from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1728, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([0,27,16]))
chi.galois_orbit()
[g,chi] = znchar(Mod(37,1728))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(1728\) | |
| Conductor: | \(576\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(48\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 576.bm | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Related number fields
| Field of values: | \(\Q(\zeta_{48})\) |
| Fixed field: | Number field defined by a degree 48 polynomial |
Characters in Galois orbit
| Character | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{1728}(37,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{48}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{5}{48}\right)\) | \(-i\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{1}{6}\right)\) |
| \(\chi_{1728}(181,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{11}{48}\right)\) | \(e\left(\frac{1}{48}\right)\) | \(-i\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{5}{48}\right)\) | \(e\left(\frac{5}{6}\right)\) |
| \(\chi_{1728}(253,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{13}{48}\right)\) | \(e\left(\frac{23}{48}\right)\) | \(i\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{19}{48}\right)\) | \(e\left(\frac{1}{6}\right)\) |
| \(\chi_{1728}(397,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{48}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{17}{48}\right)\) | \(e\left(\frac{19}{48}\right)\) | \(i\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{5}{6}\right)\) |
| \(\chi_{1728}(469,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{48}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{19}{48}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(-i\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{13}{48}\right)\) | \(e\left(\frac{1}{6}\right)\) |
| \(\chi_{1728}(613,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{23}{48}\right)\) | \(e\left(\frac{37}{48}\right)\) | \(-i\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{5}{6}\right)\) |
| \(\chi_{1728}(685,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{48}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{11}{48}\right)\) | \(i\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{1}{6}\right)\) |
| \(\chi_{1728}(829,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{29}{48}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(i\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{35}{48}\right)\) | \(e\left(\frac{5}{6}\right)\) |
| \(\chi_{1728}(901,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{35}{48}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{29}{48}\right)\) | \(-i\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{1}{48}\right)\) | \(e\left(\frac{1}{6}\right)\) |
| \(\chi_{1728}(1045,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{35}{48}\right)\) | \(e\left(\frac{25}{48}\right)\) | \(-i\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{29}{48}\right)\) | \(e\left(\frac{5}{6}\right)\) |
| \(\chi_{1728}(1117,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{48}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{47}{48}\right)\) | \(i\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{1}{6}\right)\) |
| \(\chi_{1728}(1261,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{43}{48}\right)\) | \(i\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{23}{48}\right)\) | \(e\left(\frac{5}{6}\right)\) |
| \(\chi_{1728}(1333,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{17}{48}\right)\) | \(-i\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{37}{48}\right)\) | \(e\left(\frac{1}{6}\right)\) |
| \(\chi_{1728}(1477,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{48}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{13}{48}\right)\) | \(-i\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{17}{48}\right)\) | \(e\left(\frac{5}{6}\right)\) |
| \(\chi_{1728}(1549,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{48}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{1}{48}\right)\) | \(e\left(\frac{35}{48}\right)\) | \(i\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{1}{6}\right)\) |
| \(\chi_{1728}(1693,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{48}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{5}{48}\right)\) | \(e\left(\frac{31}{48}\right)\) | \(i\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{11}{48}\right)\) | \(e\left(\frac{5}{6}\right)\) |