from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1862, base_ring=CyclotomicField(126))
M = H._module
chi = DirichletCharacter(H, M([18,112]))
chi.galois_orbit()
[g,chi] = znchar(Mod(43,1862))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(1862\) | |
| Conductor: | \(931\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(63\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 931.cc | 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 | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(23\) | \(25\) | \(27\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{1862}(43,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{44}{63}\right)\) | \(e\left(\frac{23}{63}\right)\) | \(e\left(\frac{25}{63}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{10}{63}\right)\) | \(e\left(\frac{4}{63}\right)\) | \(e\left(\frac{29}{63}\right)\) | \(e\left(\frac{13}{63}\right)\) | \(e\left(\frac{46}{63}\right)\) | \(e\left(\frac{2}{21}\right)\) |
| \(\chi_{1862}(85,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{63}\right)\) | \(e\left(\frac{25}{63}\right)\) | \(e\left(\frac{8}{63}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{41}{63}\right)\) | \(e\left(\frac{29}{63}\right)\) | \(e\left(\frac{37}{63}\right)\) | \(e\left(\frac{47}{63}\right)\) | \(e\left(\frac{50}{63}\right)\) | \(e\left(\frac{4}{21}\right)\) |
| \(\chi_{1862}(169,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{50}{63}\right)\) | \(e\left(\frac{29}{63}\right)\) | \(e\left(\frac{37}{63}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{40}{63}\right)\) | \(e\left(\frac{16}{63}\right)\) | \(e\left(\frac{53}{63}\right)\) | \(e\left(\frac{52}{63}\right)\) | \(e\left(\frac{58}{63}\right)\) | \(e\left(\frac{8}{21}\right)\) |
| \(\chi_{1862}(225,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{20}{63}\right)\) | \(e\left(\frac{62}{63}\right)\) | \(e\left(\frac{40}{63}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{16}{63}\right)\) | \(e\left(\frac{19}{63}\right)\) | \(e\left(\frac{59}{63}\right)\) | \(e\left(\frac{46}{63}\right)\) | \(e\left(\frac{61}{63}\right)\) | \(e\left(\frac{20}{21}\right)\) |
| \(\chi_{1862}(253,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{61}{63}\right)\) | \(e\left(\frac{19}{63}\right)\) | \(e\left(\frac{59}{63}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{11}{63}\right)\) | \(e\left(\frac{17}{63}\right)\) | \(e\left(\frac{13}{63}\right)\) | \(e\left(\frac{8}{63}\right)\) | \(e\left(\frac{38}{63}\right)\) | \(e\left(\frac{19}{21}\right)\) |
| \(\chi_{1862}(309,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{63}\right)\) | \(e\left(\frac{59}{63}\right)\) | \(e\left(\frac{34}{63}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{1}{63}\right)\) | \(e\left(\frac{13}{63}\right)\) | \(e\left(\frac{47}{63}\right)\) | \(e\left(\frac{58}{63}\right)\) | \(e\left(\frac{55}{63}\right)\) | \(e\left(\frac{17}{21}\right)\) |
| \(\chi_{1862}(351,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{40}{63}\right)\) | \(e\left(\frac{61}{63}\right)\) | \(e\left(\frac{17}{63}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{32}{63}\right)\) | \(e\left(\frac{38}{63}\right)\) | \(e\left(\frac{55}{63}\right)\) | \(e\left(\frac{29}{63}\right)\) | \(e\left(\frac{59}{63}\right)\) | \(e\left(\frac{19}{21}\right)\) |
| \(\chi_{1862}(365,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{63}\right)\) | \(e\left(\frac{22}{63}\right)\) | \(e\left(\frac{2}{63}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{26}{63}\right)\) | \(e\left(\frac{23}{63}\right)\) | \(e\left(\frac{25}{63}\right)\) | \(e\left(\frac{59}{63}\right)\) | \(e\left(\frac{44}{63}\right)\) | \(e\left(\frac{1}{21}\right)\) |
| \(\chi_{1862}(435,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{63}\right)\) | \(e\left(\frac{2}{63}\right)\) | \(e\left(\frac{46}{63}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{31}{63}\right)\) | \(e\left(\frac{25}{63}\right)\) | \(e\left(\frac{8}{63}\right)\) | \(e\left(\frac{34}{63}\right)\) | \(e\left(\frac{4}{63}\right)\) | \(e\left(\frac{2}{21}\right)\) |
| \(\chi_{1862}(519,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{34}{63}\right)\) | \(e\left(\frac{55}{63}\right)\) | \(e\left(\frac{5}{63}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{2}{63}\right)\) | \(e\left(\frac{26}{63}\right)\) | \(e\left(\frac{31}{63}\right)\) | \(e\left(\frac{53}{63}\right)\) | \(e\left(\frac{47}{63}\right)\) | \(e\left(\frac{13}{21}\right)\) |
| \(\chi_{1862}(575,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{53}{63}\right)\) | \(e\left(\frac{32}{63}\right)\) | \(e\left(\frac{43}{63}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{55}{63}\right)\) | \(e\left(\frac{22}{63}\right)\) | \(e\left(\frac{2}{63}\right)\) | \(e\left(\frac{40}{63}\right)\) | \(e\left(\frac{1}{63}\right)\) | \(e\left(\frac{11}{21}\right)\) |
| \(\chi_{1862}(617,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{63}\right)\) | \(e\left(\frac{34}{63}\right)\) | \(e\left(\frac{26}{63}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{23}{63}\right)\) | \(e\left(\frac{47}{63}\right)\) | \(e\left(\frac{10}{63}\right)\) | \(e\left(\frac{11}{63}\right)\) | \(e\left(\frac{5}{63}\right)\) | \(e\left(\frac{13}{21}\right)\) |
| \(\chi_{1862}(631,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{63}\right)\) | \(e\left(\frac{58}{63}\right)\) | \(e\left(\frac{11}{63}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{17}{63}\right)\) | \(e\left(\frac{32}{63}\right)\) | \(e\left(\frac{43}{63}\right)\) | \(e\left(\frac{41}{63}\right)\) | \(e\left(\frac{53}{63}\right)\) | \(e\left(\frac{16}{21}\right)\) |
| \(\chi_{1862}(701,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{59}{63}\right)\) | \(e\left(\frac{38}{63}\right)\) | \(e\left(\frac{55}{63}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{22}{63}\right)\) | \(e\left(\frac{34}{63}\right)\) | \(e\left(\frac{26}{63}\right)\) | \(e\left(\frac{16}{63}\right)\) | \(e\left(\frac{13}{63}\right)\) | \(e\left(\frac{17}{21}\right)\) |
| \(\chi_{1862}(757,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{63}\right)\) | \(e\left(\frac{8}{63}\right)\) | \(e\left(\frac{58}{63}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{61}{63}\right)\) | \(e\left(\frac{37}{63}\right)\) | \(e\left(\frac{32}{63}\right)\) | \(e\left(\frac{10}{63}\right)\) | \(e\left(\frac{16}{63}\right)\) | \(e\left(\frac{8}{21}\right)\) |
| \(\chi_{1862}(841,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{26}{63}\right)\) | \(e\left(\frac{5}{63}\right)\) | \(e\left(\frac{52}{63}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{46}{63}\right)\) | \(e\left(\frac{31}{63}\right)\) | \(e\left(\frac{20}{63}\right)\) | \(e\left(\frac{22}{63}\right)\) | \(e\left(\frac{10}{63}\right)\) | \(e\left(\frac{5}{21}\right)\) |
| \(\chi_{1862}(897,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{10}{63}\right)\) | \(e\left(\frac{31}{63}\right)\) | \(e\left(\frac{20}{63}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{8}{63}\right)\) | \(e\left(\frac{41}{63}\right)\) | \(e\left(\frac{61}{63}\right)\) | \(e\left(\frac{23}{63}\right)\) | \(e\left(\frac{62}{63}\right)\) | \(e\left(\frac{10}{21}\right)\) |
| \(\chi_{1862}(967,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{32}{63}\right)\) | \(e\left(\frac{11}{63}\right)\) | \(e\left(\frac{1}{63}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{13}{63}\right)\) | \(e\left(\frac{43}{63}\right)\) | \(e\left(\frac{44}{63}\right)\) | \(e\left(\frac{61}{63}\right)\) | \(e\left(\frac{22}{63}\right)\) | \(e\left(\frac{11}{21}\right)\) |
| \(\chi_{1862}(1023,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{63}\right)\) | \(e\left(\frac{44}{63}\right)\) | \(e\left(\frac{4}{63}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{52}{63}\right)\) | \(e\left(\frac{46}{63}\right)\) | \(e\left(\frac{50}{63}\right)\) | \(e\left(\frac{55}{63}\right)\) | \(e\left(\frac{25}{63}\right)\) | \(e\left(\frac{2}{21}\right)\) |
| \(\chi_{1862}(1051,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{63}\right)\) | \(e\left(\frac{1}{63}\right)\) | \(e\left(\frac{23}{63}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{47}{63}\right)\) | \(e\left(\frac{44}{63}\right)\) | \(e\left(\frac{4}{63}\right)\) | \(e\left(\frac{17}{63}\right)\) | \(e\left(\frac{2}{63}\right)\) | \(e\left(\frac{1}{21}\right)\) |
| \(\chi_{1862}(1107,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{62}{63}\right)\) | \(e\left(\frac{41}{63}\right)\) | \(e\left(\frac{61}{63}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{37}{63}\right)\) | \(e\left(\frac{40}{63}\right)\) | \(e\left(\frac{38}{63}\right)\) | \(e\left(\frac{4}{63}\right)\) | \(e\left(\frac{19}{63}\right)\) | \(e\left(\frac{20}{21}\right)\) |
| \(\chi_{1862}(1149,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{22}{63}\right)\) | \(e\left(\frac{43}{63}\right)\) | \(e\left(\frac{44}{63}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{5}{63}\right)\) | \(e\left(\frac{2}{63}\right)\) | \(e\left(\frac{46}{63}\right)\) | \(e\left(\frac{38}{63}\right)\) | \(e\left(\frac{23}{63}\right)\) | \(e\left(\frac{1}{21}\right)\) |
| \(\chi_{1862}(1163,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{46}{63}\right)\) | \(e\left(\frac{4}{63}\right)\) | \(e\left(\frac{29}{63}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{62}{63}\right)\) | \(e\left(\frac{50}{63}\right)\) | \(e\left(\frac{16}{63}\right)\) | \(e\left(\frac{5}{63}\right)\) | \(e\left(\frac{8}{63}\right)\) | \(e\left(\frac{4}{21}\right)\) |
| \(\chi_{1862}(1233,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{63}\right)\) | \(e\left(\frac{47}{63}\right)\) | \(e\left(\frac{10}{63}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{4}{63}\right)\) | \(e\left(\frac{52}{63}\right)\) | \(e\left(\frac{62}{63}\right)\) | \(e\left(\frac{43}{63}\right)\) | \(e\left(\frac{31}{63}\right)\) | \(e\left(\frac{5}{21}\right)\) |
| \(\chi_{1862}(1289,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{38}{63}\right)\) | \(e\left(\frac{17}{63}\right)\) | \(e\left(\frac{13}{63}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{43}{63}\right)\) | \(e\left(\frac{55}{63}\right)\) | \(e\left(\frac{5}{63}\right)\) | \(e\left(\frac{37}{63}\right)\) | \(e\left(\frac{34}{63}\right)\) | \(e\left(\frac{17}{21}\right)\) |
| \(\chi_{1862}(1317,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{16}{63}\right)\) | \(e\left(\frac{37}{63}\right)\) | \(e\left(\frac{32}{63}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{38}{63}\right)\) | \(e\left(\frac{53}{63}\right)\) | \(e\left(\frac{22}{63}\right)\) | \(e\left(\frac{62}{63}\right)\) | \(e\left(\frac{11}{63}\right)\) | \(e\left(\frac{16}{21}\right)\) |
| \(\chi_{1862}(1415,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{58}{63}\right)\) | \(e\left(\frac{16}{63}\right)\) | \(e\left(\frac{53}{63}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{59}{63}\right)\) | \(e\left(\frac{11}{63}\right)\) | \(e\left(\frac{1}{63}\right)\) | \(e\left(\frac{20}{63}\right)\) | \(e\left(\frac{32}{63}\right)\) | \(e\left(\frac{16}{21}\right)\) |
| \(\chi_{1862}(1429,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{63}\right)\) | \(e\left(\frac{40}{63}\right)\) | \(e\left(\frac{38}{63}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{53}{63}\right)\) | \(e\left(\frac{59}{63}\right)\) | \(e\left(\frac{34}{63}\right)\) | \(e\left(\frac{50}{63}\right)\) | \(e\left(\frac{17}{63}\right)\) | \(e\left(\frac{19}{21}\right)\) |
| \(\chi_{1862}(1499,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{63}\right)\) | \(e\left(\frac{20}{63}\right)\) | \(e\left(\frac{19}{63}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{58}{63}\right)\) | \(e\left(\frac{61}{63}\right)\) | \(e\left(\frac{17}{63}\right)\) | \(e\left(\frac{25}{63}\right)\) | \(e\left(\frac{40}{63}\right)\) | \(e\left(\frac{20}{21}\right)\) |
| \(\chi_{1862}(1555,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{63}\right)\) | \(e\left(\frac{53}{63}\right)\) | \(e\left(\frac{22}{63}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{34}{63}\right)\) | \(e\left(\frac{1}{63}\right)\) | \(e\left(\frac{23}{63}\right)\) | \(e\left(\frac{19}{63}\right)\) | \(e\left(\frac{43}{63}\right)\) | \(e\left(\frac{11}{21}\right)\) |
| \(\chi_{1862}(1583,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{52}{63}\right)\) | \(e\left(\frac{10}{63}\right)\) | \(e\left(\frac{41}{63}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{29}{63}\right)\) | \(e\left(\frac{62}{63}\right)\) | \(e\left(\frac{40}{63}\right)\) | \(e\left(\frac{44}{63}\right)\) | \(e\left(\frac{20}{63}\right)\) | \(e\left(\frac{10}{21}\right)\) |