{64}{195}\right)\) | \(e\left(\frac{35}{78}\right)\) | \(e\left(\frac{2}{13}\right)\) |
| \(\chi_{1859}(178,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{269}{390}\right)\) | \(e\left(\frac{181}{195}\right)\) | \(e\left(\frac{74}{195}\right)\) | \(e\left(\frac{46}{65}\right)\) | \(e\left(\frac{241}{390}\right)\) | \(e\left(\frac{313}{390}\right)\) | \(e\left(\frac{9}{130}\right)\) | \(e\left(\frac{167}{195}\right)\) | \(e\left(\frac{31}{78}\right)\) | \(e\left(\frac{4}{13}\right)\) |
| \(\chi_{1859}(204,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{251}{390}\right)\) | \(e\left(\frac{79}{195}\right)\) | \(e\left(\frac{56}{195}\right)\) | \(e\left(\frac{19}{65}\right)\) | \(e\left(\frac{19}{390}\right)\) | \(e\left(\frac{337}{390}\right)\) | \(e\left(\frac{121}{130}\right)\) | \(e\left(\frac{158}{195}\right)\) | \(e\left(\frac{73}{78}\right)\) | \(e\left(\frac{9}{13}\right)\) |
| \(\chi_{1859}(211,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{229}{390}\right)\) | \(e\left(\frac{41}{195}\right)\) | \(e\left(\frac{34}{195}\right)\) | \(e\left(\frac{51}{65}\right)\) | \(e\left(\frac{311}{390}\right)\) | \(e\left(\frac{323}{390}\right)\) | \(e\left(\frac{99}{130}\right)\) | \(e\left(\frac{82}{195}\right)\) | \(e\left(\frac{29}{78}\right)\) | \(e\left(\frac{5}{13}\right)\) |
| \(\chi_{1859}(217,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{47}{390}\right)\) | \(e\left(\frac{28}{195}\right)\) | \(e\left(\frac{47}{195}\right)\) | \(e\left(\frac{38}{65}\right)\) | \(e\left(\frac{103}{390}\right)\) | \(e\left(\frac{349}{390}\right)\) | \(e\left(\frac{47}{130}\right)\) | \(e\left(\frac{56}{195}\right)\) | \(e\left(\frac{55}{78}\right)\) | \(e\left(\frac{5}{13}\right)\) |
| \(\chi_{1859}(237,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{331}{390}\right)\) | \(e\left(\frac{164}{195}\right)\) | \(e\left(\frac{136}{195}\right)\) | \(e\left(\frac{9}{65}\right)\) | \(e\left(\frac{269}{390}\right)\) | \(e\left(\frac{317}{390}\right)\) | \(e\left(\frac{71}{130}\right)\) | \(e\left(\frac{133}{195}\right)\) | \(e\left(\frac{77}{78}\right)\) | \(e\left(\frac{7}{13}\right)\) |
| \(\chi_{1859}(250,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{187}{390}\right)\) | \(e\left(\frac{128}{195}\right)\) | \(e\left(\frac{187}{195}\right)\) | \(e\left(\frac{53}{65}\right)\) | \(e\left(\frac{53}{390}\right)\) | \(e\left(\frac{119}{390}\right)\) | \(e\left(\frac{57}{130}\right)\) | \(e\left(\frac{61}{195}\right)\) | \(e\left(\frac{23}{78}\right)\) | \(e\left(\frac{8}{13}\right)\) |
| \(\chi_{1859}(282,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{353}{390}\right)\) | \(e\left(\frac{7}{195}\right)\) | \(e\left(\frac{158}{195}\right)\) | \(e\left(\frac{42}{65}\right)\) | \(e\left(\frac{367}{390}\right)\) | \(e\left(\frac{331}{390}\right)\) | \(e\left(\frac{93}{130}\right)\) | \(e\left(\frac{14}{195}\right)\) | \(e\left(\frac{43}{78}\right)\) | \(e\left(\frac{11}{13}\right)\) |
| \(\chi_{1859}(321,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{209}{390}\right)\) | \(e\left(\frac{166}{195}\right)\) | \(e\left(\frac{14}{195}\right)\) | \(e\left(\frac{21}{65}\right)\) | \(e\left(\frac{151}{390}\right)\) | \(e\left(\frac{133}{390}\right)\) | \(e\left(\frac{79}{130}\right)\) | \(e\left(\frac{137}{195}\right)\) | \(e\left(\frac{67}{78}\right)\) | \(e\left(\frac{12}{13}\right)\) |
| \(\chi_{1859}(347,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{191}{390}\right)\) | \(e\left(\frac{64}{195}\right)\) | \(e\left(\frac{191}{195}\right)\) | \(e\left(\frac{59}{65}\right)\) | \(e\left(\frac{319}{390}\right)\) | \(e\left(\frac{157}{390}\right)\) | \(e\left(\frac{61}{130}\right)\) | \(e\left(\frac{128}{195}\right)\) | \(e\left(\frac{31}{78}\right)\) | \(e\left(\frac{4}{13}\right)\) |
| \(\chi_{1859}(354,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{49}{390}\right)\) | \(e\left(\frac{191}{195}\right)\) | \(e\left(\frac{49}{195}\right)\) | \(e\left(\frac{41}{65}\right)\) | \(e\left(\frac{41}{390}\right)\) | \(e\left(\frac{173}{390}\right)\) | \(e\left(\frac{49}{130}\right)\) | \(e\left(\frac{187}{195}\right)\) | \(e\left(\frac{59}{78}\right)\) | \(e\left(\frac{3}{13}\right)\) |
| \(\chi_{1859}(380,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{151}{390}\right)\) | \(e\left(\frac{119}{195}\right)\) | \(e\left(\frac{151}{195}\right)\) | \(e\left(\frac{64}{65}\right)\) | \(e\left(\frac{389}{390}\right)\) | \(e\left(\frac{167}{390}\right)\) | \(e\left(\frac{21}{130}\right)\) | \(e\left(\frac{43}{195}\right)\) | \(e\left(\frac{29}{78}\right)\) | \(e\left(\frac{5}{13}\right)\) |
| \(\chi_{1859}(393,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{390}\right)\) | \(e\left(\frac{83}{195}\right)\) | \(e\left(\frac{7}{195}\right)\) | \(e\left(\frac{43}{65}\right)\) | \(e\left(\frac{173}{390}\right)\) | \(e\left(\frac{359}{390}\right)\) | \(e\left(\frac{7}{130}\right)\) | \(e\left(\frac{166}{195}\right)\) | \(e\left(\frac{53}{78}\right)\) | \(e\left(\frac{6}{13}\right)\) |
| \(\chi_{1859}(425,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{293}{390}\right)\) | \(e\left(\frac{187}{195}\right)\) | \(e\left(\frac{98}{195}\right)\) | \(e\left(\frac{17}{65}\right)\) | \(e\left(\frac{277}{390}\right)\) | \(e\left(\frac{151}{390}\right)\) | \(e\left(\frac{33}{130}\right)\) | \(e\left(\frac{179}{195}\right)\) | \(e\left(\frac{1}{78}\right)\) | \(e\left(\frac{6}{13}\right)\) |
| \(\chi_{1859}(458,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{223}{390}\right)\) | \(e\left(\frac{137}{195}\right)\) | \(e\left(\frac{28}{195}\right)\) | \(e\left(\frac{42}{65}\right)\) | \(e\left(\frac{107}{390}\right)\) | \(e\left(\frac{71}{390}\right)\) | \(e\left(\frac{93}{130}\right)\) | \(e\left(\frac{79}{195}\right)\) | \(e\left(\frac{17}{78}\right)\) | \(e\left(\frac{11}{13}\right)\) |
| \(\chi_{1859}(464,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{149}{390}\right)\) | \(e\left(\frac{151}{195}\right)\) | \(e\left(\frac{149}{195}\right)\) | \(e\left(\frac{61}{65}\right)\) | \(e\left(\frac{61}{390}\right)\) | \(e\left(\frac{343}{390}\right)\) | \(e\left(\frac{19}{130}\right)\) | \(e\left(\frac{107}{195}\right)\) | \(e\left(\frac{25}{78}\right)\) | \(e\left(\frac{7}{13}\right)\) |
| \(\chi_{1859}(490,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{131}{390}\right)\) | \(e\left(\frac{49}{195}\right)\) | \(e\left(\frac{131}{195}\right)\) | \(e\left(\frac{34}{65}\right)\) | \(e\left(\frac{229}{390}\right)\) | \(e\left(\frac{367}{390}\right)\) | \(e\left(\frac{1}{130}\right)\) | \(e\left(\frac{98}{195}\right)\) | \(e\left(\frac{67}{78}\right)\) | \(e\left(\frac{12}{13}\right)\) |
| \(\chi_{1859}(497,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{259}{390}\right)\) | \(e\left(\frac{146}{195}\right)\) | \(e\left(\frac{64}{195}\right)\) | \(e\left(\frac{31}{65}\right)\) | \(e\left(\frac{161}{390}\right)\) | \(e\left(\frac{23}{390}\right)\) | \(e\left(\frac{129}{130}\right)\) | \(e\left(\frac{97}{195}\right)\) | \(e\left(\frac{11}{78}\right)\) | \(e\left(\frac{1}{13}\right)\) |
| \(\chi_{1859}(503,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{317}{390}\right)\) | \(e\left(\frac{193}{195}\right)\) | \(e\left(\frac{122}{195}\right)\) | \(e\left(\frac{53}{65}\right)\) | \(e\left(\frac{313}{390}\right)\) | \(e\left(\frac{379}{390}\right)\) | \(e\left(\frac{57}{130}\right)\) | \(e\left(\frac{191}{195}\right)\) | \(e\left(\frac{49}{78}\right)\) | \(e\left(\frac{8}{13}\right)\) |
| \(\chi_{1859}(523,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{361}{390}\right)\) | \(e\left(\frac{74}{195}\right)\) | \(e\left(\frac{166}{195}\right)\) | \(e\left(\frac{54}{65}\right)\) | \(e\left(\frac{119}{390}\right)\) | \(e\left(\frac{17}{390}\right)\) | \(e\left(\frac{101}{130}\right)\) | \(e\left(\frac{148}{195}\right)\) | \(e\left(\frac{59}{78}\right)\) | \(e\left(\frac{3}{13}\right)\) |
| \(\chi_{1859}(536,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{217}{390}\right)\) | \(e\left(\frac{38}{195}\right)\) | \(e\left(\frac{22}{195}\right)\) | \(e\left(\frac{33}{65}\right)\) | \(e\left(\frac{293}{390}\right)\) | \(e\left(\frac{209}{390}\right)\) | \(e\left(\frac{87}{130}\right)\) | \(e\left(\frac{76}{195}\right)\) | \(e\left(\frac{5}{78}\right)\) | \(e\left(\frac{4}{13}\right)\) |
| \(\chi_{1859}(568,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{233}{390}\right)\) | \(e\left(\frac{172}{195}\right)\) | \(e\left(\frac{38}{195}\right)\) | \(e\left(\frac{57}{65}\right)\) | \(e\left(\frac{187}{390}\right)\) | \(e\left(\frac{361}{390}\right)\) | \(e\left(\frac{103}{130}\right)\) | \(e\left(\frac{149}{195}\right)\) | \(e\left(\frac{37}{78}\right)\) | \(e\left(\frac{1}{13}\right)\) |
| \(\chi_{1859}(601,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{390}\right)\) | \(e\left(\frac{92}{195}\right)\) | \(e\left(\frac{43}{195}\right)\) | \(e\left(\frac{32}{65}\right)\) | \(e\left(\frac{227}{390}\right)\) | \(e\left(\frac{311}{390}\right)\) | \(e\left(\frac{43}{130}\right)\) | \(e\left(\frac{184}{195}\right)\) | \(e\left(\frac{47}{78}\right)\) | \(e\left(\frac{9}{13}\right)\) |