from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1137, base_ring=CyclotomicField(126))
M = H._module
chi = DirichletCharacter(H, M([0,106]))
chi.galois_orbit()
[g,chi] = znchar(Mod(37,1137))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(1137\) | |
| Conductor: | \(379\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(63\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 379.m | 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: | $\Q(\zeta_{63})$ |
| Fixed field: | Number field defined by a degree 63 polynomial |
First 31 of 36 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{1137}(37,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{53}{63}\right)\) | \(e\left(\frac{43}{63}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{25}{63}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{62}{63}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{20}{63}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{23}{63}\right)\) |
| \(\chi_{1137}(64,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{63}\right)\) | \(e\left(\frac{2}{63}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{29}{63}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{19}{63}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{61}{63}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{4}{63}\right)\) |
| \(\chi_{1137}(67,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{44}{63}\right)\) | \(e\left(\frac{25}{63}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{16}{63}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{17}{63}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{38}{63}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{50}{63}\right)\) |
| \(\chi_{1137}(70,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{50}{63}\right)\) | \(e\left(\frac{37}{63}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{1}{63}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{5}{63}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{26}{63}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{11}{63}\right)\) |
| \(\chi_{1137}(139,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{16}{63}\right)\) | \(e\left(\frac{32}{63}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{23}{63}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{52}{63}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{31}{63}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{1}{63}\right)\) |
| \(\chi_{1137}(142,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{63}\right)\) | \(e\left(\frac{62}{63}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{17}{63}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{22}{63}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{1}{63}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{61}{63}\right)\) |
| \(\chi_{1137}(196,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{52}{63}\right)\) | \(e\left(\frac{41}{63}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{59}{63}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{43}{63}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{22}{63}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{19}{63}\right)\) |
| \(\chi_{1137}(205,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{34}{63}\right)\) | \(e\left(\frac{5}{63}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{41}{63}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{16}{63}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{58}{63}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{10}{63}\right)\) |
| \(\chi_{1137}(232,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{63}\right)\) | \(e\left(\frac{23}{63}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{50}{63}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{61}{63}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{40}{63}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{46}{63}\right)\) |
| \(\chi_{1137}(244,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{61}{63}\right)\) | \(e\left(\frac{59}{63}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{5}{63}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{25}{63}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{4}{63}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{55}{63}\right)\) |
| \(\chi_{1137}(316,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{40}{63}\right)\) | \(e\left(\frac{17}{63}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{26}{63}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{4}{63}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{46}{63}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{34}{63}\right)\) |
| \(\chi_{1137}(331,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{55}{63}\right)\) | \(e\left(\frac{47}{63}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{20}{63}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{37}{63}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{16}{63}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{31}{63}\right)\) |
| \(\chi_{1137}(352,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{63}\right)\) | \(e\left(\frac{11}{63}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{2}{63}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{10}{63}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{52}{63}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{22}{63}\right)\) |
| \(\chi_{1137}(385,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{63}\right)\) | \(e\left(\frac{46}{63}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{37}{63}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{59}{63}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{17}{63}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{29}{63}\right)\) |
| \(\chi_{1137}(409,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{47}{63}\right)\) | \(e\left(\frac{31}{63}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{40}{63}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{11}{63}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{32}{63}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{62}{63}\right)\) |
| \(\chi_{1137}(412,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{59}{63}\right)\) | \(e\left(\frac{55}{63}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{10}{63}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{50}{63}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{8}{63}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{47}{63}\right)\) |
| \(\chi_{1137}(415,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{46}{63}\right)\) | \(e\left(\frac{29}{63}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{11}{63}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{55}{63}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{34}{63}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{58}{63}\right)\) |
| \(\chi_{1137}(529,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{63}\right)\) | \(e\left(\frac{16}{63}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{43}{63}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{26}{63}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{47}{63}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{32}{63}\right)\) |
| \(\chi_{1137}(538,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{63}\right)\) | \(e\left(\frac{58}{63}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{22}{63}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{47}{63}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{5}{63}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{53}{63}\right)\) |
| \(\chi_{1137}(544,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{20}{63}\right)\) | \(e\left(\frac{40}{63}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{13}{63}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{2}{63}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{23}{63}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{17}{63}\right)\) |
| \(\chi_{1137}(577,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{63}\right)\) | \(e\left(\frac{38}{63}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{47}{63}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{46}{63}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{25}{63}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{13}{63}\right)\) |
| \(\chi_{1137}(601,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{63}\right)\) | \(e\left(\frac{26}{63}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{62}{63}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{58}{63}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{37}{63}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{52}{63}\right)\) |
| \(\chi_{1137}(646,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{58}{63}\right)\) | \(e\left(\frac{53}{63}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{44}{63}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{31}{63}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{10}{63}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{43}{63}\right)\) |
| \(\chi_{1137}(685,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{63}\right)\) | \(e\left(\frac{4}{63}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{58}{63}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{38}{63}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{59}{63}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{8}{63}\right)\) |
| \(\chi_{1137}(772,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{26}{63}\right)\) | \(e\left(\frac{52}{63}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{61}{63}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{53}{63}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{11}{63}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{41}{63}\right)\) |
| \(\chi_{1137}(781,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{63}\right)\) | \(e\left(\frac{8}{63}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{53}{63}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{13}{63}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{55}{63}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{16}{63}\right)\) |
| \(\chi_{1137}(799,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{10}{63}\right)\) | \(e\left(\frac{20}{63}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{38}{63}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{1}{63}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{43}{63}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{40}{63}\right)\) |
| \(\chi_{1137}(835,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{62}{63}\right)\) | \(e\left(\frac{61}{63}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{34}{63}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{44}{63}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{2}{63}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{59}{63}\right)\) |
| \(\chi_{1137}(841,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{22}{63}\right)\) | \(e\left(\frac{44}{63}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{8}{63}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{40}{63}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{19}{63}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{25}{63}\right)\) |
| \(\chi_{1137}(895,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{63}\right)\) | \(e\left(\frac{19}{63}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{55}{63}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{23}{63}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{44}{63}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{38}{63}\right)\) |
| \(\chi_{1137}(925,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{38}{63}\right)\) | \(e\left(\frac{13}{63}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{31}{63}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{29}{63}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{50}{63}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{26}{63}\right)\) |