from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(927, base_ring=CyclotomicField(102))
M = H._module
chi = DirichletCharacter(H, M([34,72]))
chi.galois_orbit()
[g,chi] = znchar(Mod(13,927))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(927\) | |
| Conductor: | \(927\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(51\) | 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: | $\Q(\zeta_{51})$ |
| Fixed field: | Number field defined by a degree 51 polynomial |
First 31 of 32 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{927}(13,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{20}{51}\right)\) | \(e\left(\frac{40}{51}\right)\) | \(e\left(\frac{19}{51}\right)\) | \(e\left(\frac{8}{51}\right)\) | \(e\left(\frac{3}{17}\right)\) | \(e\left(\frac{13}{17}\right)\) | \(e\left(\frac{20}{51}\right)\) | \(e\left(\frac{25}{51}\right)\) | \(e\left(\frac{28}{51}\right)\) | \(e\left(\frac{29}{51}\right)\) |
| \(\chi_{927}(34,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{51}\right)\) | \(e\left(\frac{35}{51}\right)\) | \(e\left(\frac{23}{51}\right)\) | \(e\left(\frac{7}{51}\right)\) | \(e\left(\frac{9}{17}\right)\) | \(e\left(\frac{5}{17}\right)\) | \(e\left(\frac{43}{51}\right)\) | \(e\left(\frac{41}{51}\right)\) | \(e\left(\frac{50}{51}\right)\) | \(e\left(\frac{19}{51}\right)\) |
| \(\chi_{927}(61,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{10}{51}\right)\) | \(e\left(\frac{20}{51}\right)\) | \(e\left(\frac{35}{51}\right)\) | \(e\left(\frac{4}{51}\right)\) | \(e\left(\frac{10}{17}\right)\) | \(e\left(\frac{15}{17}\right)\) | \(e\left(\frac{10}{51}\right)\) | \(e\left(\frac{38}{51}\right)\) | \(e\left(\frac{14}{51}\right)\) | \(e\left(\frac{40}{51}\right)\) |
| \(\chi_{927}(76,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{51}\right)\) | \(e\left(\frac{31}{51}\right)\) | \(e\left(\frac{16}{51}\right)\) | \(e\left(\frac{47}{51}\right)\) | \(e\left(\frac{7}{17}\right)\) | \(e\left(\frac{2}{17}\right)\) | \(e\left(\frac{41}{51}\right)\) | \(e\left(\frac{13}{51}\right)\) | \(e\left(\frac{37}{51}\right)\) | \(e\left(\frac{11}{51}\right)\) |
| \(\chi_{927}(79,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{22}{51}\right)\) | \(e\left(\frac{44}{51}\right)\) | \(e\left(\frac{26}{51}\right)\) | \(e\left(\frac{19}{51}\right)\) | \(e\left(\frac{5}{17}\right)\) | \(e\left(\frac{16}{17}\right)\) | \(e\left(\frac{22}{51}\right)\) | \(e\left(\frac{2}{51}\right)\) | \(e\left(\frac{41}{51}\right)\) | \(e\left(\frac{37}{51}\right)\) |
| \(\chi_{927}(112,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{50}{51}\right)\) | \(e\left(\frac{49}{51}\right)\) | \(e\left(\frac{22}{51}\right)\) | \(e\left(\frac{20}{51}\right)\) | \(e\left(\frac{16}{17}\right)\) | \(e\left(\frac{7}{17}\right)\) | \(e\left(\frac{50}{51}\right)\) | \(e\left(\frac{37}{51}\right)\) | \(e\left(\frac{19}{51}\right)\) | \(e\left(\frac{47}{51}\right)\) |
| \(\chi_{927}(133,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{46}{51}\right)\) | \(e\left(\frac{41}{51}\right)\) | \(e\left(\frac{8}{51}\right)\) | \(e\left(\frac{49}{51}\right)\) | \(e\left(\frac{12}{17}\right)\) | \(e\left(\frac{1}{17}\right)\) | \(e\left(\frac{46}{51}\right)\) | \(e\left(\frac{32}{51}\right)\) | \(e\left(\frac{44}{51}\right)\) | \(e\left(\frac{31}{51}\right)\) |
| \(\chi_{927}(169,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{40}{51}\right)\) | \(e\left(\frac{29}{51}\right)\) | \(e\left(\frac{38}{51}\right)\) | \(e\left(\frac{16}{51}\right)\) | \(e\left(\frac{6}{17}\right)\) | \(e\left(\frac{9}{17}\right)\) | \(e\left(\frac{40}{51}\right)\) | \(e\left(\frac{50}{51}\right)\) | \(e\left(\frac{5}{51}\right)\) | \(e\left(\frac{7}{51}\right)\) |
| \(\chi_{927}(175,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{47}{51}\right)\) | \(e\left(\frac{43}{51}\right)\) | \(e\left(\frac{37}{51}\right)\) | \(e\left(\frac{29}{51}\right)\) | \(e\left(\frac{13}{17}\right)\) | \(e\left(\frac{11}{17}\right)\) | \(e\left(\frac{47}{51}\right)\) | \(e\left(\frac{46}{51}\right)\) | \(e\left(\frac{25}{51}\right)\) | \(e\left(\frac{35}{51}\right)\) |
| \(\chi_{927}(184,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{32}{51}\right)\) | \(e\left(\frac{13}{51}\right)\) | \(e\left(\frac{10}{51}\right)\) | \(e\left(\frac{23}{51}\right)\) | \(e\left(\frac{15}{17}\right)\) | \(e\left(\frac{14}{17}\right)\) | \(e\left(\frac{32}{51}\right)\) | \(e\left(\frac{40}{51}\right)\) | \(e\left(\frac{4}{51}\right)\) | \(e\left(\frac{26}{51}\right)\) |
| \(\chi_{927}(196,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{51}\right)\) | \(e\left(\frac{8}{51}\right)\) | \(e\left(\frac{14}{51}\right)\) | \(e\left(\frac{22}{51}\right)\) | \(e\left(\frac{4}{17}\right)\) | \(e\left(\frac{6}{17}\right)\) | \(e\left(\frac{4}{51}\right)\) | \(e\left(\frac{5}{51}\right)\) | \(e\left(\frac{26}{51}\right)\) | \(e\left(\frac{16}{51}\right)\) |
| \(\chi_{927}(214,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{51}\right)\) | \(e\left(\frac{11}{51}\right)\) | \(e\left(\frac{32}{51}\right)\) | \(e\left(\frac{43}{51}\right)\) | \(e\left(\frac{14}{17}\right)\) | \(e\left(\frac{4}{17}\right)\) | \(e\left(\frac{31}{51}\right)\) | \(e\left(\frac{26}{51}\right)\) | \(e\left(\frac{23}{51}\right)\) | \(e\left(\frac{22}{51}\right)\) |
| \(\chi_{927}(220,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{51}\right)\) | \(e\left(\frac{4}{51}\right)\) | \(e\left(\frac{7}{51}\right)\) | \(e\left(\frac{11}{51}\right)\) | \(e\left(\frac{2}{17}\right)\) | \(e\left(\frac{3}{17}\right)\) | \(e\left(\frac{2}{51}\right)\) | \(e\left(\frac{28}{51}\right)\) | \(e\left(\frac{13}{51}\right)\) | \(e\left(\frac{8}{51}\right)\) |
| \(\chi_{927}(229,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{35}{51}\right)\) | \(e\left(\frac{19}{51}\right)\) | \(e\left(\frac{46}{51}\right)\) | \(e\left(\frac{14}{51}\right)\) | \(e\left(\frac{1}{17}\right)\) | \(e\left(\frac{10}{17}\right)\) | \(e\left(\frac{35}{51}\right)\) | \(e\left(\frac{31}{51}\right)\) | \(e\left(\frac{49}{51}\right)\) | \(e\left(\frac{38}{51}\right)\) |
| \(\chi_{927}(322,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{51}\right)\) | \(e\left(\frac{23}{51}\right)\) | \(e\left(\frac{2}{51}\right)\) | \(e\left(\frac{25}{51}\right)\) | \(e\left(\frac{3}{17}\right)\) | \(e\left(\frac{13}{17}\right)\) | \(e\left(\frac{37}{51}\right)\) | \(e\left(\frac{8}{51}\right)\) | \(e\left(\frac{11}{51}\right)\) | \(e\left(\frac{46}{51}\right)\) |
| \(\chi_{927}(373,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{51}\right)\) | \(e\left(\frac{22}{51}\right)\) | \(e\left(\frac{13}{51}\right)\) | \(e\left(\frac{35}{51}\right)\) | \(e\left(\frac{11}{17}\right)\) | \(e\left(\frac{8}{17}\right)\) | \(e\left(\frac{11}{51}\right)\) | \(e\left(\frac{1}{51}\right)\) | \(e\left(\frac{46}{51}\right)\) | \(e\left(\frac{44}{51}\right)\) |
| \(\chi_{927}(385,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{51}\right)\) | \(e\left(\frac{14}{51}\right)\) | \(e\left(\frac{50}{51}\right)\) | \(e\left(\frac{13}{51}\right)\) | \(e\left(\frac{7}{17}\right)\) | \(e\left(\frac{2}{17}\right)\) | \(e\left(\frac{7}{51}\right)\) | \(e\left(\frac{47}{51}\right)\) | \(e\left(\frac{20}{51}\right)\) | \(e\left(\frac{28}{51}\right)\) |
| \(\chi_{927}(409,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{51}\right)\) | \(e\left(\frac{16}{51}\right)\) | \(e\left(\frac{28}{51}\right)\) | \(e\left(\frac{44}{51}\right)\) | \(e\left(\frac{8}{17}\right)\) | \(e\left(\frac{12}{17}\right)\) | \(e\left(\frac{8}{51}\right)\) | \(e\left(\frac{10}{51}\right)\) | \(e\left(\frac{1}{51}\right)\) | \(e\left(\frac{32}{51}\right)\) |
| \(\chi_{927}(421,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{16}{51}\right)\) | \(e\left(\frac{32}{51}\right)\) | \(e\left(\frac{5}{51}\right)\) | \(e\left(\frac{37}{51}\right)\) | \(e\left(\frac{16}{17}\right)\) | \(e\left(\frac{7}{17}\right)\) | \(e\left(\frac{16}{51}\right)\) | \(e\left(\frac{20}{51}\right)\) | \(e\left(\frac{2}{51}\right)\) | \(e\left(\frac{13}{51}\right)\) |
| \(\chi_{927}(484,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{51}\right)\) | \(e\left(\frac{26}{51}\right)\) | \(e\left(\frac{20}{51}\right)\) | \(e\left(\frac{46}{51}\right)\) | \(e\left(\frac{13}{17}\right)\) | \(e\left(\frac{11}{17}\right)\) | \(e\left(\frac{13}{51}\right)\) | \(e\left(\frac{29}{51}\right)\) | \(e\left(\frac{8}{51}\right)\) | \(e\left(\frac{1}{51}\right)\) |
| \(\chi_{927}(493,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{49}{51}\right)\) | \(e\left(\frac{47}{51}\right)\) | \(e\left(\frac{44}{51}\right)\) | \(e\left(\frac{40}{51}\right)\) | \(e\left(\frac{15}{17}\right)\) | \(e\left(\frac{14}{17}\right)\) | \(e\left(\frac{49}{51}\right)\) | \(e\left(\frac{23}{51}\right)\) | \(e\left(\frac{38}{51}\right)\) | \(e\left(\frac{43}{51}\right)\) |
| \(\chi_{927}(529,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{51}\right)\) | \(e\left(\frac{38}{51}\right)\) | \(e\left(\frac{41}{51}\right)\) | \(e\left(\frac{28}{51}\right)\) | \(e\left(\frac{2}{17}\right)\) | \(e\left(\frac{3}{17}\right)\) | \(e\left(\frac{19}{51}\right)\) | \(e\left(\frac{11}{51}\right)\) | \(e\left(\frac{47}{51}\right)\) | \(e\left(\frac{25}{51}\right)\) |
| \(\chi_{927}(538,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{51}\right)\) | \(e\left(\frac{2}{51}\right)\) | \(e\left(\frac{29}{51}\right)\) | \(e\left(\frac{31}{51}\right)\) | \(e\left(\frac{1}{17}\right)\) | \(e\left(\frac{10}{17}\right)\) | \(e\left(\frac{1}{51}\right)\) | \(e\left(\frac{14}{51}\right)\) | \(e\left(\frac{32}{51}\right)\) | \(e\left(\frac{4}{51}\right)\) |
| \(\chi_{927}(652,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{26}{51}\right)\) | \(e\left(\frac{1}{51}\right)\) | \(e\left(\frac{40}{51}\right)\) | \(e\left(\frac{41}{51}\right)\) | \(e\left(\frac{9}{17}\right)\) | \(e\left(\frac{5}{17}\right)\) | \(e\left(\frac{26}{51}\right)\) | \(e\left(\frac{7}{51}\right)\) | \(e\left(\frac{16}{51}\right)\) | \(e\left(\frac{2}{51}\right)\) |
| \(\chi_{927}(679,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{44}{51}\right)\) | \(e\left(\frac{37}{51}\right)\) | \(e\left(\frac{1}{51}\right)\) | \(e\left(\frac{38}{51}\right)\) | \(e\left(\frac{10}{17}\right)\) | \(e\left(\frac{15}{17}\right)\) | \(e\left(\frac{44}{51}\right)\) | \(e\left(\frac{4}{51}\right)\) | \(e\left(\frac{31}{51}\right)\) | \(e\left(\frac{23}{51}\right)\) |
| \(\chi_{927}(682,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{28}{51}\right)\) | \(e\left(\frac{5}{51}\right)\) | \(e\left(\frac{47}{51}\right)\) | \(e\left(\frac{1}{51}\right)\) | \(e\left(\frac{11}{17}\right)\) | \(e\left(\frac{8}{17}\right)\) | \(e\left(\frac{28}{51}\right)\) | \(e\left(\frac{35}{51}\right)\) | \(e\left(\frac{29}{51}\right)\) | \(e\left(\frac{10}{51}\right)\) |
| \(\chi_{927}(697,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{51}\right)\) | \(e\left(\frac{10}{51}\right)\) | \(e\left(\frac{43}{51}\right)\) | \(e\left(\frac{2}{51}\right)\) | \(e\left(\frac{5}{17}\right)\) | \(e\left(\frac{16}{17}\right)\) | \(e\left(\frac{5}{51}\right)\) | \(e\left(\frac{19}{51}\right)\) | \(e\left(\frac{7}{51}\right)\) | \(e\left(\frac{20}{51}\right)\) |
| \(\chi_{927}(718,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{51}\right)\) | \(e\left(\frac{50}{51}\right)\) | \(e\left(\frac{11}{51}\right)\) | \(e\left(\frac{10}{51}\right)\) | \(e\left(\frac{8}{17}\right)\) | \(e\left(\frac{12}{17}\right)\) | \(e\left(\frac{25}{51}\right)\) | \(e\left(\frac{44}{51}\right)\) | \(e\left(\frac{35}{51}\right)\) | \(e\left(\frac{49}{51}\right)\) |
| \(\chi_{927}(751,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{51}\right)\) | \(e\left(\frac{7}{51}\right)\) | \(e\left(\frac{25}{51}\right)\) | \(e\left(\frac{32}{51}\right)\) | \(e\left(\frac{12}{17}\right)\) | \(e\left(\frac{1}{17}\right)\) | \(e\left(\frac{29}{51}\right)\) | \(e\left(\frac{49}{51}\right)\) | \(e\left(\frac{10}{51}\right)\) | \(e\left(\frac{14}{51}\right)\) |
| \(\chi_{927}(787,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{51}\right)\) | \(e\left(\frac{46}{51}\right)\) | \(e\left(\frac{4}{51}\right)\) | \(e\left(\frac{50}{51}\right)\) | \(e\left(\frac{6}{17}\right)\) | \(e\left(\frac{9}{17}\right)\) | \(e\left(\frac{23}{51}\right)\) | \(e\left(\frac{16}{51}\right)\) | \(e\left(\frac{22}{51}\right)\) | \(e\left(\frac{41}{51}\right)\) |
| \(\chi_{927}(814,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{38}{51}\right)\) | \(e\left(\frac{25}{51}\right)\) | \(e\left(\frac{31}{51}\right)\) | \(e\left(\frac{5}{51}\right)\) | \(e\left(\frac{4}{17}\right)\) | \(e\left(\frac{6}{17}\right)\) | \(e\left(\frac{38}{51}\right)\) | \(e\left(\frac{22}{51}\right)\) | \(e\left(\frac{43}{51}\right)\) | \(e\left(\frac{50}{51}\right)\) |