from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2001, base_ring=CyclotomicField(308))
M = H._module
chi = DirichletCharacter(H, M([0,42,253]))
chi.galois_orbit()
[g,chi] = znchar(Mod(10,2001))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(2001\) | |
| Conductor: | \(667\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(308\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 667.x | 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: | $\Q(\zeta_{308})$ |
| Fixed field: | Number field defined by a degree 308 polynomial (not computed) |
First 31 of 120 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{2001}(10,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{308}\right)\) | \(e\left(\frac{29}{154}\right)\) | \(e\left(\frac{16}{77}\right)\) | \(e\left(\frac{69}{154}\right)\) | \(e\left(\frac{87}{308}\right)\) | \(e\left(\frac{93}{308}\right)\) | \(e\left(\frac{235}{308}\right)\) | \(e\left(\frac{107}{154}\right)\) | \(e\left(\frac{167}{308}\right)\) | \(e\left(\frac{29}{77}\right)\) |
| \(\chi_{2001}(19,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{211}{308}\right)\) | \(e\left(\frac{57}{154}\right)\) | \(e\left(\frac{58}{77}\right)\) | \(e\left(\frac{125}{154}\right)\) | \(e\left(\frac{17}{308}\right)\) | \(e\left(\frac{135}{308}\right)\) | \(e\left(\frac{53}{308}\right)\) | \(e\left(\frac{51}{154}\right)\) | \(e\left(\frac{153}{308}\right)\) | \(e\left(\frac{57}{77}\right)\) |
| \(\chi_{2001}(37,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{308}\right)\) | \(e\left(\frac{5}{154}\right)\) | \(e\left(\frac{24}{77}\right)\) | \(e\left(\frac{65}{154}\right)\) | \(e\left(\frac{15}{308}\right)\) | \(e\left(\frac{101}{308}\right)\) | \(e\left(\frac{83}{308}\right)\) | \(e\left(\frac{45}{154}\right)\) | \(e\left(\frac{135}{308}\right)\) | \(e\left(\frac{5}{77}\right)\) |
| \(\chi_{2001}(40,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{163}{308}\right)\) | \(e\left(\frac{9}{154}\right)\) | \(e\left(\frac{74}{77}\right)\) | \(e\left(\frac{117}{154}\right)\) | \(e\left(\frac{181}{308}\right)\) | \(e\left(\frac{151}{308}\right)\) | \(e\left(\frac{57}{308}\right)\) | \(e\left(\frac{81}{154}\right)\) | \(e\left(\frac{89}{308}\right)\) | \(e\left(\frac{9}{77}\right)\) |
| \(\chi_{2001}(43,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{283}{308}\right)\) | \(e\left(\frac{129}{154}\right)\) | \(e\left(\frac{34}{77}\right)\) | \(e\left(\frac{137}{154}\right)\) | \(e\left(\frac{233}{308}\right)\) | \(e\left(\frac{111}{308}\right)\) | \(e\left(\frac{201}{308}\right)\) | \(e\left(\frac{83}{154}\right)\) | \(e\left(\frac{249}{308}\right)\) | \(e\left(\frac{52}{77}\right)\) |
| \(\chi_{2001}(61,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{223}{308}\right)\) | \(e\left(\frac{69}{154}\right)\) | \(e\left(\frac{54}{77}\right)\) | \(e\left(\frac{127}{154}\right)\) | \(e\left(\frac{53}{308}\right)\) | \(e\left(\frac{131}{308}\right)\) | \(e\left(\frac{129}{308}\right)\) | \(e\left(\frac{5}{154}\right)\) | \(e\left(\frac{169}{308}\right)\) | \(e\left(\frac{69}{77}\right)\) |
| \(\chi_{2001}(76,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{308}\right)\) | \(e\left(\frac{37}{154}\right)\) | \(e\left(\frac{39}{77}\right)\) | \(e\left(\frac{19}{154}\right)\) | \(e\left(\frac{111}{308}\right)\) | \(e\left(\frac{193}{308}\right)\) | \(e\left(\frac{183}{308}\right)\) | \(e\left(\frac{25}{154}\right)\) | \(e\left(\frac{75}{308}\right)\) | \(e\left(\frac{37}{77}\right)\) |
| \(\chi_{2001}(79,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{271}{308}\right)\) | \(e\left(\frac{117}{154}\right)\) | \(e\left(\frac{38}{77}\right)\) | \(e\left(\frac{135}{154}\right)\) | \(e\left(\frac{197}{308}\right)\) | \(e\left(\frac{115}{308}\right)\) | \(e\left(\frac{125}{308}\right)\) | \(e\left(\frac{129}{154}\right)\) | \(e\left(\frac{233}{308}\right)\) | \(e\left(\frac{40}{77}\right)\) |
| \(\chi_{2001}(97,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{281}{308}\right)\) | \(e\left(\frac{127}{154}\right)\) | \(e\left(\frac{9}{77}\right)\) | \(e\left(\frac{111}{154}\right)\) | \(e\left(\frac{227}{308}\right)\) | \(e\left(\frac{9}{308}\right)\) | \(e\left(\frac{291}{308}\right)\) | \(e\left(\frac{65}{154}\right)\) | \(e\left(\frac{195}{308}\right)\) | \(e\left(\frac{50}{77}\right)\) |
| \(\chi_{2001}(106,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{71}{308}\right)\) | \(e\left(\frac{71}{154}\right)\) | \(e\left(\frac{2}{77}\right)\) | \(e\left(\frac{153}{154}\right)\) | \(e\left(\frac{213}{308}\right)\) | \(e\left(\frac{79}{308}\right)\) | \(e\left(\frac{193}{308}\right)\) | \(e\left(\frac{23}{154}\right)\) | \(e\left(\frac{69}{308}\right)\) | \(e\left(\frac{71}{77}\right)\) |
| \(\chi_{2001}(130,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{308}\right)\) | \(e\left(\frac{3}{154}\right)\) | \(e\left(\frac{76}{77}\right)\) | \(e\left(\frac{39}{154}\right)\) | \(e\left(\frac{9}{308}\right)\) | \(e\left(\frac{307}{308}\right)\) | \(e\left(\frac{173}{308}\right)\) | \(e\left(\frac{27}{154}\right)\) | \(e\left(\frac{81}{308}\right)\) | \(e\left(\frac{3}{77}\right)\) |
| \(\chi_{2001}(148,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{139}{308}\right)\) | \(e\left(\frac{139}{154}\right)\) | \(e\left(\frac{5}{77}\right)\) | \(e\left(\frac{113}{154}\right)\) | \(e\left(\frac{109}{308}\right)\) | \(e\left(\frac{159}{308}\right)\) | \(e\left(\frac{213}{308}\right)\) | \(e\left(\frac{19}{154}\right)\) | \(e\left(\frac{57}{308}\right)\) | \(e\left(\frac{62}{77}\right)\) |
| \(\chi_{2001}(166,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{215}{308}\right)\) | \(e\left(\frac{61}{154}\right)\) | \(e\left(\frac{31}{77}\right)\) | \(e\left(\frac{23}{154}\right)\) | \(e\left(\frac{29}{308}\right)\) | \(e\left(\frac{31}{308}\right)\) | \(e\left(\frac{181}{308}\right)\) | \(e\left(\frac{87}{154}\right)\) | \(e\left(\frac{261}{308}\right)\) | \(e\left(\frac{61}{77}\right)\) |
| \(\chi_{2001}(172,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{109}{308}\right)\) | \(e\left(\frac{109}{154}\right)\) | \(e\left(\frac{15}{77}\right)\) | \(e\left(\frac{31}{154}\right)\) | \(e\left(\frac{19}{308}\right)\) | \(e\left(\frac{169}{308}\right)\) | \(e\left(\frac{23}{308}\right)\) | \(e\left(\frac{57}{154}\right)\) | \(e\left(\frac{171}{308}\right)\) | \(e\left(\frac{32}{77}\right)\) |
| \(\chi_{2001}(205,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{67}{308}\right)\) | \(e\left(\frac{67}{154}\right)\) | \(e\left(\frac{29}{77}\right)\) | \(e\left(\frac{101}{154}\right)\) | \(e\left(\frac{201}{308}\right)\) | \(e\left(\frac{183}{308}\right)\) | \(e\left(\frac{65}{308}\right)\) | \(e\left(\frac{141}{154}\right)\) | \(e\left(\frac{269}{308}\right)\) | \(e\left(\frac{67}{77}\right)\) |
| \(\chi_{2001}(214,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{191}{308}\right)\) | \(e\left(\frac{37}{154}\right)\) | \(e\left(\frac{39}{77}\right)\) | \(e\left(\frac{19}{154}\right)\) | \(e\left(\frac{265}{308}\right)\) | \(e\left(\frac{39}{308}\right)\) | \(e\left(\frac{29}{308}\right)\) | \(e\left(\frac{25}{154}\right)\) | \(e\left(\frac{229}{308}\right)\) | \(e\left(\frac{37}{77}\right)\) |
| \(\chi_{2001}(217,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{227}{308}\right)\) | \(e\left(\frac{73}{154}\right)\) | \(e\left(\frac{27}{77}\right)\) | \(e\left(\frac{25}{154}\right)\) | \(e\left(\frac{65}{308}\right)\) | \(e\left(\frac{27}{308}\right)\) | \(e\left(\frac{257}{308}\right)\) | \(e\left(\frac{41}{154}\right)\) | \(e\left(\frac{277}{308}\right)\) | \(e\left(\frac{73}{77}\right)\) |
| \(\chi_{2001}(235,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{83}{308}\right)\) | \(e\left(\frac{83}{154}\right)\) | \(e\left(\frac{75}{77}\right)\) | \(e\left(\frac{1}{154}\right)\) | \(e\left(\frac{249}{308}\right)\) | \(e\left(\frac{75}{308}\right)\) | \(e\left(\frac{269}{308}\right)\) | \(e\left(\frac{131}{154}\right)\) | \(e\left(\frac{85}{308}\right)\) | \(e\left(\frac{6}{77}\right)\) |
| \(\chi_{2001}(247,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{185}{308}\right)\) | \(e\left(\frac{31}{154}\right)\) | \(e\left(\frac{41}{77}\right)\) | \(e\left(\frac{95}{154}\right)\) | \(e\left(\frac{247}{308}\right)\) | \(e\left(\frac{41}{308}\right)\) | \(e\left(\frac{299}{308}\right)\) | \(e\left(\frac{125}{154}\right)\) | \(e\left(\frac{67}{308}\right)\) | \(e\left(\frac{31}{77}\right)\) |
| \(\chi_{2001}(250,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{261}{308}\right)\) | \(e\left(\frac{107}{154}\right)\) | \(e\left(\frac{67}{77}\right)\) | \(e\left(\frac{5}{154}\right)\) | \(e\left(\frac{167}{308}\right)\) | \(e\left(\frac{221}{308}\right)\) | \(e\left(\frac{267}{308}\right)\) | \(e\left(\frac{39}{154}\right)\) | \(e\left(\frac{271}{308}\right)\) | \(e\left(\frac{30}{77}\right)\) |
| \(\chi_{2001}(304,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{171}{308}\right)\) | \(e\left(\frac{17}{154}\right)\) | \(e\left(\frac{20}{77}\right)\) | \(e\left(\frac{67}{154}\right)\) | \(e\left(\frac{205}{308}\right)\) | \(e\left(\frac{251}{308}\right)\) | \(e\left(\frac{5}{308}\right)\) | \(e\left(\frac{153}{154}\right)\) | \(e\left(\frac{305}{308}\right)\) | \(e\left(\frac{17}{77}\right)\) |
| \(\chi_{2001}(316,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{97}{308}\right)\) | \(e\left(\frac{97}{154}\right)\) | \(e\left(\frac{19}{77}\right)\) | \(e\left(\frac{29}{154}\right)\) | \(e\left(\frac{291}{308}\right)\) | \(e\left(\frac{173}{308}\right)\) | \(e\left(\frac{255}{308}\right)\) | \(e\left(\frac{103}{154}\right)\) | \(e\left(\frac{155}{308}\right)\) | \(e\left(\frac{20}{77}\right)\) |
| \(\chi_{2001}(337,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{289}{308}\right)\) | \(e\left(\frac{135}{154}\right)\) | \(e\left(\frac{32}{77}\right)\) | \(e\left(\frac{61}{154}\right)\) | \(e\left(\frac{251}{308}\right)\) | \(e\left(\frac{109}{308}\right)\) | \(e\left(\frac{239}{308}\right)\) | \(e\left(\frac{137}{154}\right)\) | \(e\left(\frac{103}{308}\right)\) | \(e\left(\frac{58}{77}\right)\) |
| \(\chi_{2001}(379,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{263}{308}\right)\) | \(e\left(\frac{109}{154}\right)\) | \(e\left(\frac{15}{77}\right)\) | \(e\left(\frac{31}{154}\right)\) | \(e\left(\frac{173}{308}\right)\) | \(e\left(\frac{15}{308}\right)\) | \(e\left(\frac{177}{308}\right)\) | \(e\left(\frac{57}{154}\right)\) | \(e\left(\frac{17}{308}\right)\) | \(e\left(\frac{32}{77}\right)\) |
| \(\chi_{2001}(385,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{229}{308}\right)\) | \(e\left(\frac{75}{154}\right)\) | \(e\left(\frac{52}{77}\right)\) | \(e\left(\frac{51}{154}\right)\) | \(e\left(\frac{71}{308}\right)\) | \(e\left(\frac{129}{308}\right)\) | \(e\left(\frac{167}{308}\right)\) | \(e\left(\frac{59}{154}\right)\) | \(e\left(\frac{23}{308}\right)\) | \(e\left(\frac{75}{77}\right)\) |
| \(\chi_{2001}(388,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{107}{308}\right)\) | \(e\left(\frac{107}{154}\right)\) | \(e\left(\frac{67}{77}\right)\) | \(e\left(\frac{5}{154}\right)\) | \(e\left(\frac{13}{308}\right)\) | \(e\left(\frac{67}{308}\right)\) | \(e\left(\frac{113}{308}\right)\) | \(e\left(\frac{39}{154}\right)\) | \(e\left(\frac{117}{308}\right)\) | \(e\left(\frac{30}{77}\right)\) |
| \(\chi_{2001}(421,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{213}{308}\right)\) | \(e\left(\frac{59}{154}\right)\) | \(e\left(\frac{6}{77}\right)\) | \(e\left(\frac{151}{154}\right)\) | \(e\left(\frac{23}{308}\right)\) | \(e\left(\frac{237}{308}\right)\) | \(e\left(\frac{271}{308}\right)\) | \(e\left(\frac{69}{154}\right)\) | \(e\left(\frac{207}{308}\right)\) | \(e\left(\frac{59}{77}\right)\) |
| \(\chi_{2001}(424,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{205}{308}\right)\) | \(e\left(\frac{51}{154}\right)\) | \(e\left(\frac{60}{77}\right)\) | \(e\left(\frac{47}{154}\right)\) | \(e\left(\frac{307}{308}\right)\) | \(e\left(\frac{137}{308}\right)\) | \(e\left(\frac{15}{308}\right)\) | \(e\left(\frac{151}{154}\right)\) | \(e\left(\frac{299}{308}\right)\) | \(e\left(\frac{51}{77}\right)\) |
| \(\chi_{2001}(433,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{277}{308}\right)\) | \(e\left(\frac{123}{154}\right)\) | \(e\left(\frac{36}{77}\right)\) | \(e\left(\frac{59}{154}\right)\) | \(e\left(\frac{215}{308}\right)\) | \(e\left(\frac{113}{308}\right)\) | \(e\left(\frac{163}{308}\right)\) | \(e\left(\frac{29}{154}\right)\) | \(e\left(\frac{87}{308}\right)\) | \(e\left(\frac{46}{77}\right)\) |
| \(\chi_{2001}(454,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{295}{308}\right)\) | \(e\left(\frac{141}{154}\right)\) | \(e\left(\frac{30}{77}\right)\) | \(e\left(\frac{139}{154}\right)\) | \(e\left(\frac{269}{308}\right)\) | \(e\left(\frac{107}{308}\right)\) | \(e\left(\frac{277}{308}\right)\) | \(e\left(\frac{37}{154}\right)\) | \(e\left(\frac{265}{308}\right)\) | \(e\left(\frac{64}{77}\right)\) |
| \(\chi_{2001}(475,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{135}{308}\right)\) | \(e\left(\frac{135}{154}\right)\) | \(e\left(\frac{32}{77}\right)\) | \(e\left(\frac{61}{154}\right)\) | \(e\left(\frac{97}{308}\right)\) | \(e\left(\frac{263}{308}\right)\) | \(e\left(\frac{85}{308}\right)\) | \(e\left(\frac{137}{154}\right)\) | \(e\left(\frac{257}{308}\right)\) | \(e\left(\frac{58}{77}\right)\) |