from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1024, base_ring=CyclotomicField(64))
M = H._module
chi = DirichletCharacter(H, M([32,41]))
chi.galois_orbit()
[g,chi] = znchar(Mod(15,1024))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(1024\) | |
| 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: | no | |
| 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 | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{1024}(15,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{64}\right)\) | \(e\left(\frac{41}{64}\right)\) | \(e\left(\frac{29}{32}\right)\) | \(e\left(\frac{27}{32}\right)\) | \(e\left(\frac{61}{64}\right)\) | \(e\left(\frac{7}{64}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{15}{64}\right)\) | \(e\left(\frac{53}{64}\right)\) |
| \(\chi_{1024}(47,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{64}\right)\) | \(e\left(\frac{43}{64}\right)\) | \(e\left(\frac{7}{32}\right)\) | \(e\left(\frac{1}{32}\right)\) | \(e\left(\frac{39}{64}\right)\) | \(e\left(\frac{37}{64}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{61}{64}\right)\) | \(e\left(\frac{15}{64}\right)\) |
| \(\chi_{1024}(79,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{64}\right)\) | \(e\left(\frac{45}{64}\right)\) | \(e\left(\frac{17}{32}\right)\) | \(e\left(\frac{7}{32}\right)\) | \(e\left(\frac{17}{64}\right)\) | \(e\left(\frac{3}{64}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{43}{64}\right)\) | \(e\left(\frac{41}{64}\right)\) |
| \(\chi_{1024}(111,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{64}\right)\) | \(e\left(\frac{47}{64}\right)\) | \(e\left(\frac{27}{32}\right)\) | \(e\left(\frac{13}{32}\right)\) | \(e\left(\frac{59}{64}\right)\) | \(e\left(\frac{33}{64}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{25}{64}\right)\) | \(e\left(\frac{3}{64}\right)\) |
| \(\chi_{1024}(143,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{64}\right)\) | \(e\left(\frac{49}{64}\right)\) | \(e\left(\frac{5}{32}\right)\) | \(e\left(\frac{19}{32}\right)\) | \(e\left(\frac{37}{64}\right)\) | \(e\left(\frac{63}{64}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{7}{64}\right)\) | \(e\left(\frac{29}{64}\right)\) |
| \(\chi_{1024}(175,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{25}{64}\right)\) | \(e\left(\frac{51}{64}\right)\) | \(e\left(\frac{15}{32}\right)\) | \(e\left(\frac{25}{32}\right)\) | \(e\left(\frac{15}{64}\right)\) | \(e\left(\frac{29}{64}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{53}{64}\right)\) | \(e\left(\frac{55}{64}\right)\) |
| \(\chi_{1024}(207,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{64}\right)\) | \(e\left(\frac{53}{64}\right)\) | \(e\left(\frac{25}{32}\right)\) | \(e\left(\frac{31}{32}\right)\) | \(e\left(\frac{57}{64}\right)\) | \(e\left(\frac{59}{64}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{35}{64}\right)\) | \(e\left(\frac{17}{64}\right)\) |
| \(\chi_{1024}(239,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{37}{64}\right)\) | \(e\left(\frac{55}{64}\right)\) | \(e\left(\frac{3}{32}\right)\) | \(e\left(\frac{5}{32}\right)\) | \(e\left(\frac{35}{64}\right)\) | \(e\left(\frac{25}{64}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{17}{64}\right)\) | \(e\left(\frac{43}{64}\right)\) |
| \(\chi_{1024}(271,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{64}\right)\) | \(e\left(\frac{57}{64}\right)\) | \(e\left(\frac{13}{32}\right)\) | \(e\left(\frac{11}{32}\right)\) | \(e\left(\frac{13}{64}\right)\) | \(e\left(\frac{55}{64}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{63}{64}\right)\) | \(e\left(\frac{5}{64}\right)\) |
| \(\chi_{1024}(303,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{49}{64}\right)\) | \(e\left(\frac{59}{64}\right)\) | \(e\left(\frac{23}{32}\right)\) | \(e\left(\frac{17}{32}\right)\) | \(e\left(\frac{55}{64}\right)\) | \(e\left(\frac{21}{64}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{45}{64}\right)\) | \(e\left(\frac{31}{64}\right)\) |
| \(\chi_{1024}(335,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{55}{64}\right)\) | \(e\left(\frac{61}{64}\right)\) | \(e\left(\frac{1}{32}\right)\) | \(e\left(\frac{23}{32}\right)\) | \(e\left(\frac{33}{64}\right)\) | \(e\left(\frac{51}{64}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{27}{64}\right)\) | \(e\left(\frac{57}{64}\right)\) |
| \(\chi_{1024}(367,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{61}{64}\right)\) | \(e\left(\frac{63}{64}\right)\) | \(e\left(\frac{11}{32}\right)\) | \(e\left(\frac{29}{32}\right)\) | \(e\left(\frac{11}{64}\right)\) | \(e\left(\frac{17}{64}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{9}{64}\right)\) | \(e\left(\frac{19}{64}\right)\) |
| \(\chi_{1024}(399,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{64}\right)\) | \(e\left(\frac{1}{64}\right)\) | \(e\left(\frac{21}{32}\right)\) | \(e\left(\frac{3}{32}\right)\) | \(e\left(\frac{53}{64}\right)\) | \(e\left(\frac{47}{64}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{55}{64}\right)\) | \(e\left(\frac{45}{64}\right)\) |
| \(\chi_{1024}(431,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{9}{64}\right)\) | \(e\left(\frac{3}{64}\right)\) | \(e\left(\frac{31}{32}\right)\) | \(e\left(\frac{9}{32}\right)\) | \(e\left(\frac{31}{64}\right)\) | \(e\left(\frac{13}{64}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{37}{64}\right)\) | \(e\left(\frac{7}{64}\right)\) |
| \(\chi_{1024}(463,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{15}{64}\right)\) | \(e\left(\frac{5}{64}\right)\) | \(e\left(\frac{9}{32}\right)\) | \(e\left(\frac{15}{32}\right)\) | \(e\left(\frac{9}{64}\right)\) | \(e\left(\frac{43}{64}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{19}{64}\right)\) | \(e\left(\frac{33}{64}\right)\) |
| \(\chi_{1024}(495,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{21}{64}\right)\) | \(e\left(\frac{7}{64}\right)\) | \(e\left(\frac{19}{32}\right)\) | \(e\left(\frac{21}{32}\right)\) | \(e\left(\frac{51}{64}\right)\) | \(e\left(\frac{9}{64}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{1}{64}\right)\) | \(e\left(\frac{59}{64}\right)\) |
| \(\chi_{1024}(527,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{27}{64}\right)\) | \(e\left(\frac{9}{64}\right)\) | \(e\left(\frac{29}{32}\right)\) | \(e\left(\frac{27}{32}\right)\) | \(e\left(\frac{29}{64}\right)\) | \(e\left(\frac{39}{64}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{47}{64}\right)\) | \(e\left(\frac{21}{64}\right)\) |
| \(\chi_{1024}(559,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{33}{64}\right)\) | \(e\left(\frac{11}{64}\right)\) | \(e\left(\frac{7}{32}\right)\) | \(e\left(\frac{1}{32}\right)\) | \(e\left(\frac{7}{64}\right)\) | \(e\left(\frac{5}{64}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{29}{64}\right)\) | \(e\left(\frac{47}{64}\right)\) |
| \(\chi_{1024}(591,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{39}{64}\right)\) | \(e\left(\frac{13}{64}\right)\) | \(e\left(\frac{17}{32}\right)\) | \(e\left(\frac{7}{32}\right)\) | \(e\left(\frac{49}{64}\right)\) | \(e\left(\frac{35}{64}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{11}{64}\right)\) | \(e\left(\frac{9}{64}\right)\) |
| \(\chi_{1024}(623,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{45}{64}\right)\) | \(e\left(\frac{15}{64}\right)\) | \(e\left(\frac{27}{32}\right)\) | \(e\left(\frac{13}{32}\right)\) | \(e\left(\frac{27}{64}\right)\) | \(e\left(\frac{1}{64}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{57}{64}\right)\) | \(e\left(\frac{35}{64}\right)\) |
| \(\chi_{1024}(655,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{51}{64}\right)\) | \(e\left(\frac{17}{64}\right)\) | \(e\left(\frac{5}{32}\right)\) | \(e\left(\frac{19}{32}\right)\) | \(e\left(\frac{5}{64}\right)\) | \(e\left(\frac{31}{64}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{39}{64}\right)\) | \(e\left(\frac{61}{64}\right)\) |
| \(\chi_{1024}(687,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{57}{64}\right)\) | \(e\left(\frac{19}{64}\right)\) | \(e\left(\frac{15}{32}\right)\) | \(e\left(\frac{25}{32}\right)\) | \(e\left(\frac{47}{64}\right)\) | \(e\left(\frac{61}{64}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{21}{64}\right)\) | \(e\left(\frac{23}{64}\right)\) |
| \(\chi_{1024}(719,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{63}{64}\right)\) | \(e\left(\frac{21}{64}\right)\) | \(e\left(\frac{25}{32}\right)\) | \(e\left(\frac{31}{32}\right)\) | \(e\left(\frac{25}{64}\right)\) | \(e\left(\frac{27}{64}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{3}{64}\right)\) | \(e\left(\frac{49}{64}\right)\) |
| \(\chi_{1024}(751,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{64}\right)\) | \(e\left(\frac{23}{64}\right)\) | \(e\left(\frac{3}{32}\right)\) | \(e\left(\frac{5}{32}\right)\) | \(e\left(\frac{3}{64}\right)\) | \(e\left(\frac{57}{64}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{49}{64}\right)\) | \(e\left(\frac{11}{64}\right)\) |
| \(\chi_{1024}(783,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{64}\right)\) | \(e\left(\frac{25}{64}\right)\) | \(e\left(\frac{13}{32}\right)\) | \(e\left(\frac{11}{32}\right)\) | \(e\left(\frac{45}{64}\right)\) | \(e\left(\frac{23}{64}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{31}{64}\right)\) | \(e\left(\frac{37}{64}\right)\) |
| \(\chi_{1024}(815,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{64}\right)\) | \(e\left(\frac{27}{64}\right)\) | \(e\left(\frac{23}{32}\right)\) | \(e\left(\frac{17}{32}\right)\) | \(e\left(\frac{23}{64}\right)\) | \(e\left(\frac{53}{64}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{13}{64}\right)\) | \(e\left(\frac{63}{64}\right)\) |
| \(\chi_{1024}(847,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{64}\right)\) | \(e\left(\frac{29}{64}\right)\) | \(e\left(\frac{1}{32}\right)\) | \(e\left(\frac{23}{32}\right)\) | \(e\left(\frac{1}{64}\right)\) | \(e\left(\frac{19}{64}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{59}{64}\right)\) | \(e\left(\frac{25}{64}\right)\) |
| \(\chi_{1024}(879,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{64}\right)\) | \(e\left(\frac{31}{64}\right)\) | \(e\left(\frac{11}{32}\right)\) | \(e\left(\frac{29}{32}\right)\) | \(e\left(\frac{43}{64}\right)\) | \(e\left(\frac{49}{64}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{41}{64}\right)\) | \(e\left(\frac{51}{64}\right)\) |
| \(\chi_{1024}(911,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{35}{64}\right)\) | \(e\left(\frac{33}{64}\right)\) | \(e\left(\frac{21}{32}\right)\) | \(e\left(\frac{3}{32}\right)\) | \(e\left(\frac{21}{64}\right)\) | \(e\left(\frac{15}{64}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{23}{64}\right)\) | \(e\left(\frac{13}{64}\right)\) |
| \(\chi_{1024}(943,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{41}{64}\right)\) | \(e\left(\frac{35}{64}\right)\) | \(e\left(\frac{31}{32}\right)\) | \(e\left(\frac{9}{32}\right)\) | \(e\left(\frac{63}{64}\right)\) | \(e\left(\frac{45}{64}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{5}{64}\right)\) | \(e\left(\frac{39}{64}\right)\) |
| \(\chi_{1024}(975,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{47}{64}\right)\) | \(e\left(\frac{37}{64}\right)\) | \(e\left(\frac{9}{32}\right)\) | \(e\left(\frac{15}{32}\right)\) | \(e\left(\frac{41}{64}\right)\) | \(e\left(\frac{11}{64}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{51}{64}\right)\) | \(e\left(\frac{1}{64}\right)\) |