from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1280, base_ring=CyclotomicField(64))
M = H._module
chi = DirichletCharacter(H, M([0,47,48]))
chi.galois_orbit()
[g,chi] = znchar(Mod(13,1280))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(1280\) | |
| Conductor: | \(1280\) | 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: | 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 | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{1280}(13,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{61}{64}\right)\) | \(e\left(\frac{3}{32}\right)\) | \(e\left(\frac{29}{32}\right)\) | \(e\left(\frac{27}{64}\right)\) | \(e\left(\frac{49}{64}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{25}{64}\right)\) | \(e\left(\frac{3}{64}\right)\) | \(e\left(\frac{17}{32}\right)\) | \(e\left(\frac{55}{64}\right)\) |
| \(\chi_{1280}(37,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{27}{64}\right)\) | \(e\left(\frac{5}{32}\right)\) | \(e\left(\frac{27}{32}\right)\) | \(e\left(\frac{13}{64}\right)\) | \(e\left(\frac{7}{64}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{31}{64}\right)\) | \(e\left(\frac{37}{64}\right)\) | \(e\left(\frac{7}{32}\right)\) | \(e\left(\frac{17}{64}\right)\) |
| \(\chi_{1280}(93,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{49}{64}\right)\) | \(e\left(\frac{15}{32}\right)\) | \(e\left(\frac{17}{32}\right)\) | \(e\left(\frac{7}{64}\right)\) | \(e\left(\frac{53}{64}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{61}{64}\right)\) | \(e\left(\frac{15}{64}\right)\) | \(e\left(\frac{21}{32}\right)\) | \(e\left(\frac{19}{64}\right)\) |
| \(\chi_{1280}(117,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{47}{64}\right)\) | \(e\left(\frac{17}{32}\right)\) | \(e\left(\frac{15}{32}\right)\) | \(e\left(\frac{25}{64}\right)\) | \(e\left(\frac{43}{64}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{35}{64}\right)\) | \(e\left(\frac{17}{64}\right)\) | \(e\left(\frac{11}{32}\right)\) | \(e\left(\frac{13}{64}\right)\) |
| \(\chi_{1280}(173,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{37}{64}\right)\) | \(e\left(\frac{27}{32}\right)\) | \(e\left(\frac{5}{32}\right)\) | \(e\left(\frac{51}{64}\right)\) | \(e\left(\frac{57}{64}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{33}{64}\right)\) | \(e\left(\frac{27}{64}\right)\) | \(e\left(\frac{25}{32}\right)\) | \(e\left(\frac{47}{64}\right)\) |
| \(\chi_{1280}(197,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{64}\right)\) | \(e\left(\frac{29}{32}\right)\) | \(e\left(\frac{3}{32}\right)\) | \(e\left(\frac{37}{64}\right)\) | \(e\left(\frac{15}{64}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{39}{64}\right)\) | \(e\left(\frac{61}{64}\right)\) | \(e\left(\frac{15}{32}\right)\) | \(e\left(\frac{9}{64}\right)\) |
| \(\chi_{1280}(253,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{25}{64}\right)\) | \(e\left(\frac{7}{32}\right)\) | \(e\left(\frac{25}{32}\right)\) | \(e\left(\frac{31}{64}\right)\) | \(e\left(\frac{61}{64}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{5}{64}\right)\) | \(e\left(\frac{39}{64}\right)\) | \(e\left(\frac{29}{32}\right)\) | \(e\left(\frac{11}{64}\right)\) |
| \(\chi_{1280}(277,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{64}\right)\) | \(e\left(\frac{9}{32}\right)\) | \(e\left(\frac{23}{32}\right)\) | \(e\left(\frac{49}{64}\right)\) | \(e\left(\frac{51}{64}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{43}{64}\right)\) | \(e\left(\frac{41}{64}\right)\) | \(e\left(\frac{19}{32}\right)\) | \(e\left(\frac{5}{64}\right)\) |
| \(\chi_{1280}(333,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{64}\right)\) | \(e\left(\frac{19}{32}\right)\) | \(e\left(\frac{13}{32}\right)\) | \(e\left(\frac{11}{64}\right)\) | \(e\left(\frac{1}{64}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{41}{64}\right)\) | \(e\left(\frac{51}{64}\right)\) | \(e\left(\frac{1}{32}\right)\) | \(e\left(\frac{39}{64}\right)\) |
| \(\chi_{1280}(357,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{64}\right)\) | \(e\left(\frac{21}{32}\right)\) | \(e\left(\frac{11}{32}\right)\) | \(e\left(\frac{61}{64}\right)\) | \(e\left(\frac{23}{64}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{47}{64}\right)\) | \(e\left(\frac{21}{64}\right)\) | \(e\left(\frac{23}{32}\right)\) | \(e\left(\frac{1}{64}\right)\) |
| \(\chi_{1280}(413,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{64}\right)\) | \(e\left(\frac{31}{32}\right)\) | \(e\left(\frac{1}{32}\right)\) | \(e\left(\frac{55}{64}\right)\) | \(e\left(\frac{5}{64}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{13}{64}\right)\) | \(e\left(\frac{63}{64}\right)\) | \(e\left(\frac{5}{32}\right)\) | \(e\left(\frac{3}{64}\right)\) |
| \(\chi_{1280}(437,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{63}{64}\right)\) | \(e\left(\frac{1}{32}\right)\) | \(e\left(\frac{31}{32}\right)\) | \(e\left(\frac{9}{64}\right)\) | \(e\left(\frac{59}{64}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{51}{64}\right)\) | \(e\left(\frac{1}{64}\right)\) | \(e\left(\frac{27}{32}\right)\) | \(e\left(\frac{61}{64}\right)\) |
| \(\chi_{1280}(493,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{53}{64}\right)\) | \(e\left(\frac{11}{32}\right)\) | \(e\left(\frac{21}{32}\right)\) | \(e\left(\frac{35}{64}\right)\) | \(e\left(\frac{9}{64}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{49}{64}\right)\) | \(e\left(\frac{11}{64}\right)\) | \(e\left(\frac{9}{32}\right)\) | \(e\left(\frac{31}{64}\right)\) |
| \(\chi_{1280}(517,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{64}\right)\) | \(e\left(\frac{13}{32}\right)\) | \(e\left(\frac{19}{32}\right)\) | \(e\left(\frac{21}{64}\right)\) | \(e\left(\frac{31}{64}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{55}{64}\right)\) | \(e\left(\frac{45}{64}\right)\) | \(e\left(\frac{31}{32}\right)\) | \(e\left(\frac{57}{64}\right)\) |
| \(\chi_{1280}(573,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{41}{64}\right)\) | \(e\left(\frac{23}{32}\right)\) | \(e\left(\frac{9}{32}\right)\) | \(e\left(\frac{15}{64}\right)\) | \(e\left(\frac{13}{64}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{21}{64}\right)\) | \(e\left(\frac{23}{64}\right)\) | \(e\left(\frac{13}{32}\right)\) | \(e\left(\frac{59}{64}\right)\) |
| \(\chi_{1280}(597,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{39}{64}\right)\) | \(e\left(\frac{25}{32}\right)\) | \(e\left(\frac{7}{32}\right)\) | \(e\left(\frac{33}{64}\right)\) | \(e\left(\frac{3}{64}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{59}{64}\right)\) | \(e\left(\frac{25}{64}\right)\) | \(e\left(\frac{3}{32}\right)\) | \(e\left(\frac{53}{64}\right)\) |
| \(\chi_{1280}(653,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{64}\right)\) | \(e\left(\frac{3}{32}\right)\) | \(e\left(\frac{29}{32}\right)\) | \(e\left(\frac{59}{64}\right)\) | \(e\left(\frac{17}{64}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{57}{64}\right)\) | \(e\left(\frac{35}{64}\right)\) | \(e\left(\frac{17}{32}\right)\) | \(e\left(\frac{23}{64}\right)\) |
| \(\chi_{1280}(677,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{64}\right)\) | \(e\left(\frac{5}{32}\right)\) | \(e\left(\frac{27}{32}\right)\) | \(e\left(\frac{45}{64}\right)\) | \(e\left(\frac{39}{64}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{63}{64}\right)\) | \(e\left(\frac{5}{64}\right)\) | \(e\left(\frac{7}{32}\right)\) | \(e\left(\frac{49}{64}\right)\) |
| \(\chi_{1280}(733,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{64}\right)\) | \(e\left(\frac{15}{32}\right)\) | \(e\left(\frac{17}{32}\right)\) | \(e\left(\frac{39}{64}\right)\) | \(e\left(\frac{21}{64}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{29}{64}\right)\) | \(e\left(\frac{47}{64}\right)\) | \(e\left(\frac{21}{32}\right)\) | \(e\left(\frac{51}{64}\right)\) |
| \(\chi_{1280}(757,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{15}{64}\right)\) | \(e\left(\frac{17}{32}\right)\) | \(e\left(\frac{15}{32}\right)\) | \(e\left(\frac{57}{64}\right)\) | \(e\left(\frac{11}{64}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{3}{64}\right)\) | \(e\left(\frac{49}{64}\right)\) | \(e\left(\frac{11}{32}\right)\) | \(e\left(\frac{45}{64}\right)\) |
| \(\chi_{1280}(813,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{64}\right)\) | \(e\left(\frac{27}{32}\right)\) | \(e\left(\frac{5}{32}\right)\) | \(e\left(\frac{19}{64}\right)\) | \(e\left(\frac{25}{64}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{1}{64}\right)\) | \(e\left(\frac{59}{64}\right)\) | \(e\left(\frac{25}{32}\right)\) | \(e\left(\frac{15}{64}\right)\) |
| \(\chi_{1280}(837,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{35}{64}\right)\) | \(e\left(\frac{29}{32}\right)\) | \(e\left(\frac{3}{32}\right)\) | \(e\left(\frac{5}{64}\right)\) | \(e\left(\frac{47}{64}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{7}{64}\right)\) | \(e\left(\frac{29}{64}\right)\) | \(e\left(\frac{15}{32}\right)\) | \(e\left(\frac{41}{64}\right)\) |
| \(\chi_{1280}(893,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{57}{64}\right)\) | \(e\left(\frac{7}{32}\right)\) | \(e\left(\frac{25}{32}\right)\) | \(e\left(\frac{63}{64}\right)\) | \(e\left(\frac{29}{64}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{37}{64}\right)\) | \(e\left(\frac{7}{64}\right)\) | \(e\left(\frac{29}{32}\right)\) | \(e\left(\frac{43}{64}\right)\) |
| \(\chi_{1280}(917,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{55}{64}\right)\) | \(e\left(\frac{9}{32}\right)\) | \(e\left(\frac{23}{32}\right)\) | \(e\left(\frac{17}{64}\right)\) | \(e\left(\frac{19}{64}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{11}{64}\right)\) | \(e\left(\frac{9}{64}\right)\) | \(e\left(\frac{19}{32}\right)\) | \(e\left(\frac{37}{64}\right)\) |
| \(\chi_{1280}(973,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{45}{64}\right)\) | \(e\left(\frac{19}{32}\right)\) | \(e\left(\frac{13}{32}\right)\) | \(e\left(\frac{43}{64}\right)\) | \(e\left(\frac{33}{64}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{9}{64}\right)\) | \(e\left(\frac{19}{64}\right)\) | \(e\left(\frac{1}{32}\right)\) | \(e\left(\frac{7}{64}\right)\) |
| \(\chi_{1280}(997,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{64}\right)\) | \(e\left(\frac{21}{32}\right)\) | \(e\left(\frac{11}{32}\right)\) | \(e\left(\frac{29}{64}\right)\) | \(e\left(\frac{55}{64}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{15}{64}\right)\) | \(e\left(\frac{53}{64}\right)\) | \(e\left(\frac{23}{32}\right)\) | \(e\left(\frac{33}{64}\right)\) |
| \(\chi_{1280}(1053,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{33}{64}\right)\) | \(e\left(\frac{31}{32}\right)\) | \(e\left(\frac{1}{32}\right)\) | \(e\left(\frac{23}{64}\right)\) | \(e\left(\frac{37}{64}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{45}{64}\right)\) | \(e\left(\frac{31}{64}\right)\) | \(e\left(\frac{5}{32}\right)\) | \(e\left(\frac{35}{64}\right)\) |
| \(\chi_{1280}(1077,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{64}\right)\) | \(e\left(\frac{1}{32}\right)\) | \(e\left(\frac{31}{32}\right)\) | \(e\left(\frac{41}{64}\right)\) | \(e\left(\frac{27}{64}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{19}{64}\right)\) | \(e\left(\frac{33}{64}\right)\) | \(e\left(\frac{27}{32}\right)\) | \(e\left(\frac{29}{64}\right)\) |
| \(\chi_{1280}(1133,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{21}{64}\right)\) | \(e\left(\frac{11}{32}\right)\) | \(e\left(\frac{21}{32}\right)\) | \(e\left(\frac{3}{64}\right)\) | \(e\left(\frac{41}{64}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{17}{64}\right)\) | \(e\left(\frac{43}{64}\right)\) | \(e\left(\frac{9}{32}\right)\) | \(e\left(\frac{63}{64}\right)\) |
| \(\chi_{1280}(1157,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{51}{64}\right)\) | \(e\left(\frac{13}{32}\right)\) | \(e\left(\frac{19}{32}\right)\) | \(e\left(\frac{53}{64}\right)\) | \(e\left(\frac{63}{64}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{23}{64}\right)\) | \(e\left(\frac{13}{64}\right)\) | \(e\left(\frac{31}{32}\right)\) | \(e\left(\frac{25}{64}\right)\) |
| \(\chi_{1280}(1213,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{9}{64}\right)\) | \(e\left(\frac{23}{32}\right)\) | \(e\left(\frac{9}{32}\right)\) | \(e\left(\frac{47}{64}\right)\) | \(e\left(\frac{45}{64}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{53}{64}\right)\) | \(e\left(\frac{55}{64}\right)\) | \(e\left(\frac{13}{32}\right)\) | \(e\left(\frac{27}{64}\right)\) |