from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(417, base_ring=CyclotomicField(138))
M = H._module
chi = DirichletCharacter(H, M([0,2]))
chi.galois_orbit()
[g,chi] = znchar(Mod(4,417))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(417\) | |
| Conductor: | \(139\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(69\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 139.g | 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: | $\Q(\zeta_{69})$ |
| Fixed field: | Number field defined by a degree 69 polynomial |
First 31 of 44 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{417}(4,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{69}\right)\) | \(e\left(\frac{2}{69}\right)\) | \(e\left(\frac{17}{69}\right)\) | \(e\left(\frac{50}{69}\right)\) | \(e\left(\frac{1}{23}\right)\) | \(e\left(\frac{6}{23}\right)\) | \(e\left(\frac{7}{69}\right)\) | \(e\left(\frac{64}{69}\right)\) | \(e\left(\frac{17}{23}\right)\) | \(e\left(\frac{4}{69}\right)\) |
| \(\chi_{417}(7,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{69}\right)\) | \(e\left(\frac{50}{69}\right)\) | \(e\left(\frac{11}{69}\right)\) | \(e\left(\frac{8}{69}\right)\) | \(e\left(\frac{2}{23}\right)\) | \(e\left(\frac{12}{23}\right)\) | \(e\left(\frac{37}{69}\right)\) | \(e\left(\frac{13}{69}\right)\) | \(e\left(\frac{11}{23}\right)\) | \(e\left(\frac{31}{69}\right)\) |
| \(\chi_{417}(13,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{32}{69}\right)\) | \(e\left(\frac{64}{69}\right)\) | \(e\left(\frac{61}{69}\right)\) | \(e\left(\frac{13}{69}\right)\) | \(e\left(\frac{9}{23}\right)\) | \(e\left(\frac{8}{23}\right)\) | \(e\left(\frac{17}{69}\right)\) | \(e\left(\frac{47}{69}\right)\) | \(e\left(\frac{15}{23}\right)\) | \(e\left(\frac{59}{69}\right)\) |
| \(\chi_{417}(16,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{69}\right)\) | \(e\left(\frac{4}{69}\right)\) | \(e\left(\frac{34}{69}\right)\) | \(e\left(\frac{31}{69}\right)\) | \(e\left(\frac{2}{23}\right)\) | \(e\left(\frac{12}{23}\right)\) | \(e\left(\frac{14}{69}\right)\) | \(e\left(\frac{59}{69}\right)\) | \(e\left(\frac{11}{23}\right)\) | \(e\left(\frac{8}{69}\right)\) |
| \(\chi_{417}(25,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{69}\right)\) | \(e\left(\frac{34}{69}\right)\) | \(e\left(\frac{13}{69}\right)\) | \(e\left(\frac{22}{69}\right)\) | \(e\left(\frac{17}{23}\right)\) | \(e\left(\frac{10}{23}\right)\) | \(e\left(\frac{50}{69}\right)\) | \(e\left(\frac{53}{69}\right)\) | \(e\left(\frac{13}{23}\right)\) | \(e\left(\frac{68}{69}\right)\) |
| \(\chi_{417}(28,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{26}{69}\right)\) | \(e\left(\frac{52}{69}\right)\) | \(e\left(\frac{28}{69}\right)\) | \(e\left(\frac{58}{69}\right)\) | \(e\left(\frac{3}{23}\right)\) | \(e\left(\frac{18}{23}\right)\) | \(e\left(\frac{44}{69}\right)\) | \(e\left(\frac{8}{69}\right)\) | \(e\left(\frac{5}{23}\right)\) | \(e\left(\frac{35}{69}\right)\) |
| \(\chi_{417}(31,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{28}{69}\right)\) | \(e\left(\frac{56}{69}\right)\) | \(e\left(\frac{62}{69}\right)\) | \(e\left(\frac{20}{69}\right)\) | \(e\left(\frac{5}{23}\right)\) | \(e\left(\frac{7}{23}\right)\) | \(e\left(\frac{58}{69}\right)\) | \(e\left(\frac{67}{69}\right)\) | \(e\left(\frac{16}{23}\right)\) | \(e\left(\frac{43}{69}\right)\) |
| \(\chi_{417}(37,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{40}{69}\right)\) | \(e\left(\frac{11}{69}\right)\) | \(e\left(\frac{59}{69}\right)\) | \(e\left(\frac{68}{69}\right)\) | \(e\left(\frac{17}{23}\right)\) | \(e\left(\frac{10}{23}\right)\) | \(e\left(\frac{4}{69}\right)\) | \(e\left(\frac{7}{69}\right)\) | \(e\left(\frac{13}{23}\right)\) | \(e\left(\frac{22}{69}\right)\) |
| \(\chi_{417}(46,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{14}{69}\right)\) | \(e\left(\frac{28}{69}\right)\) | \(e\left(\frac{31}{69}\right)\) | \(e\left(\frac{10}{69}\right)\) | \(e\left(\frac{14}{23}\right)\) | \(e\left(\frac{15}{23}\right)\) | \(e\left(\frac{29}{69}\right)\) | \(e\left(\frac{68}{69}\right)\) | \(e\left(\frac{8}{23}\right)\) | \(e\left(\frac{56}{69}\right)\) |
| \(\chi_{417}(49,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{50}{69}\right)\) | \(e\left(\frac{31}{69}\right)\) | \(e\left(\frac{22}{69}\right)\) | \(e\left(\frac{16}{69}\right)\) | \(e\left(\frac{4}{23}\right)\) | \(e\left(\frac{1}{23}\right)\) | \(e\left(\frac{5}{69}\right)\) | \(e\left(\frac{26}{69}\right)\) | \(e\left(\frac{22}{23}\right)\) | \(e\left(\frac{62}{69}\right)\) |
| \(\chi_{417}(67,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{69}\right)\) | \(e\left(\frac{16}{69}\right)\) | \(e\left(\frac{67}{69}\right)\) | \(e\left(\frac{55}{69}\right)\) | \(e\left(\frac{8}{23}\right)\) | \(e\left(\frac{2}{23}\right)\) | \(e\left(\frac{56}{69}\right)\) | \(e\left(\frac{29}{69}\right)\) | \(e\left(\frac{21}{23}\right)\) | \(e\left(\frac{32}{69}\right)\) |
| \(\chi_{417}(118,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{69}\right)\) | \(e\left(\frac{22}{69}\right)\) | \(e\left(\frac{49}{69}\right)\) | \(e\left(\frac{67}{69}\right)\) | \(e\left(\frac{11}{23}\right)\) | \(e\left(\frac{20}{23}\right)\) | \(e\left(\frac{8}{69}\right)\) | \(e\left(\frac{14}{69}\right)\) | \(e\left(\frac{3}{23}\right)\) | \(e\left(\frac{44}{69}\right)\) |
| \(\chi_{417}(121,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{69}\right)\) | \(e\left(\frac{14}{69}\right)\) | \(e\left(\frac{50}{69}\right)\) | \(e\left(\frac{5}{69}\right)\) | \(e\left(\frac{7}{23}\right)\) | \(e\left(\frac{19}{23}\right)\) | \(e\left(\frac{49}{69}\right)\) | \(e\left(\frac{34}{69}\right)\) | \(e\left(\frac{4}{23}\right)\) | \(e\left(\frac{28}{69}\right)\) |
| \(\chi_{417}(124,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{69}\right)\) | \(e\left(\frac{58}{69}\right)\) | \(e\left(\frac{10}{69}\right)\) | \(e\left(\frac{1}{69}\right)\) | \(e\left(\frac{6}{23}\right)\) | \(e\left(\frac{13}{23}\right)\) | \(e\left(\frac{65}{69}\right)\) | \(e\left(\frac{62}{69}\right)\) | \(e\left(\frac{10}{23}\right)\) | \(e\left(\frac{47}{69}\right)\) |
| \(\chi_{417}(127,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{56}{69}\right)\) | \(e\left(\frac{43}{69}\right)\) | \(e\left(\frac{55}{69}\right)\) | \(e\left(\frac{40}{69}\right)\) | \(e\left(\frac{10}{23}\right)\) | \(e\left(\frac{14}{23}\right)\) | \(e\left(\frac{47}{69}\right)\) | \(e\left(\frac{65}{69}\right)\) | \(e\left(\frac{9}{23}\right)\) | \(e\left(\frac{17}{69}\right)\) |
| \(\chi_{417}(136,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{55}{69}\right)\) | \(e\left(\frac{41}{69}\right)\) | \(e\left(\frac{38}{69}\right)\) | \(e\left(\frac{59}{69}\right)\) | \(e\left(\frac{9}{23}\right)\) | \(e\left(\frac{8}{23}\right)\) | \(e\left(\frac{40}{69}\right)\) | \(e\left(\frac{1}{69}\right)\) | \(e\left(\frac{15}{23}\right)\) | \(e\left(\frac{13}{69}\right)\) |
| \(\chi_{417}(148,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{69}\right)\) | \(e\left(\frac{13}{69}\right)\) | \(e\left(\frac{7}{69}\right)\) | \(e\left(\frac{49}{69}\right)\) | \(e\left(\frac{18}{23}\right)\) | \(e\left(\frac{16}{23}\right)\) | \(e\left(\frac{11}{69}\right)\) | \(e\left(\frac{2}{69}\right)\) | \(e\left(\frac{7}{23}\right)\) | \(e\left(\frac{26}{69}\right)\) |
| \(\chi_{417}(163,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{22}{69}\right)\) | \(e\left(\frac{44}{69}\right)\) | \(e\left(\frac{29}{69}\right)\) | \(e\left(\frac{65}{69}\right)\) | \(e\left(\frac{22}{23}\right)\) | \(e\left(\frac{17}{23}\right)\) | \(e\left(\frac{16}{69}\right)\) | \(e\left(\frac{28}{69}\right)\) | \(e\left(\frac{6}{23}\right)\) | \(e\left(\frac{19}{69}\right)\) |
| \(\chi_{417}(169,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{64}{69}\right)\) | \(e\left(\frac{59}{69}\right)\) | \(e\left(\frac{53}{69}\right)\) | \(e\left(\frac{26}{69}\right)\) | \(e\left(\frac{18}{23}\right)\) | \(e\left(\frac{16}{23}\right)\) | \(e\left(\frac{34}{69}\right)\) | \(e\left(\frac{25}{69}\right)\) | \(e\left(\frac{7}{23}\right)\) | \(e\left(\frac{49}{69}\right)\) |
| \(\chi_{417}(190,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{69}\right)\) | \(e\left(\frac{10}{69}\right)\) | \(e\left(\frac{16}{69}\right)\) | \(e\left(\frac{43}{69}\right)\) | \(e\left(\frac{5}{23}\right)\) | \(e\left(\frac{7}{23}\right)\) | \(e\left(\frac{35}{69}\right)\) | \(e\left(\frac{44}{69}\right)\) | \(e\left(\frac{16}{23}\right)\) | \(e\left(\frac{20}{69}\right)\) |
| \(\chi_{417}(193,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{62}{69}\right)\) | \(e\left(\frac{55}{69}\right)\) | \(e\left(\frac{19}{69}\right)\) | \(e\left(\frac{64}{69}\right)\) | \(e\left(\frac{16}{23}\right)\) | \(e\left(\frac{4}{23}\right)\) | \(e\left(\frac{20}{69}\right)\) | \(e\left(\frac{35}{69}\right)\) | \(e\left(\frac{19}{23}\right)\) | \(e\left(\frac{41}{69}\right)\) |
| \(\chi_{417}(205,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{59}{69}\right)\) | \(e\left(\frac{49}{69}\right)\) | \(e\left(\frac{37}{69}\right)\) | \(e\left(\frac{52}{69}\right)\) | \(e\left(\frac{13}{23}\right)\) | \(e\left(\frac{9}{23}\right)\) | \(e\left(\frac{68}{69}\right)\) | \(e\left(\frac{50}{69}\right)\) | \(e\left(\frac{14}{23}\right)\) | \(e\left(\frac{29}{69}\right)\) |
| \(\chi_{417}(208,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{34}{69}\right)\) | \(e\left(\frac{68}{69}\right)\) | \(e\left(\frac{26}{69}\right)\) | \(e\left(\frac{44}{69}\right)\) | \(e\left(\frac{11}{23}\right)\) | \(e\left(\frac{20}{23}\right)\) | \(e\left(\frac{31}{69}\right)\) | \(e\left(\frac{37}{69}\right)\) | \(e\left(\frac{3}{23}\right)\) | \(e\left(\frac{67}{69}\right)\) |
| \(\chi_{417}(217,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{53}{69}\right)\) | \(e\left(\frac{37}{69}\right)\) | \(e\left(\frac{4}{69}\right)\) | \(e\left(\frac{28}{69}\right)\) | \(e\left(\frac{7}{23}\right)\) | \(e\left(\frac{19}{23}\right)\) | \(e\left(\frac{26}{69}\right)\) | \(e\left(\frac{11}{69}\right)\) | \(e\left(\frac{4}{23}\right)\) | \(e\left(\frac{5}{69}\right)\) |
| \(\chi_{417}(220,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{69}\right)\) | \(e\left(\frac{26}{69}\right)\) | \(e\left(\frac{14}{69}\right)\) | \(e\left(\frac{29}{69}\right)\) | \(e\left(\frac{13}{23}\right)\) | \(e\left(\frac{9}{23}\right)\) | \(e\left(\frac{22}{69}\right)\) | \(e\left(\frac{4}{69}\right)\) | \(e\left(\frac{14}{23}\right)\) | \(e\left(\frac{52}{69}\right)\) |
| \(\chi_{417}(238,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{10}{69}\right)\) | \(e\left(\frac{20}{69}\right)\) | \(e\left(\frac{32}{69}\right)\) | \(e\left(\frac{17}{69}\right)\) | \(e\left(\frac{10}{23}\right)\) | \(e\left(\frac{14}{23}\right)\) | \(e\left(\frac{1}{69}\right)\) | \(e\left(\frac{19}{69}\right)\) | \(e\left(\frac{9}{23}\right)\) | \(e\left(\frac{40}{69}\right)\) |
| \(\chi_{417}(256,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{69}\right)\) | \(e\left(\frac{8}{69}\right)\) | \(e\left(\frac{68}{69}\right)\) | \(e\left(\frac{62}{69}\right)\) | \(e\left(\frac{4}{23}\right)\) | \(e\left(\frac{1}{23}\right)\) | \(e\left(\frac{28}{69}\right)\) | \(e\left(\frac{49}{69}\right)\) | \(e\left(\frac{22}{23}\right)\) | \(e\left(\frac{16}{69}\right)\) |
| \(\chi_{417}(259,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{65}{69}\right)\) | \(e\left(\frac{61}{69}\right)\) | \(e\left(\frac{1}{69}\right)\) | \(e\left(\frac{7}{69}\right)\) | \(e\left(\frac{19}{23}\right)\) | \(e\left(\frac{22}{23}\right)\) | \(e\left(\frac{41}{69}\right)\) | \(e\left(\frac{20}{69}\right)\) | \(e\left(\frac{1}{23}\right)\) | \(e\left(\frac{53}{69}\right)\) |
| \(\chi_{417}(283,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{69}\right)\) | \(e\left(\frac{17}{69}\right)\) | \(e\left(\frac{41}{69}\right)\) | \(e\left(\frac{11}{69}\right)\) | \(e\left(\frac{20}{23}\right)\) | \(e\left(\frac{5}{23}\right)\) | \(e\left(\frac{25}{69}\right)\) | \(e\left(\frac{61}{69}\right)\) | \(e\left(\frac{18}{23}\right)\) | \(e\left(\frac{34}{69}\right)\) |
| \(\chi_{417}(289,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{38}{69}\right)\) | \(e\left(\frac{7}{69}\right)\) | \(e\left(\frac{25}{69}\right)\) | \(e\left(\frac{37}{69}\right)\) | \(e\left(\frac{15}{23}\right)\) | \(e\left(\frac{21}{23}\right)\) | \(e\left(\frac{59}{69}\right)\) | \(e\left(\frac{17}{69}\right)\) | \(e\left(\frac{2}{23}\right)\) | \(e\left(\frac{14}{69}\right)\) |
| \(\chi_{417}(298,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{44}{69}\right)\) | \(e\left(\frac{19}{69}\right)\) | \(e\left(\frac{58}{69}\right)\) | \(e\left(\frac{61}{69}\right)\) | \(e\left(\frac{21}{23}\right)\) | \(e\left(\frac{11}{23}\right)\) | \(e\left(\frac{32}{69}\right)\) | \(e\left(\frac{56}{69}\right)\) | \(e\left(\frac{12}{23}\right)\) | \(e\left(\frac{38}{69}\right)\) |