from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(768, base_ring=CyclotomicField(64))
M = H._module
chi = DirichletCharacter(H, M([32,23,0]))
chi.galois_orbit()
[g,chi] = znchar(Mod(19,768))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
Modulus: | \(768\) | |
Conductor: | \(256\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(64\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from 256.n | 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: | $\Q(\zeta_{64})$ |
Fixed field: | Number field defined by a degree 64 polynomial |
First 31 of 32 characters in Galois orbit
Character | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
---|---|---|---|---|---|---|---|---|---|---|---|---|
\(\chi_{768}(19,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{64}\right)\) | \(e\left(\frac{3}{32}\right)\) | \(e\left(\frac{3}{64}\right)\) | \(e\left(\frac{57}{64}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{49}{64}\right)\) | \(e\left(\frac{17}{32}\right)\) | \(e\left(\frac{23}{32}\right)\) | \(e\left(\frac{13}{64}\right)\) | \(e\left(\frac{3}{8}\right)\) |
\(\chi_{768}(43,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{61}{64}\right)\) | \(e\left(\frac{1}{32}\right)\) | \(e\left(\frac{33}{64}\right)\) | \(e\left(\frac{51}{64}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{27}{64}\right)\) | \(e\left(\frac{27}{32}\right)\) | \(e\left(\frac{29}{32}\right)\) | \(e\left(\frac{15}{64}\right)\) | \(e\left(\frac{1}{8}\right)\) |
\(\chi_{768}(67,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{51}{64}\right)\) | \(e\left(\frac{15}{32}\right)\) | \(e\left(\frac{15}{64}\right)\) | \(e\left(\frac{29}{64}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{53}{64}\right)\) | \(e\left(\frac{21}{32}\right)\) | \(e\left(\frac{19}{32}\right)\) | \(e\left(\frac{1}{64}\right)\) | \(e\left(\frac{7}{8}\right)\) |
\(\chi_{768}(91,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{57}{64}\right)\) | \(e\left(\frac{13}{32}\right)\) | \(e\left(\frac{13}{64}\right)\) | \(e\left(\frac{55}{64}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{63}{64}\right)\) | \(e\left(\frac{31}{32}\right)\) | \(e\left(\frac{25}{32}\right)\) | \(e\left(\frac{35}{64}\right)\) | \(e\left(\frac{5}{8}\right)\) |
\(\chi_{768}(115,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{15}{64}\right)\) | \(e\left(\frac{27}{32}\right)\) | \(e\left(\frac{27}{64}\right)\) | \(e\left(\frac{1}{64}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{57}{64}\right)\) | \(e\left(\frac{25}{32}\right)\) | \(e\left(\frac{15}{32}\right)\) | \(e\left(\frac{53}{64}\right)\) | \(e\left(\frac{3}{8}\right)\) |
\(\chi_{768}(139,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{53}{64}\right)\) | \(e\left(\frac{25}{32}\right)\) | \(e\left(\frac{57}{64}\right)\) | \(e\left(\frac{59}{64}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{35}{64}\right)\) | \(e\left(\frac{3}{32}\right)\) | \(e\left(\frac{21}{32}\right)\) | \(e\left(\frac{55}{64}\right)\) | \(e\left(\frac{1}{8}\right)\) |
\(\chi_{768}(163,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{64}\right)\) | \(e\left(\frac{7}{32}\right)\) | \(e\left(\frac{39}{64}\right)\) | \(e\left(\frac{37}{64}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{61}{64}\right)\) | \(e\left(\frac{29}{32}\right)\) | \(e\left(\frac{11}{32}\right)\) | \(e\left(\frac{41}{64}\right)\) | \(e\left(\frac{7}{8}\right)\) |
\(\chi_{768}(187,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{49}{64}\right)\) | \(e\left(\frac{5}{32}\right)\) | \(e\left(\frac{37}{64}\right)\) | \(e\left(\frac{63}{64}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{7}{64}\right)\) | \(e\left(\frac{7}{32}\right)\) | \(e\left(\frac{17}{32}\right)\) | \(e\left(\frac{11}{64}\right)\) | \(e\left(\frac{5}{8}\right)\) |
\(\chi_{768}(211,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{64}\right)\) | \(e\left(\frac{19}{32}\right)\) | \(e\left(\frac{51}{64}\right)\) | \(e\left(\frac{9}{64}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{1}{64}\right)\) | \(e\left(\frac{1}{32}\right)\) | \(e\left(\frac{7}{32}\right)\) | \(e\left(\frac{29}{64}\right)\) | \(e\left(\frac{3}{8}\right)\) |
\(\chi_{768}(235,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{45}{64}\right)\) | \(e\left(\frac{17}{32}\right)\) | \(e\left(\frac{17}{64}\right)\) | \(e\left(\frac{3}{64}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{43}{64}\right)\) | \(e\left(\frac{11}{32}\right)\) | \(e\left(\frac{13}{32}\right)\) | \(e\left(\frac{31}{64}\right)\) | \(e\left(\frac{1}{8}\right)\) |
\(\chi_{768}(259,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{35}{64}\right)\) | \(e\left(\frac{31}{32}\right)\) | \(e\left(\frac{63}{64}\right)\) | \(e\left(\frac{45}{64}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{5}{64}\right)\) | \(e\left(\frac{5}{32}\right)\) | \(e\left(\frac{3}{32}\right)\) | \(e\left(\frac{17}{64}\right)\) | \(e\left(\frac{7}{8}\right)\) |
\(\chi_{768}(283,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{41}{64}\right)\) | \(e\left(\frac{29}{32}\right)\) | \(e\left(\frac{61}{64}\right)\) | \(e\left(\frac{7}{64}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{15}{64}\right)\) | \(e\left(\frac{15}{32}\right)\) | \(e\left(\frac{9}{32}\right)\) | \(e\left(\frac{51}{64}\right)\) | \(e\left(\frac{5}{8}\right)\) |
\(\chi_{768}(307,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{63}{64}\right)\) | \(e\left(\frac{11}{32}\right)\) | \(e\left(\frac{11}{64}\right)\) | \(e\left(\frac{17}{64}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{9}{64}\right)\) | \(e\left(\frac{9}{32}\right)\) | \(e\left(\frac{31}{32}\right)\) | \(e\left(\frac{5}{64}\right)\) | \(e\left(\frac{3}{8}\right)\) |
\(\chi_{768}(331,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{37}{64}\right)\) | \(e\left(\frac{9}{32}\right)\) | \(e\left(\frac{41}{64}\right)\) | \(e\left(\frac{11}{64}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{51}{64}\right)\) | \(e\left(\frac{19}{32}\right)\) | \(e\left(\frac{5}{32}\right)\) | \(e\left(\frac{7}{64}\right)\) | \(e\left(\frac{1}{8}\right)\) |
\(\chi_{768}(355,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{27}{64}\right)\) | \(e\left(\frac{23}{32}\right)\) | \(e\left(\frac{23}{64}\right)\) | \(e\left(\frac{53}{64}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{13}{64}\right)\) | \(e\left(\frac{13}{32}\right)\) | \(e\left(\frac{27}{32}\right)\) | \(e\left(\frac{57}{64}\right)\) | \(e\left(\frac{7}{8}\right)\) |
\(\chi_{768}(379,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{33}{64}\right)\) | \(e\left(\frac{21}{32}\right)\) | \(e\left(\frac{21}{64}\right)\) | \(e\left(\frac{15}{64}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{23}{64}\right)\) | \(e\left(\frac{23}{32}\right)\) | \(e\left(\frac{1}{32}\right)\) | \(e\left(\frac{27}{64}\right)\) | \(e\left(\frac{5}{8}\right)\) |
\(\chi_{768}(403,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{55}{64}\right)\) | \(e\left(\frac{3}{32}\right)\) | \(e\left(\frac{35}{64}\right)\) | \(e\left(\frac{25}{64}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{17}{64}\right)\) | \(e\left(\frac{17}{32}\right)\) | \(e\left(\frac{23}{32}\right)\) | \(e\left(\frac{45}{64}\right)\) | \(e\left(\frac{3}{8}\right)\) |
\(\chi_{768}(427,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{64}\right)\) | \(e\left(\frac{1}{32}\right)\) | \(e\left(\frac{1}{64}\right)\) | \(e\left(\frac{19}{64}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{59}{64}\right)\) | \(e\left(\frac{27}{32}\right)\) | \(e\left(\frac{29}{32}\right)\) | \(e\left(\frac{47}{64}\right)\) | \(e\left(\frac{1}{8}\right)\) |
\(\chi_{768}(451,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{64}\right)\) | \(e\left(\frac{15}{32}\right)\) | \(e\left(\frac{47}{64}\right)\) | \(e\left(\frac{61}{64}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{21}{64}\right)\) | \(e\left(\frac{21}{32}\right)\) | \(e\left(\frac{19}{32}\right)\) | \(e\left(\frac{33}{64}\right)\) | \(e\left(\frac{7}{8}\right)\) |
\(\chi_{768}(475,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{25}{64}\right)\) | \(e\left(\frac{13}{32}\right)\) | \(e\left(\frac{45}{64}\right)\) | \(e\left(\frac{23}{64}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{31}{64}\right)\) | \(e\left(\frac{31}{32}\right)\) | \(e\left(\frac{25}{32}\right)\) | \(e\left(\frac{3}{64}\right)\) | \(e\left(\frac{5}{8}\right)\) |
\(\chi_{768}(499,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{47}{64}\right)\) | \(e\left(\frac{27}{32}\right)\) | \(e\left(\frac{59}{64}\right)\) | \(e\left(\frac{33}{64}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{25}{64}\right)\) | \(e\left(\frac{25}{32}\right)\) | \(e\left(\frac{15}{32}\right)\) | \(e\left(\frac{21}{64}\right)\) | \(e\left(\frac{3}{8}\right)\) |
\(\chi_{768}(523,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{21}{64}\right)\) | \(e\left(\frac{25}{32}\right)\) | \(e\left(\frac{25}{64}\right)\) | \(e\left(\frac{27}{64}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{3}{64}\right)\) | \(e\left(\frac{3}{32}\right)\) | \(e\left(\frac{21}{32}\right)\) | \(e\left(\frac{23}{64}\right)\) | \(e\left(\frac{1}{8}\right)\) |
\(\chi_{768}(547,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{64}\right)\) | \(e\left(\frac{7}{32}\right)\) | \(e\left(\frac{7}{64}\right)\) | \(e\left(\frac{5}{64}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{29}{64}\right)\) | \(e\left(\frac{29}{32}\right)\) | \(e\left(\frac{11}{32}\right)\) | \(e\left(\frac{9}{64}\right)\) | \(e\left(\frac{7}{8}\right)\) |
\(\chi_{768}(571,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{64}\right)\) | \(e\left(\frac{5}{32}\right)\) | \(e\left(\frac{5}{64}\right)\) | \(e\left(\frac{31}{64}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{39}{64}\right)\) | \(e\left(\frac{7}{32}\right)\) | \(e\left(\frac{17}{32}\right)\) | \(e\left(\frac{43}{64}\right)\) | \(e\left(\frac{5}{8}\right)\) |
\(\chi_{768}(595,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{39}{64}\right)\) | \(e\left(\frac{19}{32}\right)\) | \(e\left(\frac{19}{64}\right)\) | \(e\left(\frac{41}{64}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{33}{64}\right)\) | \(e\left(\frac{1}{32}\right)\) | \(e\left(\frac{7}{32}\right)\) | \(e\left(\frac{61}{64}\right)\) | \(e\left(\frac{3}{8}\right)\) |
\(\chi_{768}(619,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{64}\right)\) | \(e\left(\frac{17}{32}\right)\) | \(e\left(\frac{49}{64}\right)\) | \(e\left(\frac{35}{64}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{11}{64}\right)\) | \(e\left(\frac{11}{32}\right)\) | \(e\left(\frac{13}{32}\right)\) | \(e\left(\frac{63}{64}\right)\) | \(e\left(\frac{1}{8}\right)\) |
\(\chi_{768}(643,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{64}\right)\) | \(e\left(\frac{31}{32}\right)\) | \(e\left(\frac{31}{64}\right)\) | \(e\left(\frac{13}{64}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{37}{64}\right)\) | \(e\left(\frac{5}{32}\right)\) | \(e\left(\frac{3}{32}\right)\) | \(e\left(\frac{49}{64}\right)\) | \(e\left(\frac{7}{8}\right)\) |
\(\chi_{768}(667,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{9}{64}\right)\) | \(e\left(\frac{29}{32}\right)\) | \(e\left(\frac{29}{64}\right)\) | \(e\left(\frac{39}{64}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{47}{64}\right)\) | \(e\left(\frac{15}{32}\right)\) | \(e\left(\frac{9}{32}\right)\) | \(e\left(\frac{19}{64}\right)\) | \(e\left(\frac{5}{8}\right)\) |
\(\chi_{768}(691,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{64}\right)\) | \(e\left(\frac{11}{32}\right)\) | \(e\left(\frac{43}{64}\right)\) | \(e\left(\frac{49}{64}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{41}{64}\right)\) | \(e\left(\frac{9}{32}\right)\) | \(e\left(\frac{31}{32}\right)\) | \(e\left(\frac{37}{64}\right)\) | \(e\left(\frac{3}{8}\right)\) |
\(\chi_{768}(715,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{64}\right)\) | \(e\left(\frac{9}{32}\right)\) | \(e\left(\frac{9}{64}\right)\) | \(e\left(\frac{43}{64}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{19}{64}\right)\) | \(e\left(\frac{19}{32}\right)\) | \(e\left(\frac{5}{32}\right)\) | \(e\left(\frac{39}{64}\right)\) | \(e\left(\frac{1}{8}\right)\) |
\(\chi_{768}(739,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{64}\right)\) | \(e\left(\frac{23}{32}\right)\) | \(e\left(\frac{55}{64}\right)\) | \(e\left(\frac{21}{64}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{45}{64}\right)\) | \(e\left(\frac{13}{32}\right)\) | \(e\left(\frac{27}{32}\right)\) | \(e\left(\frac{25}{64}\right)\) | \(e\left(\frac{7}{8}\right)\) |