from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2011, base_ring=CyclotomicField(402))
M = H._module
chi = DirichletCharacter(H, M([98]))
chi.galois_orbit()
[g,chi] = znchar(Mod(4,2011))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(2011\) | |
| Conductor: | \(2011\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(201\) | 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: | $\Q(\zeta_{201})$ |
| Fixed field: | Number field defined by a degree 201 polynomial (not computed) |
First 31 of 132 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{2011}(4,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{146}{201}\right)\) | \(e\left(\frac{49}{201}\right)\) | \(e\left(\frac{91}{201}\right)\) | \(e\left(\frac{127}{201}\right)\) | \(e\left(\frac{65}{67}\right)\) | \(e\left(\frac{103}{201}\right)\) | \(e\left(\frac{12}{67}\right)\) | \(e\left(\frac{98}{201}\right)\) | \(e\left(\frac{24}{67}\right)\) | \(e\left(\frac{161}{201}\right)\) |
| \(\chi_{2011}(16,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{91}{201}\right)\) | \(e\left(\frac{98}{201}\right)\) | \(e\left(\frac{182}{201}\right)\) | \(e\left(\frac{53}{201}\right)\) | \(e\left(\frac{63}{67}\right)\) | \(e\left(\frac{5}{201}\right)\) | \(e\left(\frac{24}{67}\right)\) | \(e\left(\frac{196}{201}\right)\) | \(e\left(\frac{48}{67}\right)\) | \(e\left(\frac{121}{201}\right)\) |
| \(\chi_{2011}(30,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{161}{201}\right)\) | \(e\left(\frac{127}{201}\right)\) | \(e\left(\frac{121}{201}\right)\) | \(e\left(\frac{1}{201}\right)\) | \(e\left(\frac{29}{67}\right)\) | \(e\left(\frac{148}{201}\right)\) | \(e\left(\frac{27}{67}\right)\) | \(e\left(\frac{53}{201}\right)\) | \(e\left(\frac{54}{67}\right)\) | \(e\left(\frac{44}{201}\right)\) |
| \(\chi_{2011}(115,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{10}{201}\right)\) | \(e\left(\frac{119}{201}\right)\) | \(e\left(\frac{20}{201}\right)\) | \(e\left(\frac{50}{201}\right)\) | \(e\left(\frac{43}{67}\right)\) | \(e\left(\frac{164}{201}\right)\) | \(e\left(\frac{10}{67}\right)\) | \(e\left(\frac{37}{201}\right)\) | \(e\left(\frac{20}{67}\right)\) | \(e\left(\frac{190}{201}\right)\) |
| \(\chi_{2011}(117,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{94}{201}\right)\) | \(e\left(\frac{194}{201}\right)\) | \(e\left(\frac{188}{201}\right)\) | \(e\left(\frac{68}{201}\right)\) | \(e\left(\frac{29}{67}\right)\) | \(e\left(\frac{14}{201}\right)\) | \(e\left(\frac{27}{67}\right)\) | \(e\left(\frac{187}{201}\right)\) | \(e\left(\frac{54}{67}\right)\) | \(e\left(\frac{178}{201}\right)\) |
| \(\chi_{2011}(120,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{106}{201}\right)\) | \(e\left(\frac{176}{201}\right)\) | \(e\left(\frac{11}{201}\right)\) | \(e\left(\frac{128}{201}\right)\) | \(e\left(\frac{27}{67}\right)\) | \(e\left(\frac{50}{201}\right)\) | \(e\left(\frac{39}{67}\right)\) | \(e\left(\frac{151}{201}\right)\) | \(e\left(\frac{11}{67}\right)\) | \(e\left(\frac{4}{201}\right)\) |
| \(\chi_{2011}(134,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{113}{201}\right)\) | \(e\left(\frac{199}{201}\right)\) | \(e\left(\frac{25}{201}\right)\) | \(e\left(\frac{163}{201}\right)\) | \(e\left(\frac{37}{67}\right)\) | \(e\left(\frac{4}{201}\right)\) | \(e\left(\frac{46}{67}\right)\) | \(e\left(\frac{197}{201}\right)\) | \(e\left(\frac{25}{67}\right)\) | \(e\left(\frac{137}{201}\right)\) |
| \(\chi_{2011}(189,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{125}{201}\right)\) | \(e\left(\frac{181}{201}\right)\) | \(e\left(\frac{49}{201}\right)\) | \(e\left(\frac{22}{201}\right)\) | \(e\left(\frac{35}{67}\right)\) | \(e\left(\frac{40}{201}\right)\) | \(e\left(\frac{58}{67}\right)\) | \(e\left(\frac{161}{201}\right)\) | \(e\left(\frac{49}{67}\right)\) | \(e\left(\frac{164}{201}\right)\) |
| \(\chi_{2011}(193,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{62}{201}\right)\) | \(e\left(\frac{175}{201}\right)\) | \(e\left(\frac{124}{201}\right)\) | \(e\left(\frac{109}{201}\right)\) | \(e\left(\frac{12}{67}\right)\) | \(e\left(\frac{52}{201}\right)\) | \(e\left(\frac{62}{67}\right)\) | \(e\left(\frac{149}{201}\right)\) | \(e\left(\frac{57}{67}\right)\) | \(e\left(\frac{173}{201}\right)\) |
| \(\chi_{2011}(194,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{115}{201}\right)\) | \(e\left(\frac{62}{201}\right)\) | \(e\left(\frac{29}{201}\right)\) | \(e\left(\frac{173}{201}\right)\) | \(e\left(\frac{59}{67}\right)\) | \(e\left(\frac{77}{201}\right)\) | \(e\left(\frac{48}{67}\right)\) | \(e\left(\frac{124}{201}\right)\) | \(e\left(\frac{29}{67}\right)\) | \(e\left(\frac{175}{201}\right)\) |
| \(\chi_{2011}(209,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{32}{201}\right)\) | \(e\left(\frac{19}{201}\right)\) | \(e\left(\frac{64}{201}\right)\) | \(e\left(\frac{160}{201}\right)\) | \(e\left(\frac{17}{67}\right)\) | \(e\left(\frac{163}{201}\right)\) | \(e\left(\frac{32}{67}\right)\) | \(e\left(\frac{38}{201}\right)\) | \(e\left(\frac{64}{67}\right)\) | \(e\left(\frac{5}{201}\right)\) |
| \(\chi_{2011}(211,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{194}{201}\right)\) | \(e\left(\frac{178}{201}\right)\) | \(e\left(\frac{187}{201}\right)\) | \(e\left(\frac{166}{201}\right)\) | \(e\left(\frac{57}{67}\right)\) | \(e\left(\frac{46}{201}\right)\) | \(e\left(\frac{60}{67}\right)\) | \(e\left(\frac{155}{201}\right)\) | \(e\left(\frac{53}{67}\right)\) | \(e\left(\frac{68}{201}\right)\) |
| \(\chi_{2011}(225,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{176}{201}\right)\) | \(e\left(\frac{4}{201}\right)\) | \(e\left(\frac{151}{201}\right)\) | \(e\left(\frac{76}{201}\right)\) | \(e\left(\frac{60}{67}\right)\) | \(e\left(\frac{193}{201}\right)\) | \(e\left(\frac{42}{67}\right)\) | \(e\left(\frac{8}{201}\right)\) | \(e\left(\frac{17}{67}\right)\) | \(e\left(\frac{128}{201}\right)\) |
| \(\chi_{2011}(237,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{193}{201}\right)\) | \(e\left(\frac{146}{201}\right)\) | \(e\left(\frac{185}{201}\right)\) | \(e\left(\frac{161}{201}\right)\) | \(e\left(\frac{46}{67}\right)\) | \(e\left(\frac{110}{201}\right)\) | \(e\left(\frac{59}{67}\right)\) | \(e\left(\frac{91}{201}\right)\) | \(e\left(\frac{51}{67}\right)\) | \(e\left(\frac{49}{201}\right)\) |
| \(\chi_{2011}(256,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{182}{201}\right)\) | \(e\left(\frac{196}{201}\right)\) | \(e\left(\frac{163}{201}\right)\) | \(e\left(\frac{106}{201}\right)\) | \(e\left(\frac{59}{67}\right)\) | \(e\left(\frac{10}{201}\right)\) | \(e\left(\frac{48}{67}\right)\) | \(e\left(\frac{191}{201}\right)\) | \(e\left(\frac{29}{67}\right)\) | \(e\left(\frac{41}{201}\right)\) |
| \(\chi_{2011}(279,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{187}{201}\right)\) | \(e\left(\frac{155}{201}\right)\) | \(e\left(\frac{173}{201}\right)\) | \(e\left(\frac{131}{201}\right)\) | \(e\left(\frac{47}{67}\right)\) | \(e\left(\frac{92}{201}\right)\) | \(e\left(\frac{53}{67}\right)\) | \(e\left(\frac{109}{201}\right)\) | \(e\left(\frac{39}{67}\right)\) | \(e\left(\frac{136}{201}\right)\) |
| \(\chi_{2011}(286,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{98}{201}\right)\) | \(e\left(\frac{121}{201}\right)\) | \(e\left(\frac{196}{201}\right)\) | \(e\left(\frac{88}{201}\right)\) | \(e\left(\frac{6}{67}\right)\) | \(e\left(\frac{160}{201}\right)\) | \(e\left(\frac{31}{67}\right)\) | \(e\left(\frac{41}{201}\right)\) | \(e\left(\frac{62}{67}\right)\) | \(e\left(\frac{53}{201}\right)\) |
| \(\chi_{2011}(294,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{100}{201}\right)\) | \(e\left(\frac{185}{201}\right)\) | \(e\left(\frac{200}{201}\right)\) | \(e\left(\frac{98}{201}\right)\) | \(e\left(\frac{28}{67}\right)\) | \(e\left(\frac{32}{201}\right)\) | \(e\left(\frac{33}{67}\right)\) | \(e\left(\frac{169}{201}\right)\) | \(e\left(\frac{66}{67}\right)\) | \(e\left(\frac{91}{201}\right)\) |
| \(\chi_{2011}(296,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{201}\right)\) | \(e\left(\frac{142}{201}\right)\) | \(e\left(\frac{34}{201}\right)\) | \(e\left(\frac{85}{201}\right)\) | \(e\left(\frac{53}{67}\right)\) | \(e\left(\frac{118}{201}\right)\) | \(e\left(\frac{17}{67}\right)\) | \(e\left(\frac{83}{201}\right)\) | \(e\left(\frac{34}{67}\right)\) | \(e\left(\frac{122}{201}\right)\) |
| \(\chi_{2011}(297,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{154}{201}\right)\) | \(e\left(\frac{104}{201}\right)\) | \(e\left(\frac{107}{201}\right)\) | \(e\left(\frac{167}{201}\right)\) | \(e\left(\frac{19}{67}\right)\) | \(e\left(\frac{194}{201}\right)\) | \(e\left(\frac{20}{67}\right)\) | \(e\left(\frac{7}{201}\right)\) | \(e\left(\frac{40}{67}\right)\) | \(e\left(\frac{112}{201}\right)\) |
| \(\chi_{2011}(311,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{201}\right)\) | \(e\left(\frac{188}{201}\right)\) | \(e\left(\frac{62}{201}\right)\) | \(e\left(\frac{155}{201}\right)\) | \(e\left(\frac{6}{67}\right)\) | \(e\left(\frac{26}{201}\right)\) | \(e\left(\frac{31}{67}\right)\) | \(e\left(\frac{175}{201}\right)\) | \(e\left(\frac{62}{67}\right)\) | \(e\left(\frac{187}{201}\right)\) |
| \(\chi_{2011}(323,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{201}\right)\) | \(e\left(\frac{151}{201}\right)\) | \(e\left(\frac{22}{201}\right)\) | \(e\left(\frac{55}{201}\right)\) | \(e\left(\frac{54}{67}\right)\) | \(e\left(\frac{100}{201}\right)\) | \(e\left(\frac{11}{67}\right)\) | \(e\left(\frac{101}{201}\right)\) | \(e\left(\frac{22}{67}\right)\) | \(e\left(\frac{8}{201}\right)\) |
| \(\chi_{2011}(339,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{50}{201}\right)\) | \(e\left(\frac{193}{201}\right)\) | \(e\left(\frac{100}{201}\right)\) | \(e\left(\frac{49}{201}\right)\) | \(e\left(\frac{14}{67}\right)\) | \(e\left(\frac{16}{201}\right)\) | \(e\left(\frac{50}{67}\right)\) | \(e\left(\frac{185}{201}\right)\) | \(e\left(\frac{33}{67}\right)\) | \(e\left(\frac{146}{201}\right)\) |
| \(\chi_{2011}(350,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{151}{201}\right)\) | \(e\left(\frac{8}{201}\right)\) | \(e\left(\frac{101}{201}\right)\) | \(e\left(\frac{152}{201}\right)\) | \(e\left(\frac{53}{67}\right)\) | \(e\left(\frac{185}{201}\right)\) | \(e\left(\frac{17}{67}\right)\) | \(e\left(\frac{16}{201}\right)\) | \(e\left(\frac{34}{67}\right)\) | \(e\left(\frac{55}{201}\right)\) |
| \(\chi_{2011}(371,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{152}{201}\right)\) | \(e\left(\frac{40}{201}\right)\) | \(e\left(\frac{103}{201}\right)\) | \(e\left(\frac{157}{201}\right)\) | \(e\left(\frac{64}{67}\right)\) | \(e\left(\frac{121}{201}\right)\) | \(e\left(\frac{18}{67}\right)\) | \(e\left(\frac{80}{201}\right)\) | \(e\left(\frac{36}{67}\right)\) | \(e\left(\frac{74}{201}\right)\) |
| \(\chi_{2011}(412,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{140}{201}\right)\) | \(e\left(\frac{58}{201}\right)\) | \(e\left(\frac{79}{201}\right)\) | \(e\left(\frac{97}{201}\right)\) | \(e\left(\frac{66}{67}\right)\) | \(e\left(\frac{85}{201}\right)\) | \(e\left(\frac{6}{67}\right)\) | \(e\left(\frac{116}{201}\right)\) | \(e\left(\frac{12}{67}\right)\) | \(e\left(\frac{47}{201}\right)\) |
| \(\chi_{2011}(426,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{148}{201}\right)\) | \(e\left(\frac{113}{201}\right)\) | \(e\left(\frac{95}{201}\right)\) | \(e\left(\frac{137}{201}\right)\) | \(e\left(\frac{20}{67}\right)\) | \(e\left(\frac{176}{201}\right)\) | \(e\left(\frac{14}{67}\right)\) | \(e\left(\frac{25}{201}\right)\) | \(e\left(\frac{28}{67}\right)\) | \(e\left(\frac{199}{201}\right)\) |
| \(\chi_{2011}(442,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{77}{201}\right)\) | \(e\left(\frac{52}{201}\right)\) | \(e\left(\frac{154}{201}\right)\) | \(e\left(\frac{184}{201}\right)\) | \(e\left(\frac{43}{67}\right)\) | \(e\left(\frac{97}{201}\right)\) | \(e\left(\frac{10}{67}\right)\) | \(e\left(\frac{104}{201}\right)\) | \(e\left(\frac{20}{67}\right)\) | \(e\left(\frac{56}{201}\right)\) |
| \(\chi_{2011}(459,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{133}{201}\right)\) | \(e\left(\frac{35}{201}\right)\) | \(e\left(\frac{65}{201}\right)\) | \(e\left(\frac{62}{201}\right)\) | \(e\left(\frac{56}{67}\right)\) | \(e\left(\frac{131}{201}\right)\) | \(e\left(\frac{66}{67}\right)\) | \(e\left(\frac{70}{201}\right)\) | \(e\left(\frac{65}{67}\right)\) | \(e\left(\frac{115}{201}\right)\) |
| \(\chi_{2011}(466,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{82}{201}\right)\) | \(e\left(\frac{11}{201}\right)\) | \(e\left(\frac{164}{201}\right)\) | \(e\left(\frac{8}{201}\right)\) | \(e\left(\frac{31}{67}\right)\) | \(e\left(\frac{179}{201}\right)\) | \(e\left(\frac{15}{67}\right)\) | \(e\left(\frac{22}{201}\right)\) | \(e\left(\frac{30}{67}\right)\) | \(e\left(\frac{151}{201}\right)\) |
| \(\chi_{2011}(467,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{80}{201}\right)\) | \(e\left(\frac{148}{201}\right)\) | \(e\left(\frac{160}{201}\right)\) | \(e\left(\frac{199}{201}\right)\) | \(e\left(\frac{9}{67}\right)\) | \(e\left(\frac{106}{201}\right)\) | \(e\left(\frac{13}{67}\right)\) | \(e\left(\frac{95}{201}\right)\) | \(e\left(\frac{26}{67}\right)\) | \(e\left(\frac{113}{201}\right)\) |