from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3072, base_ring=CyclotomicField(64))
M = H._module
chi = DirichletCharacter(H, M([0,37,0]))
chi.galois_orbit()
[g,chi] = znchar(Mod(49,3072))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(3072\) | |
| 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.m | 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: | $\Q(\zeta_{64})$ |
| Fixed field: | Number field defined by a degree 64 polynomial |
First 31 of 32 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{3072}(49,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{64}\right)\) | \(e\left(\frac{25}{32}\right)\) | \(e\left(\frac{9}{64}\right)\) | \(e\left(\frac{11}{64}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{19}{64}\right)\) | \(e\left(\frac{3}{32}\right)\) | \(e\left(\frac{5}{32}\right)\) | \(e\left(\frac{7}{64}\right)\) | \(e\left(\frac{5}{8}\right)\) |
| \(\chi_{3072}(145,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{64}\right)\) | \(e\left(\frac{27}{32}\right)\) | \(e\left(\frac{11}{64}\right)\) | \(e\left(\frac{49}{64}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{9}{64}\right)\) | \(e\left(\frac{25}{32}\right)\) | \(e\left(\frac{31}{32}\right)\) | \(e\left(\frac{37}{64}\right)\) | \(e\left(\frac{7}{8}\right)\) |
| \(\chi_{3072}(241,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{25}{64}\right)\) | \(e\left(\frac{29}{32}\right)\) | \(e\left(\frac{13}{64}\right)\) | \(e\left(\frac{23}{64}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{63}{64}\right)\) | \(e\left(\frac{15}{32}\right)\) | \(e\left(\frac{25}{32}\right)\) | \(e\left(\frac{3}{64}\right)\) | \(e\left(\frac{1}{8}\right)\) |
| \(\chi_{3072}(337,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{64}\right)\) | \(e\left(\frac{31}{32}\right)\) | \(e\left(\frac{15}{64}\right)\) | \(e\left(\frac{61}{64}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{53}{64}\right)\) | \(e\left(\frac{5}{32}\right)\) | \(e\left(\frac{19}{32}\right)\) | \(e\left(\frac{33}{64}\right)\) | \(e\left(\frac{3}{8}\right)\) |
| \(\chi_{3072}(433,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{64}\right)\) | \(e\left(\frac{1}{32}\right)\) | \(e\left(\frac{17}{64}\right)\) | \(e\left(\frac{35}{64}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{43}{64}\right)\) | \(e\left(\frac{27}{32}\right)\) | \(e\left(\frac{13}{32}\right)\) | \(e\left(\frac{63}{64}\right)\) | \(e\left(\frac{5}{8}\right)\) |
| \(\chi_{3072}(529,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{64}\right)\) | \(e\left(\frac{3}{32}\right)\) | \(e\left(\frac{19}{64}\right)\) | \(e\left(\frac{9}{64}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{33}{64}\right)\) | \(e\left(\frac{17}{32}\right)\) | \(e\left(\frac{7}{32}\right)\) | \(e\left(\frac{29}{64}\right)\) | \(e\left(\frac{7}{8}\right)\) |
| \(\chi_{3072}(625,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{64}\right)\) | \(e\left(\frac{5}{32}\right)\) | \(e\left(\frac{21}{64}\right)\) | \(e\left(\frac{47}{64}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{23}{64}\right)\) | \(e\left(\frac{7}{32}\right)\) | \(e\left(\frac{1}{32}\right)\) | \(e\left(\frac{59}{64}\right)\) | \(e\left(\frac{1}{8}\right)\) |
| \(\chi_{3072}(721,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{59}{64}\right)\) | \(e\left(\frac{7}{32}\right)\) | \(e\left(\frac{23}{64}\right)\) | \(e\left(\frac{21}{64}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{13}{64}\right)\) | \(e\left(\frac{29}{32}\right)\) | \(e\left(\frac{27}{32}\right)\) | \(e\left(\frac{25}{64}\right)\) | \(e\left(\frac{3}{8}\right)\) |
| \(\chi_{3072}(817,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{53}{64}\right)\) | \(e\left(\frac{9}{32}\right)\) | \(e\left(\frac{25}{64}\right)\) | \(e\left(\frac{59}{64}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{3}{64}\right)\) | \(e\left(\frac{19}{32}\right)\) | \(e\left(\frac{21}{32}\right)\) | \(e\left(\frac{55}{64}\right)\) | \(e\left(\frac{5}{8}\right)\) |
| \(\chi_{3072}(913,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{47}{64}\right)\) | \(e\left(\frac{11}{32}\right)\) | \(e\left(\frac{27}{64}\right)\) | \(e\left(\frac{33}{64}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{57}{64}\right)\) | \(e\left(\frac{9}{32}\right)\) | \(e\left(\frac{15}{32}\right)\) | \(e\left(\frac{21}{64}\right)\) | \(e\left(\frac{7}{8}\right)\) |
| \(\chi_{3072}(1009,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{64}\right)\) | \(e\left(\frac{13}{32}\right)\) | \(e\left(\frac{29}{64}\right)\) | \(e\left(\frac{7}{64}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{47}{64}\right)\) | \(e\left(\frac{31}{32}\right)\) | \(e\left(\frac{9}{32}\right)\) | \(e\left(\frac{51}{64}\right)\) | \(e\left(\frac{1}{8}\right)\) |
| \(\chi_{3072}(1105,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{35}{64}\right)\) | \(e\left(\frac{15}{32}\right)\) | \(e\left(\frac{31}{64}\right)\) | \(e\left(\frac{45}{64}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{37}{64}\right)\) | \(e\left(\frac{21}{32}\right)\) | \(e\left(\frac{3}{32}\right)\) | \(e\left(\frac{17}{64}\right)\) | \(e\left(\frac{3}{8}\right)\) |
| \(\chi_{3072}(1201,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{64}\right)\) | \(e\left(\frac{17}{32}\right)\) | \(e\left(\frac{33}{64}\right)\) | \(e\left(\frac{19}{64}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{27}{64}\right)\) | \(e\left(\frac{11}{32}\right)\) | \(e\left(\frac{29}{32}\right)\) | \(e\left(\frac{47}{64}\right)\) | \(e\left(\frac{5}{8}\right)\) |
| \(\chi_{3072}(1297,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{64}\right)\) | \(e\left(\frac{19}{32}\right)\) | \(e\left(\frac{35}{64}\right)\) | \(e\left(\frac{57}{64}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{17}{64}\right)\) | \(e\left(\frac{1}{32}\right)\) | \(e\left(\frac{23}{32}\right)\) | \(e\left(\frac{13}{64}\right)\) | \(e\left(\frac{7}{8}\right)\) |
| \(\chi_{3072}(1393,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{64}\right)\) | \(e\left(\frac{21}{32}\right)\) | \(e\left(\frac{37}{64}\right)\) | \(e\left(\frac{31}{64}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{7}{64}\right)\) | \(e\left(\frac{23}{32}\right)\) | \(e\left(\frac{17}{32}\right)\) | \(e\left(\frac{43}{64}\right)\) | \(e\left(\frac{1}{8}\right)\) |
| \(\chi_{3072}(1489,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{64}\right)\) | \(e\left(\frac{23}{32}\right)\) | \(e\left(\frac{39}{64}\right)\) | \(e\left(\frac{5}{64}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{61}{64}\right)\) | \(e\left(\frac{13}{32}\right)\) | \(e\left(\frac{11}{32}\right)\) | \(e\left(\frac{9}{64}\right)\) | \(e\left(\frac{3}{8}\right)\) |
| \(\chi_{3072}(1585,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{64}\right)\) | \(e\left(\frac{25}{32}\right)\) | \(e\left(\frac{41}{64}\right)\) | \(e\left(\frac{43}{64}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{51}{64}\right)\) | \(e\left(\frac{3}{32}\right)\) | \(e\left(\frac{5}{32}\right)\) | \(e\left(\frac{39}{64}\right)\) | \(e\left(\frac{5}{8}\right)\) |
| \(\chi_{3072}(1681,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{63}{64}\right)\) | \(e\left(\frac{27}{32}\right)\) | \(e\left(\frac{43}{64}\right)\) | \(e\left(\frac{17}{64}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{41}{64}\right)\) | \(e\left(\frac{25}{32}\right)\) | \(e\left(\frac{31}{32}\right)\) | \(e\left(\frac{5}{64}\right)\) | \(e\left(\frac{7}{8}\right)\) |
| \(\chi_{3072}(1777,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{57}{64}\right)\) | \(e\left(\frac{29}{32}\right)\) | \(e\left(\frac{45}{64}\right)\) | \(e\left(\frac{55}{64}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{31}{64}\right)\) | \(e\left(\frac{15}{32}\right)\) | \(e\left(\frac{25}{32}\right)\) | \(e\left(\frac{35}{64}\right)\) | \(e\left(\frac{1}{8}\right)\) |
| \(\chi_{3072}(1873,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{51}{64}\right)\) | \(e\left(\frac{31}{32}\right)\) | \(e\left(\frac{47}{64}\right)\) | \(e\left(\frac{29}{64}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{21}{64}\right)\) | \(e\left(\frac{5}{32}\right)\) | \(e\left(\frac{19}{32}\right)\) | \(e\left(\frac{1}{64}\right)\) | \(e\left(\frac{3}{8}\right)\) |
| \(\chi_{3072}(1969,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{45}{64}\right)\) | \(e\left(\frac{1}{32}\right)\) | \(e\left(\frac{49}{64}\right)\) | \(e\left(\frac{3}{64}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{11}{64}\right)\) | \(e\left(\frac{27}{32}\right)\) | \(e\left(\frac{13}{32}\right)\) | \(e\left(\frac{31}{64}\right)\) | \(e\left(\frac{5}{8}\right)\) |
| \(\chi_{3072}(2065,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{39}{64}\right)\) | \(e\left(\frac{3}{32}\right)\) | \(e\left(\frac{51}{64}\right)\) | \(e\left(\frac{41}{64}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{1}{64}\right)\) | \(e\left(\frac{17}{32}\right)\) | \(e\left(\frac{7}{32}\right)\) | \(e\left(\frac{61}{64}\right)\) | \(e\left(\frac{7}{8}\right)\) |
| \(\chi_{3072}(2161,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{33}{64}\right)\) | \(e\left(\frac{5}{32}\right)\) | \(e\left(\frac{53}{64}\right)\) | \(e\left(\frac{15}{64}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{55}{64}\right)\) | \(e\left(\frac{7}{32}\right)\) | \(e\left(\frac{1}{32}\right)\) | \(e\left(\frac{27}{64}\right)\) | \(e\left(\frac{1}{8}\right)\) |
| \(\chi_{3072}(2257,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{27}{64}\right)\) | \(e\left(\frac{7}{32}\right)\) | \(e\left(\frac{55}{64}\right)\) | \(e\left(\frac{53}{64}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{45}{64}\right)\) | \(e\left(\frac{29}{32}\right)\) | \(e\left(\frac{27}{32}\right)\) | \(e\left(\frac{57}{64}\right)\) | \(e\left(\frac{3}{8}\right)\) |
| \(\chi_{3072}(2353,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{21}{64}\right)\) | \(e\left(\frac{9}{32}\right)\) | \(e\left(\frac{57}{64}\right)\) | \(e\left(\frac{27}{64}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{35}{64}\right)\) | \(e\left(\frac{19}{32}\right)\) | \(e\left(\frac{21}{32}\right)\) | \(e\left(\frac{23}{64}\right)\) | \(e\left(\frac{5}{8}\right)\) |
| \(\chi_{3072}(2449,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{15}{64}\right)\) | \(e\left(\frac{11}{32}\right)\) | \(e\left(\frac{59}{64}\right)\) | \(e\left(\frac{1}{64}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{25}{64}\right)\) | \(e\left(\frac{9}{32}\right)\) | \(e\left(\frac{15}{32}\right)\) | \(e\left(\frac{53}{64}\right)\) | \(e\left(\frac{7}{8}\right)\) |
| \(\chi_{3072}(2545,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{64}\right)\) | \(e\left(\frac{13}{32}\right)\) | \(e\left(\frac{61}{64}\right)\) | \(e\left(\frac{39}{64}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{15}{64}\right)\) | \(e\left(\frac{31}{32}\right)\) | \(e\left(\frac{9}{32}\right)\) | \(e\left(\frac{19}{64}\right)\) | \(e\left(\frac{1}{8}\right)\) |
| \(\chi_{3072}(2641,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{64}\right)\) | \(e\left(\frac{15}{32}\right)\) | \(e\left(\frac{63}{64}\right)\) | \(e\left(\frac{13}{64}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{5}{64}\right)\) | \(e\left(\frac{21}{32}\right)\) | \(e\left(\frac{3}{32}\right)\) | \(e\left(\frac{49}{64}\right)\) | \(e\left(\frac{3}{8}\right)\) |
| \(\chi_{3072}(2737,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{61}{64}\right)\) | \(e\left(\frac{17}{32}\right)\) | \(e\left(\frac{1}{64}\right)\) | \(e\left(\frac{51}{64}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{59}{64}\right)\) | \(e\left(\frac{11}{32}\right)\) | \(e\left(\frac{29}{32}\right)\) | \(e\left(\frac{15}{64}\right)\) | \(e\left(\frac{5}{8}\right)\) |
| \(\chi_{3072}(2833,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{55}{64}\right)\) | \(e\left(\frac{19}{32}\right)\) | \(e\left(\frac{3}{64}\right)\) | \(e\left(\frac{25}{64}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{49}{64}\right)\) | \(e\left(\frac{1}{32}\right)\) | \(e\left(\frac{23}{32}\right)\) | \(e\left(\frac{45}{64}\right)\) | \(e\left(\frac{7}{8}\right)\) |
| \(\chi_{3072}(2929,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{49}{64}\right)\) | \(e\left(\frac{21}{32}\right)\) | \(e\left(\frac{5}{64}\right)\) | \(e\left(\frac{63}{64}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{39}{64}\right)\) | \(e\left(\frac{23}{32}\right)\) | \(e\left(\frac{17}{32}\right)\) | \(e\left(\frac{11}{64}\right)\) | \(e\left(\frac{1}{8}\right)\) |