from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2001, base_ring=CyclotomicField(154))
M = H._module
chi = DirichletCharacter(H, M([77,140,33]))
chi.galois_orbit()
[g,chi] = znchar(Mod(35,2001))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(2001\) | |
| Conductor: | \(2001\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(154\) | 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: | $\Q(\zeta_{77})$ |
| Fixed field: | Number field defined by a degree 154 polynomial (not computed) |
First 31 of 60 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{2001}(35,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{41}{77}\right)\) | \(e\left(\frac{5}{77}\right)\) | \(e\left(\frac{19}{154}\right)\) | \(e\left(\frac{65}{77}\right)\) | \(e\left(\frac{46}{77}\right)\) | \(e\left(\frac{101}{154}\right)\) | \(e\left(\frac{3}{77}\right)\) | \(e\left(\frac{45}{77}\right)\) | \(e\left(\frac{29}{77}\right)\) | \(e\left(\frac{10}{77}\right)\) |
| \(\chi_{2001}(62,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{77}\right)\) | \(e\left(\frac{46}{77}\right)\) | \(e\left(\frac{67}{154}\right)\) | \(e\left(\frac{59}{77}\right)\) | \(e\left(\frac{69}{77}\right)\) | \(e\left(\frac{113}{154}\right)\) | \(e\left(\frac{43}{77}\right)\) | \(e\left(\frac{29}{77}\right)\) | \(e\left(\frac{5}{77}\right)\) | \(e\left(\frac{15}{77}\right)\) |
| \(\chi_{2001}(71,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{25}{77}\right)\) | \(e\left(\frac{50}{77}\right)\) | \(e\left(\frac{113}{154}\right)\) | \(e\left(\frac{34}{77}\right)\) | \(e\left(\frac{75}{77}\right)\) | \(e\left(\frac{9}{154}\right)\) | \(e\left(\frac{30}{77}\right)\) | \(e\left(\frac{65}{77}\right)\) | \(e\left(\frac{59}{77}\right)\) | \(e\left(\frac{23}{77}\right)\) |
| \(\chi_{2001}(167,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{77}\right)\) | \(e\left(\frac{10}{77}\right)\) | \(e\left(\frac{115}{154}\right)\) | \(e\left(\frac{53}{77}\right)\) | \(e\left(\frac{15}{77}\right)\) | \(e\left(\frac{125}{154}\right)\) | \(e\left(\frac{6}{77}\right)\) | \(e\left(\frac{13}{77}\right)\) | \(e\left(\frac{58}{77}\right)\) | \(e\left(\frac{20}{77}\right)\) |
| \(\chi_{2001}(179,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{77}\right)\) | \(e\left(\frac{58}{77}\right)\) | \(e\left(\frac{51}{154}\right)\) | \(e\left(\frac{61}{77}\right)\) | \(e\left(\frac{10}{77}\right)\) | \(e\left(\frac{109}{154}\right)\) | \(e\left(\frac{4}{77}\right)\) | \(e\left(\frac{60}{77}\right)\) | \(e\left(\frac{13}{77}\right)\) | \(e\left(\frac{39}{77}\right)\) |
| \(\chi_{2001}(209,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{69}{77}\right)\) | \(e\left(\frac{61}{77}\right)\) | \(e\left(\frac{47}{154}\right)\) | \(e\left(\frac{23}{77}\right)\) | \(e\left(\frac{53}{77}\right)\) | \(e\left(\frac{31}{154}\right)\) | \(e\left(\frac{52}{77}\right)\) | \(e\left(\frac{10}{77}\right)\) | \(e\left(\frac{15}{77}\right)\) | \(e\left(\frac{45}{77}\right)\) |
| \(\chi_{2001}(236,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{16}{77}\right)\) | \(e\left(\frac{32}{77}\right)\) | \(e\left(\frac{137}{154}\right)\) | \(e\left(\frac{31}{77}\right)\) | \(e\left(\frac{48}{77}\right)\) | \(e\left(\frac{15}{154}\right)\) | \(e\left(\frac{50}{77}\right)\) | \(e\left(\frac{57}{77}\right)\) | \(e\left(\frac{47}{77}\right)\) | \(e\left(\frac{64}{77}\right)\) |
| \(\chi_{2001}(266,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{77}\right)\) | \(e\left(\frac{9}{77}\right)\) | \(e\left(\frac{65}{154}\right)\) | \(e\left(\frac{40}{77}\right)\) | \(e\left(\frac{52}{77}\right)\) | \(e\left(\frac{151}{154}\right)\) | \(e\left(\frac{67}{77}\right)\) | \(e\left(\frac{4}{77}\right)\) | \(e\left(\frac{6}{77}\right)\) | \(e\left(\frac{18}{77}\right)\) |
| \(\chi_{2001}(353,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{64}{77}\right)\) | \(e\left(\frac{51}{77}\right)\) | \(e\left(\frac{9}{154}\right)\) | \(e\left(\frac{47}{77}\right)\) | \(e\left(\frac{38}{77}\right)\) | \(e\left(\frac{137}{154}\right)\) | \(e\left(\frac{46}{77}\right)\) | \(e\left(\frac{74}{77}\right)\) | \(e\left(\frac{34}{77}\right)\) | \(e\left(\frac{25}{77}\right)\) |
| \(\chi_{2001}(386,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{73}{77}\right)\) | \(e\left(\frac{69}{77}\right)\) | \(e\left(\frac{139}{154}\right)\) | \(e\left(\frac{50}{77}\right)\) | \(e\left(\frac{65}{77}\right)\) | \(e\left(\frac{131}{154}\right)\) | \(e\left(\frac{26}{77}\right)\) | \(e\left(\frac{5}{77}\right)\) | \(e\left(\frac{46}{77}\right)\) | \(e\left(\frac{61}{77}\right)\) |
| \(\chi_{2001}(440,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{57}{77}\right)\) | \(e\left(\frac{37}{77}\right)\) | \(e\left(\frac{79}{154}\right)\) | \(e\left(\frac{19}{77}\right)\) | \(e\left(\frac{17}{77}\right)\) | \(e\left(\frac{39}{154}\right)\) | \(e\left(\frac{53}{77}\right)\) | \(e\left(\frac{25}{77}\right)\) | \(e\left(\frac{76}{77}\right)\) | \(e\left(\frac{74}{77}\right)\) |
| \(\chi_{2001}(473,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{10}{77}\right)\) | \(e\left(\frac{20}{77}\right)\) | \(e\left(\frac{153}{154}\right)\) | \(e\left(\frac{29}{77}\right)\) | \(e\left(\frac{30}{77}\right)\) | \(e\left(\frac{19}{154}\right)\) | \(e\left(\frac{12}{77}\right)\) | \(e\left(\frac{26}{77}\right)\) | \(e\left(\frac{39}{77}\right)\) | \(e\left(\frac{40}{77}\right)\) |
| \(\chi_{2001}(515,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{26}{77}\right)\) | \(e\left(\frac{52}{77}\right)\) | \(e\left(\frac{59}{154}\right)\) | \(e\left(\frac{60}{77}\right)\) | \(e\left(\frac{1}{77}\right)\) | \(e\left(\frac{111}{154}\right)\) | \(e\left(\frac{62}{77}\right)\) | \(e\left(\frac{6}{77}\right)\) | \(e\left(\frac{9}{77}\right)\) | \(e\left(\frac{27}{77}\right)\) |
| \(\chi_{2001}(560,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{77}\right)\) | \(e\left(\frac{62}{77}\right)\) | \(e\left(\frac{97}{154}\right)\) | \(e\left(\frac{36}{77}\right)\) | \(e\left(\frac{16}{77}\right)\) | \(e\left(\frac{5}{154}\right)\) | \(e\left(\frac{68}{77}\right)\) | \(e\left(\frac{19}{77}\right)\) | \(e\left(\frac{67}{77}\right)\) | \(e\left(\frac{47}{77}\right)\) |
| \(\chi_{2001}(584,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{37}{77}\right)\) | \(e\left(\frac{74}{77}\right)\) | \(e\left(\frac{81}{154}\right)\) | \(e\left(\frac{38}{77}\right)\) | \(e\left(\frac{34}{77}\right)\) | \(e\left(\frac{1}{154}\right)\) | \(e\left(\frac{29}{77}\right)\) | \(e\left(\frac{50}{77}\right)\) | \(e\left(\frac{75}{77}\right)\) | \(e\left(\frac{71}{77}\right)\) |
| \(\chi_{2001}(593,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{18}{77}\right)\) | \(e\left(\frac{36}{77}\right)\) | \(e\left(\frac{29}{154}\right)\) | \(e\left(\frac{6}{77}\right)\) | \(e\left(\frac{54}{77}\right)\) | \(e\left(\frac{65}{154}\right)\) | \(e\left(\frac{37}{77}\right)\) | \(e\left(\frac{16}{77}\right)\) | \(e\left(\frac{24}{77}\right)\) | \(e\left(\frac{72}{77}\right)\) |
| \(\chi_{2001}(602,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{61}{77}\right)\) | \(e\left(\frac{45}{77}\right)\) | \(e\left(\frac{17}{154}\right)\) | \(e\left(\frac{46}{77}\right)\) | \(e\left(\frac{29}{77}\right)\) | \(e\left(\frac{139}{154}\right)\) | \(e\left(\frac{27}{77}\right)\) | \(e\left(\frac{20}{77}\right)\) | \(e\left(\frac{30}{77}\right)\) | \(e\left(\frac{13}{77}\right)\) |
| \(\chi_{2001}(614,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{77}\right)\) | \(e\left(\frac{2}{77}\right)\) | \(e\left(\frac{23}{154}\right)\) | \(e\left(\frac{26}{77}\right)\) | \(e\left(\frac{3}{77}\right)\) | \(e\left(\frac{25}{154}\right)\) | \(e\left(\frac{32}{77}\right)\) | \(e\left(\frac{18}{77}\right)\) | \(e\left(\frac{27}{77}\right)\) | \(e\left(\frac{4}{77}\right)\) |
| \(\chi_{2001}(647,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{24}{77}\right)\) | \(e\left(\frac{48}{77}\right)\) | \(e\left(\frac{13}{154}\right)\) | \(e\left(\frac{8}{77}\right)\) | \(e\left(\frac{72}{77}\right)\) | \(e\left(\frac{61}{154}\right)\) | \(e\left(\frac{75}{77}\right)\) | \(e\left(\frac{47}{77}\right)\) | \(e\left(\frac{32}{77}\right)\) | \(e\left(\frac{19}{77}\right)\) |
| \(\chi_{2001}(671,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{72}{77}\right)\) | \(e\left(\frac{67}{77}\right)\) | \(e\left(\frac{39}{154}\right)\) | \(e\left(\frac{24}{77}\right)\) | \(e\left(\frac{62}{77}\right)\) | \(e\left(\frac{29}{154}\right)\) | \(e\left(\frac{71}{77}\right)\) | \(e\left(\frac{64}{77}\right)\) | \(e\left(\frac{19}{77}\right)\) | \(e\left(\frac{57}{77}\right)\) |
| \(\chi_{2001}(680,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{32}{77}\right)\) | \(e\left(\frac{64}{77}\right)\) | \(e\left(\frac{43}{154}\right)\) | \(e\left(\frac{62}{77}\right)\) | \(e\left(\frac{19}{77}\right)\) | \(e\left(\frac{107}{154}\right)\) | \(e\left(\frac{23}{77}\right)\) | \(e\left(\frac{37}{77}\right)\) | \(e\left(\frac{17}{77}\right)\) | \(e\left(\frac{51}{77}\right)\) |
| \(\chi_{2001}(731,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{62}{77}\right)\) | \(e\left(\frac{47}{77}\right)\) | \(e\left(\frac{117}{154}\right)\) | \(e\left(\frac{72}{77}\right)\) | \(e\left(\frac{32}{77}\right)\) | \(e\left(\frac{87}{154}\right)\) | \(e\left(\frac{59}{77}\right)\) | \(e\left(\frac{38}{77}\right)\) | \(e\left(\frac{57}{77}\right)\) | \(e\left(\frac{17}{77}\right)\) |
| \(\chi_{2001}(767,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{53}{77}\right)\) | \(e\left(\frac{29}{77}\right)\) | \(e\left(\frac{141}{154}\right)\) | \(e\left(\frac{69}{77}\right)\) | \(e\left(\frac{5}{77}\right)\) | \(e\left(\frac{93}{154}\right)\) | \(e\left(\frac{2}{77}\right)\) | \(e\left(\frac{30}{77}\right)\) | \(e\left(\frac{45}{77}\right)\) | \(e\left(\frac{58}{77}\right)\) |
| \(\chi_{2001}(788,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{71}{77}\right)\) | \(e\left(\frac{65}{77}\right)\) | \(e\left(\frac{93}{154}\right)\) | \(e\left(\frac{75}{77}\right)\) | \(e\left(\frac{59}{77}\right)\) | \(e\left(\frac{81}{154}\right)\) | \(e\left(\frac{39}{77}\right)\) | \(e\left(\frac{46}{77}\right)\) | \(e\left(\frac{69}{77}\right)\) | \(e\left(\frac{53}{77}\right)\) |
| \(\chi_{2001}(818,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{76}{77}\right)\) | \(e\left(\frac{75}{77}\right)\) | \(e\left(\frac{131}{154}\right)\) | \(e\left(\frac{51}{77}\right)\) | \(e\left(\frac{74}{77}\right)\) | \(e\left(\frac{129}{154}\right)\) | \(e\left(\frac{45}{77}\right)\) | \(e\left(\frac{59}{77}\right)\) | \(e\left(\frac{50}{77}\right)\) | \(e\left(\frac{73}{77}\right)\) |
| \(\chi_{2001}(821,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{45}{77}\right)\) | \(e\left(\frac{13}{77}\right)\) | \(e\left(\frac{111}{154}\right)\) | \(e\left(\frac{15}{77}\right)\) | \(e\left(\frac{58}{77}\right)\) | \(e\left(\frac{47}{154}\right)\) | \(e\left(\frac{54}{77}\right)\) | \(e\left(\frac{40}{77}\right)\) | \(e\left(\frac{60}{77}\right)\) | \(e\left(\frac{26}{77}\right)\) |
| \(\chi_{2001}(854,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{46}{77}\right)\) | \(e\left(\frac{15}{77}\right)\) | \(e\left(\frac{57}{154}\right)\) | \(e\left(\frac{41}{77}\right)\) | \(e\left(\frac{61}{77}\right)\) | \(e\left(\frac{149}{154}\right)\) | \(e\left(\frac{9}{77}\right)\) | \(e\left(\frac{58}{77}\right)\) | \(e\left(\frac{10}{77}\right)\) | \(e\left(\frac{30}{77}\right)\) |
| \(\chi_{2001}(863,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{77}\right)\) | \(e\left(\frac{38}{77}\right)\) | \(e\left(\frac{129}{154}\right)\) | \(e\left(\frac{32}{77}\right)\) | \(e\left(\frac{57}{77}\right)\) | \(e\left(\frac{13}{154}\right)\) | \(e\left(\frac{69}{77}\right)\) | \(e\left(\frac{34}{77}\right)\) | \(e\left(\frac{51}{77}\right)\) | \(e\left(\frac{76}{77}\right)\) |
| \(\chi_{2001}(905,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{20}{77}\right)\) | \(e\left(\frac{40}{77}\right)\) | \(e\left(\frac{75}{154}\right)\) | \(e\left(\frac{58}{77}\right)\) | \(e\left(\frac{60}{77}\right)\) | \(e\left(\frac{115}{154}\right)\) | \(e\left(\frac{24}{77}\right)\) | \(e\left(\frac{52}{77}\right)\) | \(e\left(\frac{1}{77}\right)\) | \(e\left(\frac{3}{77}\right)\) |
| \(\chi_{2001}(932,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{30}{77}\right)\) | \(e\left(\frac{60}{77}\right)\) | \(e\left(\frac{151}{154}\right)\) | \(e\left(\frac{10}{77}\right)\) | \(e\left(\frac{13}{77}\right)\) | \(e\left(\frac{57}{154}\right)\) | \(e\left(\frac{36}{77}\right)\) | \(e\left(\frac{1}{77}\right)\) | \(e\left(\frac{40}{77}\right)\) | \(e\left(\frac{43}{77}\right)\) |
| \(\chi_{2001}(992,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{77}\right)\) | \(e\left(\frac{26}{77}\right)\) | \(e\left(\frac{145}{154}\right)\) | \(e\left(\frac{30}{77}\right)\) | \(e\left(\frac{39}{77}\right)\) | \(e\left(\frac{17}{154}\right)\) | \(e\left(\frac{31}{77}\right)\) | \(e\left(\frac{3}{77}\right)\) | \(e\left(\frac{43}{77}\right)\) | \(e\left(\frac{52}{77}\right)\) |