from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(256, base_ring=CyclotomicField(32))
M = H._module
chi = DirichletCharacter(H, M([0,3]))
chi.galois_orbit()
[g,chi] = znchar(Mod(9,256))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(256\) | |
| Conductor: | \(128\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(32\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 128.k | 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: | \(\Q(\zeta_{32})\) |
| Fixed field: | \(\Q(\zeta_{128})^+\) |
Characters in Galois orbit
| Character | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{256}(9,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{32}\right)\) | \(e\left(\frac{3}{32}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{31}{32}\right)\) | \(e\left(\frac{13}{32}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{5}{32}\right)\) | \(e\left(\frac{7}{32}\right)\) |
| \(\chi_{256}(25,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{32}\right)\) | \(e\left(\frac{1}{32}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{21}{32}\right)\) | \(e\left(\frac{15}{32}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{23}{32}\right)\) | \(e\left(\frac{13}{32}\right)\) |
| \(\chi_{256}(41,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{32}\right)\) | \(e\left(\frac{31}{32}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{11}{32}\right)\) | \(e\left(\frac{17}{32}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{9}{32}\right)\) | \(e\left(\frac{19}{32}\right)\) |
| \(\chi_{256}(57,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{32}\right)\) | \(e\left(\frac{29}{32}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{1}{32}\right)\) | \(e\left(\frac{19}{32}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{27}{32}\right)\) | \(e\left(\frac{25}{32}\right)\) |
| \(\chi_{256}(73,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{32}\right)\) | \(e\left(\frac{27}{32}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{23}{32}\right)\) | \(e\left(\frac{21}{32}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{13}{32}\right)\) | \(e\left(\frac{31}{32}\right)\) |
| \(\chi_{256}(89,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{32}\right)\) | \(e\left(\frac{25}{32}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{13}{32}\right)\) | \(e\left(\frac{23}{32}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{31}{32}\right)\) | \(e\left(\frac{5}{32}\right)\) |
| \(\chi_{256}(105,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{32}\right)\) | \(e\left(\frac{23}{32}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{3}{32}\right)\) | \(e\left(\frac{25}{32}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{17}{32}\right)\) | \(e\left(\frac{11}{32}\right)\) |
| \(\chi_{256}(121,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{32}\right)\) | \(e\left(\frac{21}{32}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{25}{32}\right)\) | \(e\left(\frac{27}{32}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{3}{32}\right)\) | \(e\left(\frac{17}{32}\right)\) |
| \(\chi_{256}(137,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{32}\right)\) | \(e\left(\frac{19}{32}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{15}{32}\right)\) | \(e\left(\frac{29}{32}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{21}{32}\right)\) | \(e\left(\frac{23}{32}\right)\) |
| \(\chi_{256}(153,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{32}\right)\) | \(e\left(\frac{17}{32}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{5}{32}\right)\) | \(e\left(\frac{31}{32}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{7}{32}\right)\) | \(e\left(\frac{29}{32}\right)\) |
| \(\chi_{256}(169,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{32}\right)\) | \(e\left(\frac{15}{32}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{27}{32}\right)\) | \(e\left(\frac{1}{32}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{25}{32}\right)\) | \(e\left(\frac{3}{32}\right)\) |
| \(\chi_{256}(185,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{32}\right)\) | \(e\left(\frac{13}{32}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{17}{32}\right)\) | \(e\left(\frac{3}{32}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{11}{32}\right)\) | \(e\left(\frac{9}{32}\right)\) |
| \(\chi_{256}(201,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{32}\right)\) | \(e\left(\frac{11}{32}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{7}{32}\right)\) | \(e\left(\frac{5}{32}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{29}{32}\right)\) | \(e\left(\frac{15}{32}\right)\) |
| \(\chi_{256}(217,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{27}{32}\right)\) | \(e\left(\frac{9}{32}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{29}{32}\right)\) | \(e\left(\frac{7}{32}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{15}{32}\right)\) | \(e\left(\frac{21}{32}\right)\) |
| \(\chi_{256}(233,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{21}{32}\right)\) | \(e\left(\frac{7}{32}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{19}{32}\right)\) | \(e\left(\frac{9}{32}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{1}{32}\right)\) | \(e\left(\frac{27}{32}\right)\) |
| \(\chi_{256}(249,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{15}{32}\right)\) | \(e\left(\frac{5}{32}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{9}{32}\right)\) | \(e\left(\frac{11}{32}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{19}{32}\right)\) | \(e\left(\frac{1}{32}\right)\) |