from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(667, base_ring=CyclotomicField(154))
M = H._module
chi = DirichletCharacter(H, M([56,22]))
chi.galois_orbit()
[g,chi] = znchar(Mod(16,667))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(667\) | |
| Conductor: | \(667\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(77\) | 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: | $\Q(\zeta_{77})$ |
| Fixed field: | Number field defined by a degree 77 polynomial |
First 31 of 60 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{667}(16,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{67}{77}\right)\) | \(e\left(\frac{41}{77}\right)\) | \(e\left(\frac{57}{77}\right)\) | \(e\left(\frac{39}{77}\right)\) | \(e\left(\frac{31}{77}\right)\) | \(e\left(\frac{48}{77}\right)\) | \(e\left(\frac{47}{77}\right)\) | \(e\left(\frac{5}{77}\right)\) | \(e\left(\frac{29}{77}\right)\) | \(e\left(\frac{65}{77}\right)\) |
| \(\chi_{667}(25,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{58}{77}\right)\) | \(e\left(\frac{24}{77}\right)\) | \(e\left(\frac{39}{77}\right)\) | \(e\left(\frac{51}{77}\right)\) | \(e\left(\frac{5}{77}\right)\) | \(e\left(\frac{45}{77}\right)\) | \(e\left(\frac{20}{77}\right)\) | \(e\left(\frac{48}{77}\right)\) | \(e\left(\frac{32}{77}\right)\) | \(e\left(\frac{8}{77}\right)\) |
| \(\chi_{667}(36,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{54}{77}\right)\) | \(e\left(\frac{25}{77}\right)\) | \(e\left(\frac{31}{77}\right)\) | \(e\left(\frac{5}{77}\right)\) | \(e\left(\frac{2}{77}\right)\) | \(e\left(\frac{18}{77}\right)\) | \(e\left(\frac{8}{77}\right)\) | \(e\left(\frac{50}{77}\right)\) | \(e\left(\frac{59}{77}\right)\) | \(e\left(\frac{34}{77}\right)\) |
| \(\chi_{667}(49,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{24}{77}\right)\) | \(e\left(\frac{71}{77}\right)\) | \(e\left(\frac{48}{77}\right)\) | \(e\left(\frac{45}{77}\right)\) | \(e\left(\frac{18}{77}\right)\) | \(e\left(\frac{8}{77}\right)\) | \(e\left(\frac{72}{77}\right)\) | \(e\left(\frac{65}{77}\right)\) | \(e\left(\frac{69}{77}\right)\) | \(e\left(\frac{75}{77}\right)\) |
| \(\chi_{667}(52,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{27}{77}\right)\) | \(e\left(\frac{51}{77}\right)\) | \(e\left(\frac{54}{77}\right)\) | \(e\left(\frac{41}{77}\right)\) | \(e\left(\frac{1}{77}\right)\) | \(e\left(\frac{9}{77}\right)\) | \(e\left(\frac{4}{77}\right)\) | \(e\left(\frac{25}{77}\right)\) | \(e\left(\frac{68}{77}\right)\) | \(e\left(\frac{17}{77}\right)\) |
| \(\chi_{667}(54,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{77}\right)\) | \(e\left(\frac{17}{77}\right)\) | \(e\left(\frac{18}{77}\right)\) | \(e\left(\frac{65}{77}\right)\) | \(e\left(\frac{26}{77}\right)\) | \(e\left(\frac{3}{77}\right)\) | \(e\left(\frac{27}{77}\right)\) | \(e\left(\frac{34}{77}\right)\) | \(e\left(\frac{74}{77}\right)\) | \(e\left(\frac{57}{77}\right)\) |
| \(\chi_{667}(78,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{59}{77}\right)\) | \(e\left(\frac{43}{77}\right)\) | \(e\left(\frac{41}{77}\right)\) | \(e\left(\frac{24}{77}\right)\) | \(e\left(\frac{25}{77}\right)\) | \(e\left(\frac{71}{77}\right)\) | \(e\left(\frac{23}{77}\right)\) | \(e\left(\frac{9}{77}\right)\) | \(e\left(\frac{6}{77}\right)\) | \(e\left(\frac{40}{77}\right)\) |
| \(\chi_{667}(81,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{77}\right)\) | \(e\left(\frac{9}{77}\right)\) | \(e\left(\frac{5}{77}\right)\) | \(e\left(\frac{48}{77}\right)\) | \(e\left(\frac{50}{77}\right)\) | \(e\left(\frac{65}{77}\right)\) | \(e\left(\frac{46}{77}\right)\) | \(e\left(\frac{18}{77}\right)\) | \(e\left(\frac{12}{77}\right)\) | \(e\left(\frac{3}{77}\right)\) |
| \(\chi_{667}(82,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{77}\right)\) | \(e\left(\frac{47}{77}\right)\) | \(e\left(\frac{9}{77}\right)\) | \(e\left(\frac{71}{77}\right)\) | \(e\left(\frac{13}{77}\right)\) | \(e\left(\frac{40}{77}\right)\) | \(e\left(\frac{52}{77}\right)\) | \(e\left(\frac{17}{77}\right)\) | \(e\left(\frac{37}{77}\right)\) | \(e\left(\frac{67}{77}\right)\) |
| \(\chi_{667}(94,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{47}{77}\right)\) | \(e\left(\frac{46}{77}\right)\) | \(e\left(\frac{17}{77}\right)\) | \(e\left(\frac{40}{77}\right)\) | \(e\left(\frac{16}{77}\right)\) | \(e\left(\frac{67}{77}\right)\) | \(e\left(\frac{64}{77}\right)\) | \(e\left(\frac{15}{77}\right)\) | \(e\left(\frac{10}{77}\right)\) | \(e\left(\frac{41}{77}\right)\) |
| \(\chi_{667}(110,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{62}{77}\right)\) | \(e\left(\frac{23}{77}\right)\) | \(e\left(\frac{47}{77}\right)\) | \(e\left(\frac{20}{77}\right)\) | \(e\left(\frac{8}{77}\right)\) | \(e\left(\frac{72}{77}\right)\) | \(e\left(\frac{32}{77}\right)\) | \(e\left(\frac{46}{77}\right)\) | \(e\left(\frac{5}{77}\right)\) | \(e\left(\frac{59}{77}\right)\) |
| \(\chi_{667}(123,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{75}{77}\right)\) | \(e\left(\frac{39}{77}\right)\) | \(e\left(\frac{73}{77}\right)\) | \(e\left(\frac{54}{77}\right)\) | \(e\left(\frac{37}{77}\right)\) | \(e\left(\frac{25}{77}\right)\) | \(e\left(\frac{71}{77}\right)\) | \(e\left(\frac{1}{77}\right)\) | \(e\left(\frac{52}{77}\right)\) | \(e\left(\frac{13}{77}\right)\) |
| \(\chi_{667}(140,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{36}{77}\right)\) | \(e\left(\frac{68}{77}\right)\) | \(e\left(\frac{72}{77}\right)\) | \(e\left(\frac{29}{77}\right)\) | \(e\left(\frac{27}{77}\right)\) | \(e\left(\frac{12}{77}\right)\) | \(e\left(\frac{31}{77}\right)\) | \(e\left(\frac{59}{77}\right)\) | \(e\left(\frac{65}{77}\right)\) | \(e\left(\frac{74}{77}\right)\) |
| \(\chi_{667}(141,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{77}\right)\) | \(e\left(\frac{38}{77}\right)\) | \(e\left(\frac{4}{77}\right)\) | \(e\left(\frac{23}{77}\right)\) | \(e\left(\frac{40}{77}\right)\) | \(e\left(\frac{52}{77}\right)\) | \(e\left(\frac{6}{77}\right)\) | \(e\left(\frac{76}{77}\right)\) | \(e\left(\frac{25}{77}\right)\) | \(e\left(\frac{64}{77}\right)\) |
| \(\chi_{667}(165,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{77}\right)\) | \(e\left(\frac{15}{77}\right)\) | \(e\left(\frac{34}{77}\right)\) | \(e\left(\frac{3}{77}\right)\) | \(e\left(\frac{32}{77}\right)\) | \(e\left(\frac{57}{77}\right)\) | \(e\left(\frac{51}{77}\right)\) | \(e\left(\frac{30}{77}\right)\) | \(e\left(\frac{20}{77}\right)\) | \(e\left(\frac{5}{77}\right)\) |
| \(\chi_{667}(169,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{64}{77}\right)\) | \(e\left(\frac{61}{77}\right)\) | \(e\left(\frac{51}{77}\right)\) | \(e\left(\frac{43}{77}\right)\) | \(e\left(\frac{48}{77}\right)\) | \(e\left(\frac{47}{77}\right)\) | \(e\left(\frac{38}{77}\right)\) | \(e\left(\frac{45}{77}\right)\) | \(e\left(\frac{30}{77}\right)\) | \(e\left(\frac{46}{77}\right)\) |
| \(\chi_{667}(170,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{77}\right)\) | \(e\left(\frac{10}{77}\right)\) | \(e\left(\frac{74}{77}\right)\) | \(e\left(\frac{2}{77}\right)\) | \(e\left(\frac{47}{77}\right)\) | \(e\left(\frac{38}{77}\right)\) | \(e\left(\frac{34}{77}\right)\) | \(e\left(\frac{20}{77}\right)\) | \(e\left(\frac{39}{77}\right)\) | \(e\left(\frac{29}{77}\right)\) |
| \(\chi_{667}(190,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{60}{77}\right)\) | \(e\left(\frac{62}{77}\right)\) | \(e\left(\frac{43}{77}\right)\) | \(e\left(\frac{74}{77}\right)\) | \(e\left(\frac{45}{77}\right)\) | \(e\left(\frac{20}{77}\right)\) | \(e\left(\frac{26}{77}\right)\) | \(e\left(\frac{47}{77}\right)\) | \(e\left(\frac{57}{77}\right)\) | \(e\left(\frac{72}{77}\right)\) |
| \(\chi_{667}(197,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{76}{77}\right)\) | \(e\left(\frac{58}{77}\right)\) | \(e\left(\frac{75}{77}\right)\) | \(e\left(\frac{27}{77}\right)\) | \(e\left(\frac{57}{77}\right)\) | \(e\left(\frac{51}{77}\right)\) | \(e\left(\frac{74}{77}\right)\) | \(e\left(\frac{39}{77}\right)\) | \(e\left(\frac{26}{77}\right)\) | \(e\left(\frac{45}{77}\right)\) |
| \(\chi_{667}(210,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{68}{77}\right)\) | \(e\left(\frac{60}{77}\right)\) | \(e\left(\frac{59}{77}\right)\) | \(e\left(\frac{12}{77}\right)\) | \(e\left(\frac{51}{77}\right)\) | \(e\left(\frac{74}{77}\right)\) | \(e\left(\frac{50}{77}\right)\) | \(e\left(\frac{43}{77}\right)\) | \(e\left(\frac{3}{77}\right)\) | \(e\left(\frac{20}{77}\right)\) |
| \(\chi_{667}(219,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{74}{77}\right)\) | \(e\left(\frac{20}{77}\right)\) | \(e\left(\frac{71}{77}\right)\) | \(e\left(\frac{4}{77}\right)\) | \(e\left(\frac{17}{77}\right)\) | \(e\left(\frac{76}{77}\right)\) | \(e\left(\frac{68}{77}\right)\) | \(e\left(\frac{40}{77}\right)\) | \(e\left(\frac{1}{77}\right)\) | \(e\left(\frac{58}{77}\right)\) |
| \(\chi_{667}(223,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{45}{77}\right)\) | \(e\left(\frac{8}{77}\right)\) | \(e\left(\frac{13}{77}\right)\) | \(e\left(\frac{17}{77}\right)\) | \(e\left(\frac{53}{77}\right)\) | \(e\left(\frac{15}{77}\right)\) | \(e\left(\frac{58}{77}\right)\) | \(e\left(\frac{16}{77}\right)\) | \(e\left(\frac{62}{77}\right)\) | \(e\left(\frac{54}{77}\right)\) |
| \(\chi_{667}(239,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{26}{77}\right)\) | \(e\left(\frac{32}{77}\right)\) | \(e\left(\frac{52}{77}\right)\) | \(e\left(\frac{68}{77}\right)\) | \(e\left(\frac{58}{77}\right)\) | \(e\left(\frac{60}{77}\right)\) | \(e\left(\frac{1}{77}\right)\) | \(e\left(\frac{64}{77}\right)\) | \(e\left(\frac{17}{77}\right)\) | \(e\left(\frac{62}{77}\right)\) |
| \(\chi_{667}(248,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{18}{77}\right)\) | \(e\left(\frac{34}{77}\right)\) | \(e\left(\frac{36}{77}\right)\) | \(e\left(\frac{53}{77}\right)\) | \(e\left(\frac{52}{77}\right)\) | \(e\left(\frac{6}{77}\right)\) | \(e\left(\frac{54}{77}\right)\) | \(e\left(\frac{68}{77}\right)\) | \(e\left(\frac{71}{77}\right)\) | \(e\left(\frac{37}{77}\right)\) |
| \(\chi_{667}(255,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{69}{77}\right)\) | \(e\left(\frac{2}{77}\right)\) | \(e\left(\frac{61}{77}\right)\) | \(e\left(\frac{62}{77}\right)\) | \(e\left(\frac{71}{77}\right)\) | \(e\left(\frac{23}{77}\right)\) | \(e\left(\frac{53}{77}\right)\) | \(e\left(\frac{4}{77}\right)\) | \(e\left(\frac{54}{77}\right)\) | \(e\left(\frac{52}{77}\right)\) |
| \(\chi_{667}(256,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{57}{77}\right)\) | \(e\left(\frac{5}{77}\right)\) | \(e\left(\frac{37}{77}\right)\) | \(e\left(\frac{1}{77}\right)\) | \(e\left(\frac{62}{77}\right)\) | \(e\left(\frac{19}{77}\right)\) | \(e\left(\frac{17}{77}\right)\) | \(e\left(\frac{10}{77}\right)\) | \(e\left(\frac{58}{77}\right)\) | \(e\left(\frac{53}{77}\right)\) |
| \(\chi_{667}(257,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{72}{77}\right)\) | \(e\left(\frac{59}{77}\right)\) | \(e\left(\frac{67}{77}\right)\) | \(e\left(\frac{58}{77}\right)\) | \(e\left(\frac{54}{77}\right)\) | \(e\left(\frac{24}{77}\right)\) | \(e\left(\frac{62}{77}\right)\) | \(e\left(\frac{41}{77}\right)\) | \(e\left(\frac{53}{77}\right)\) | \(e\left(\frac{71}{77}\right)\) |
| \(\chi_{667}(284,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{20}{77}\right)\) | \(e\left(\frac{72}{77}\right)\) | \(e\left(\frac{40}{77}\right)\) | \(e\left(\frac{76}{77}\right)\) | \(e\left(\frac{15}{77}\right)\) | \(e\left(\frac{58}{77}\right)\) | \(e\left(\frac{60}{77}\right)\) | \(e\left(\frac{67}{77}\right)\) | \(e\left(\frac{19}{77}\right)\) | \(e\left(\frac{24}{77}\right)\) |
| \(\chi_{667}(285,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{15}{77}\right)\) | \(e\left(\frac{54}{77}\right)\) | \(e\left(\frac{30}{77}\right)\) | \(e\left(\frac{57}{77}\right)\) | \(e\left(\frac{69}{77}\right)\) | \(e\left(\frac{5}{77}\right)\) | \(e\left(\frac{45}{77}\right)\) | \(e\left(\frac{31}{77}\right)\) | \(e\left(\frac{72}{77}\right)\) | \(e\left(\frac{18}{77}\right)\) |
| \(\chi_{667}(315,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{77}\right)\) | \(e\left(\frac{52}{77}\right)\) | \(e\left(\frac{46}{77}\right)\) | \(e\left(\frac{72}{77}\right)\) | \(e\left(\frac{75}{77}\right)\) | \(e\left(\frac{59}{77}\right)\) | \(e\left(\frac{69}{77}\right)\) | \(e\left(\frac{27}{77}\right)\) | \(e\left(\frac{18}{77}\right)\) | \(e\left(\frac{43}{77}\right)\) |
| \(\chi_{667}(326,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{61}{77}\right)\) | \(e\left(\frac{4}{77}\right)\) | \(e\left(\frac{45}{77}\right)\) | \(e\left(\frac{47}{77}\right)\) | \(e\left(\frac{65}{77}\right)\) | \(e\left(\frac{46}{77}\right)\) | \(e\left(\frac{29}{77}\right)\) | \(e\left(\frac{8}{77}\right)\) | \(e\left(\frac{31}{77}\right)\) | \(e\left(\frac{27}{77}\right)\) |