from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6930, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,45,10,42]))
chi.galois_orbit()
[g,chi] = znchar(Mod(73,6930))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(6930\) | |
| Conductor: | \(385\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 385.bs | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Related number fields
| Field of values: | \(\Q(\zeta_{60})\) |
| Fixed field: | Number field defined by a degree 60 polynomial |
Characters in Galois orbit
| Character | \(-1\) | \(1\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) | \(47\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{6930}(73,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(-i\) | \(e\left(\frac{11}{60}\right)\) |
| \(\chi_{6930}(523,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(-i\) | \(e\left(\frac{7}{60}\right)\) |
| \(\chi_{6930}(1207,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(i\) | \(e\left(\frac{29}{60}\right)\) |
| \(\chi_{6930}(1333,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(-i\) | \(e\left(\frac{23}{60}\right)\) |
| \(\chi_{6930}(1657,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(i\) | \(e\left(\frac{1}{60}\right)\) |
| \(\chi_{6930}(2593,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(-i\) | \(e\left(\frac{59}{60}\right)\) |
| \(\chi_{6930}(2917,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(i\) | \(e\left(\frac{13}{60}\right)\) |
| \(\chi_{6930}(3043,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(-i\) | \(e\left(\frac{31}{60}\right)\) |
| \(\chi_{6930}(3097,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(i\) | \(e\left(\frac{17}{60}\right)\) |
| \(\chi_{6930}(4177,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(i\) | \(e\left(\frac{49}{60}\right)\) |
| \(\chi_{6930}(4303,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(-i\) | \(e\left(\frac{43}{60}\right)\) |
| \(\chi_{6930}(4483,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(-i\) | \(e\left(\frac{47}{60}\right)\) |
| \(\chi_{6930}(5563,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(-i\) | \(e\left(\frac{19}{60}\right)\) |
| \(\chi_{6930}(5617,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(i\) | \(e\left(\frac{41}{60}\right)\) |
| \(\chi_{6930}(6067,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(i\) | \(e\left(\frac{37}{60}\right)\) |
| \(\chi_{6930}(6877,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(i\) | \(e\left(\frac{53}{60}\right)\) |