from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(256, base_ring=CyclotomicField(64))
M = H._module
chi = DirichletCharacter(H, M([0,1]))
chi.galois_orbit()
[g,chi] = znchar(Mod(5,256))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(256\) | |
| Conductor: | \(256\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(64\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | 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: | $\Q(\zeta_{64})$ |
| Fixed field: | Number field defined by a degree 64 polynomial |
First 31 of 32 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{256}(5,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{35}{64}\right)\) | \(e\left(\frac{1}{64}\right)\) | \(e\left(\frac{5}{32}\right)\) | \(e\left(\frac{3}{32}\right)\) | \(e\left(\frac{21}{64}\right)\) | \(e\left(\frac{47}{64}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{23}{64}\right)\) | \(e\left(\frac{45}{64}\right)\) |
| \(\chi_{256}(13,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{45}{64}\right)\) | \(e\left(\frac{47}{64}\right)\) | \(e\left(\frac{11}{32}\right)\) | \(e\left(\frac{13}{32}\right)\) | \(e\left(\frac{27}{64}\right)\) | \(e\left(\frac{33}{64}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{57}{64}\right)\) | \(e\left(\frac{3}{64}\right)\) |
| \(\chi_{256}(21,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{39}{64}\right)\) | \(e\left(\frac{45}{64}\right)\) | \(e\left(\frac{1}{32}\right)\) | \(e\left(\frac{7}{32}\right)\) | \(e\left(\frac{49}{64}\right)\) | \(e\left(\frac{3}{64}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{11}{64}\right)\) | \(e\left(\frac{41}{64}\right)\) |
| \(\chi_{256}(29,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{64}\right)\) | \(e\left(\frac{59}{64}\right)\) | \(e\left(\frac{7}{32}\right)\) | \(e\left(\frac{17}{32}\right)\) | \(e\left(\frac{23}{64}\right)\) | \(e\left(\frac{21}{64}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{13}{64}\right)\) | \(e\left(\frac{31}{64}\right)\) |
| \(\chi_{256}(37,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{64}\right)\) | \(e\left(\frac{25}{64}\right)\) | \(e\left(\frac{29}{32}\right)\) | \(e\left(\frac{11}{32}\right)\) | \(e\left(\frac{13}{64}\right)\) | \(e\left(\frac{23}{64}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{63}{64}\right)\) | \(e\left(\frac{37}{64}\right)\) |
| \(\chi_{256}(45,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{53}{64}\right)\) | \(e\left(\frac{7}{64}\right)\) | \(e\left(\frac{3}{32}\right)\) | \(e\left(\frac{21}{32}\right)\) | \(e\left(\frac{19}{64}\right)\) | \(e\left(\frac{9}{64}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{33}{64}\right)\) | \(e\left(\frac{59}{64}\right)\) |
| \(\chi_{256}(53,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{47}{64}\right)\) | \(e\left(\frac{5}{64}\right)\) | \(e\left(\frac{25}{32}\right)\) | \(e\left(\frac{15}{32}\right)\) | \(e\left(\frac{41}{64}\right)\) | \(e\left(\frac{43}{64}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{51}{64}\right)\) | \(e\left(\frac{33}{64}\right)\) |
| \(\chi_{256}(61,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{64}\right)\) | \(e\left(\frac{19}{64}\right)\) | \(e\left(\frac{31}{32}\right)\) | \(e\left(\frac{25}{32}\right)\) | \(e\left(\frac{15}{64}\right)\) | \(e\left(\frac{61}{64}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{53}{64}\right)\) | \(e\left(\frac{23}{64}\right)\) |
| \(\chi_{256}(69,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{51}{64}\right)\) | \(e\left(\frac{49}{64}\right)\) | \(e\left(\frac{21}{32}\right)\) | \(e\left(\frac{19}{32}\right)\) | \(e\left(\frac{5}{64}\right)\) | \(e\left(\frac{63}{64}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{39}{64}\right)\) | \(e\left(\frac{29}{64}\right)\) |
| \(\chi_{256}(77,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{61}{64}\right)\) | \(e\left(\frac{31}{64}\right)\) | \(e\left(\frac{27}{32}\right)\) | \(e\left(\frac{29}{32}\right)\) | \(e\left(\frac{11}{64}\right)\) | \(e\left(\frac{49}{64}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{9}{64}\right)\) | \(e\left(\frac{51}{64}\right)\) |
| \(\chi_{256}(85,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{55}{64}\right)\) | \(e\left(\frac{29}{64}\right)\) | \(e\left(\frac{17}{32}\right)\) | \(e\left(\frac{23}{32}\right)\) | \(e\left(\frac{33}{64}\right)\) | \(e\left(\frac{19}{64}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{27}{64}\right)\) | \(e\left(\frac{25}{64}\right)\) |
| \(\chi_{256}(93,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{33}{64}\right)\) | \(e\left(\frac{43}{64}\right)\) | \(e\left(\frac{23}{32}\right)\) | \(e\left(\frac{1}{32}\right)\) | \(e\left(\frac{7}{64}\right)\) | \(e\left(\frac{37}{64}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{29}{64}\right)\) | \(e\left(\frac{15}{64}\right)\) |
| \(\chi_{256}(101,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{59}{64}\right)\) | \(e\left(\frac{9}{64}\right)\) | \(e\left(\frac{13}{32}\right)\) | \(e\left(\frac{27}{32}\right)\) | \(e\left(\frac{61}{64}\right)\) | \(e\left(\frac{39}{64}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{15}{64}\right)\) | \(e\left(\frac{21}{64}\right)\) |
| \(\chi_{256}(109,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{64}\right)\) | \(e\left(\frac{55}{64}\right)\) | \(e\left(\frac{19}{32}\right)\) | \(e\left(\frac{5}{32}\right)\) | \(e\left(\frac{3}{64}\right)\) | \(e\left(\frac{25}{64}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{49}{64}\right)\) | \(e\left(\frac{43}{64}\right)\) |
| \(\chi_{256}(117,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{63}{64}\right)\) | \(e\left(\frac{53}{64}\right)\) | \(e\left(\frac{9}{32}\right)\) | \(e\left(\frac{31}{32}\right)\) | \(e\left(\frac{25}{64}\right)\) | \(e\left(\frac{59}{64}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{3}{64}\right)\) | \(e\left(\frac{17}{64}\right)\) |
| \(\chi_{256}(125,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{64}\right)\) | \(e\left(\frac{3}{64}\right)\) | \(e\left(\frac{15}{32}\right)\) | \(e\left(\frac{9}{32}\right)\) | \(e\left(\frac{63}{64}\right)\) | \(e\left(\frac{13}{64}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{5}{64}\right)\) | \(e\left(\frac{7}{64}\right)\) |
| \(\chi_{256}(133,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{64}\right)\) | \(e\left(\frac{33}{64}\right)\) | \(e\left(\frac{5}{32}\right)\) | \(e\left(\frac{3}{32}\right)\) | \(e\left(\frac{53}{64}\right)\) | \(e\left(\frac{15}{64}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{55}{64}\right)\) | \(e\left(\frac{13}{64}\right)\) |
| \(\chi_{256}(141,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{64}\right)\) | \(e\left(\frac{15}{64}\right)\) | \(e\left(\frac{11}{32}\right)\) | \(e\left(\frac{13}{32}\right)\) | \(e\left(\frac{59}{64}\right)\) | \(e\left(\frac{1}{64}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{25}{64}\right)\) | \(e\left(\frac{35}{64}\right)\) |
| \(\chi_{256}(149,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{64}\right)\) | \(e\left(\frac{13}{64}\right)\) | \(e\left(\frac{1}{32}\right)\) | \(e\left(\frac{7}{32}\right)\) | \(e\left(\frac{17}{64}\right)\) | \(e\left(\frac{35}{64}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{43}{64}\right)\) | \(e\left(\frac{9}{64}\right)\) |
| \(\chi_{256}(157,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{49}{64}\right)\) | \(e\left(\frac{27}{64}\right)\) | \(e\left(\frac{7}{32}\right)\) | \(e\left(\frac{17}{32}\right)\) | \(e\left(\frac{55}{64}\right)\) | \(e\left(\frac{53}{64}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{45}{64}\right)\) | \(e\left(\frac{63}{64}\right)\) |
| \(\chi_{256}(165,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{64}\right)\) | \(e\left(\frac{57}{64}\right)\) | \(e\left(\frac{29}{32}\right)\) | \(e\left(\frac{11}{32}\right)\) | \(e\left(\frac{45}{64}\right)\) | \(e\left(\frac{55}{64}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{31}{64}\right)\) | \(e\left(\frac{5}{64}\right)\) |
| \(\chi_{256}(173,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{21}{64}\right)\) | \(e\left(\frac{39}{64}\right)\) | \(e\left(\frac{3}{32}\right)\) | \(e\left(\frac{21}{32}\right)\) | \(e\left(\frac{51}{64}\right)\) | \(e\left(\frac{41}{64}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{1}{64}\right)\) | \(e\left(\frac{27}{64}\right)\) |
| \(\chi_{256}(181,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{15}{64}\right)\) | \(e\left(\frac{37}{64}\right)\) | \(e\left(\frac{25}{32}\right)\) | \(e\left(\frac{15}{32}\right)\) | \(e\left(\frac{9}{64}\right)\) | \(e\left(\frac{11}{64}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{19}{64}\right)\) | \(e\left(\frac{1}{64}\right)\) |
| \(\chi_{256}(189,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{57}{64}\right)\) | \(e\left(\frac{51}{64}\right)\) | \(e\left(\frac{31}{32}\right)\) | \(e\left(\frac{25}{32}\right)\) | \(e\left(\frac{47}{64}\right)\) | \(e\left(\frac{29}{64}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{21}{64}\right)\) | \(e\left(\frac{55}{64}\right)\) |
| \(\chi_{256}(197,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{64}\right)\) | \(e\left(\frac{17}{64}\right)\) | \(e\left(\frac{21}{32}\right)\) | \(e\left(\frac{19}{32}\right)\) | \(e\left(\frac{37}{64}\right)\) | \(e\left(\frac{31}{64}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{7}{64}\right)\) | \(e\left(\frac{61}{64}\right)\) |
| \(\chi_{256}(205,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{64}\right)\) | \(e\left(\frac{63}{64}\right)\) | \(e\left(\frac{27}{32}\right)\) | \(e\left(\frac{29}{32}\right)\) | \(e\left(\frac{43}{64}\right)\) | \(e\left(\frac{17}{64}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{41}{64}\right)\) | \(e\left(\frac{19}{64}\right)\) |
| \(\chi_{256}(213,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{64}\right)\) | \(e\left(\frac{61}{64}\right)\) | \(e\left(\frac{17}{32}\right)\) | \(e\left(\frac{23}{32}\right)\) | \(e\left(\frac{1}{64}\right)\) | \(e\left(\frac{51}{64}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{59}{64}\right)\) | \(e\left(\frac{57}{64}\right)\) |
| \(\chi_{256}(221,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{64}\right)\) | \(e\left(\frac{11}{64}\right)\) | \(e\left(\frac{23}{32}\right)\) | \(e\left(\frac{1}{32}\right)\) | \(e\left(\frac{39}{64}\right)\) | \(e\left(\frac{5}{64}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{61}{64}\right)\) | \(e\left(\frac{47}{64}\right)\) |
| \(\chi_{256}(229,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{27}{64}\right)\) | \(e\left(\frac{41}{64}\right)\) | \(e\left(\frac{13}{32}\right)\) | \(e\left(\frac{27}{32}\right)\) | \(e\left(\frac{29}{64}\right)\) | \(e\left(\frac{7}{64}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{47}{64}\right)\) | \(e\left(\frac{53}{64}\right)\) |
| \(\chi_{256}(237,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{64}\right)\) | \(e\left(\frac{23}{64}\right)\) | \(e\left(\frac{19}{32}\right)\) | \(e\left(\frac{5}{32}\right)\) | \(e\left(\frac{35}{64}\right)\) | \(e\left(\frac{57}{64}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{17}{64}\right)\) | \(e\left(\frac{11}{64}\right)\) |
| \(\chi_{256}(245,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{64}\right)\) | \(e\left(\frac{21}{64}\right)\) | \(e\left(\frac{9}{32}\right)\) | \(e\left(\frac{31}{32}\right)\) | \(e\left(\frac{57}{64}\right)\) | \(e\left(\frac{27}{64}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{35}{64}\right)\) | \(e\left(\frac{49}{64}\right)\) |