from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(257, base_ring=CyclotomicField(256))
M = H._module
chi = DirichletCharacter(H, M([1]))
chi.galois_orbit()
[g,chi] = znchar(Mod(3,257))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(257\) | |
| Conductor: | \(257\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(256\) | 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: | $\Q(\zeta_{256})$ |
| Fixed field: | Number field defined by a degree 256 polynomial (not computed) |
First 31 of 128 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{257}(3,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{1}{256}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{55}{256}\right)\) | \(e\left(\frac{49}{256}\right)\) | \(e\left(\frac{85}{256}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{1}{128}\right)\) | \(e\left(\frac{103}{256}\right)\) | \(e\left(\frac{49}{64}\right)\) |
| \(\chi_{257}(5,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{55}{256}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{209}{256}\right)\) | \(e\left(\frac{135}{256}\right)\) | \(e\left(\frac{67}{256}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{55}{128}\right)\) | \(e\left(\frac{33}{256}\right)\) | \(e\left(\frac{7}{64}\right)\) |
| \(\chi_{257}(6,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{49}{256}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{135}{256}\right)\) | \(e\left(\frac{97}{256}\right)\) | \(e\left(\frac{69}{256}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{49}{128}\right)\) | \(e\left(\frac{183}{256}\right)\) | \(e\left(\frac{33}{64}\right)\) |
| \(\chi_{257}(7,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{85}{256}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{67}{256}\right)\) | \(e\left(\frac{69}{256}\right)\) | \(e\left(\frac{57}{256}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{85}{128}\right)\) | \(e\left(\frac{51}{256}\right)\) | \(e\left(\frac{5}{64}\right)\) |
| \(\chi_{257}(10,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{103}{256}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{33}{256}\right)\) | \(e\left(\frac{183}{256}\right)\) | \(e\left(\frac{51}{256}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{103}{128}\right)\) | \(e\left(\frac{113}{256}\right)\) | \(e\left(\frac{55}{64}\right)\) |
| \(\chi_{257}(12,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{97}{256}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{215}{256}\right)\) | \(e\left(\frac{145}{256}\right)\) | \(e\left(\frac{53}{256}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{97}{128}\right)\) | \(e\left(\frac{7}{256}\right)\) | \(e\left(\frac{17}{64}\right)\) |
| \(\chi_{257}(14,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{133}{256}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{147}{256}\right)\) | \(e\left(\frac{117}{256}\right)\) | \(e\left(\frac{41}{256}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{5}{128}\right)\) | \(e\left(\frac{131}{256}\right)\) | \(e\left(\frac{53}{64}\right)\) |
| \(\chi_{257}(19,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{125}{256}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{219}{256}\right)\) | \(e\left(\frac{237}{256}\right)\) | \(e\left(\frac{129}{256}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{125}{128}\right)\) | \(e\left(\frac{75}{256}\right)\) | \(e\left(\frac{45}{64}\right)\) |
| \(\chi_{257}(20,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{151}{256}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{113}{256}\right)\) | \(e\left(\frac{231}{256}\right)\) | \(e\left(\frac{35}{256}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{23}{128}\right)\) | \(e\left(\frac{193}{256}\right)\) | \(e\left(\frac{39}{64}\right)\) |
| \(\chi_{257}(24,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{145}{256}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{39}{256}\right)\) | \(e\left(\frac{193}{256}\right)\) | \(e\left(\frac{37}{256}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{17}{128}\right)\) | \(e\left(\frac{87}{256}\right)\) | \(e\left(\frac{1}{64}\right)\) |
| \(\chi_{257}(27,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{3}{256}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{165}{256}\right)\) | \(e\left(\frac{147}{256}\right)\) | \(e\left(\frac{255}{256}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{3}{128}\right)\) | \(e\left(\frac{53}{256}\right)\) | \(e\left(\frac{19}{64}\right)\) |
| \(\chi_{257}(28,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{181}{256}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{227}{256}\right)\) | \(e\left(\frac{165}{256}\right)\) | \(e\left(\frac{25}{256}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{53}{128}\right)\) | \(e\left(\frac{211}{256}\right)\) | \(e\left(\frac{37}{64}\right)\) |
| \(\chi_{257}(33,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{197}{256}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{83}{256}\right)\) | \(e\left(\frac{181}{256}\right)\) | \(e\left(\frac{105}{256}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{69}{128}\right)\) | \(e\left(\frac{67}{256}\right)\) | \(e\left(\frac{53}{64}\right)\) |
| \(\chi_{257}(37,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{219}{256}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{13}{256}\right)\) | \(e\left(\frac{235}{256}\right)\) | \(e\left(\frac{183}{256}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{91}{128}\right)\) | \(e\left(\frac{29}{256}\right)\) | \(e\left(\frac{43}{64}\right)\) |
| \(\chi_{257}(38,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{173}{256}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{43}{256}\right)\) | \(e\left(\frac{29}{256}\right)\) | \(e\left(\frac{113}{256}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{45}{128}\right)\) | \(e\left(\frac{155}{256}\right)\) | \(e\left(\frac{29}{64}\right)\) |
| \(\chi_{257}(39,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{107}{256}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{253}{256}\right)\) | \(e\left(\frac{123}{256}\right)\) | \(e\left(\frac{135}{256}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{107}{128}\right)\) | \(e\left(\frac{13}{256}\right)\) | \(e\left(\frac{59}{64}\right)\) |
| \(\chi_{257}(40,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{199}{256}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{193}{256}\right)\) | \(e\left(\frac{23}{256}\right)\) | \(e\left(\frac{19}{256}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{71}{128}\right)\) | \(e\left(\frac{17}{256}\right)\) | \(e\left(\frac{23}{64}\right)\) |
| \(\chi_{257}(41,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{19}{256}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{21}{256}\right)\) | \(e\left(\frac{163}{256}\right)\) | \(e\left(\frac{79}{256}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{19}{128}\right)\) | \(e\left(\frac{165}{256}\right)\) | \(e\left(\frac{35}{64}\right)\) |
| \(\chi_{257}(43,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{207}{256}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{121}{256}\right)\) | \(e\left(\frac{159}{256}\right)\) | \(e\left(\frac{187}{256}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{79}{128}\right)\) | \(e\left(\frac{73}{256}\right)\) | \(e\left(\frac{31}{64}\right)\) |
| \(\chi_{257}(45,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{57}{256}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{63}{256}\right)\) | \(e\left(\frac{233}{256}\right)\) | \(e\left(\frac{237}{256}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{57}{128}\right)\) | \(e\left(\frac{239}{256}\right)\) | \(e\left(\frac{41}{64}\right)\) |
| \(\chi_{257}(47,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{61}{256}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{27}{256}\right)\) | \(e\left(\frac{173}{256}\right)\) | \(e\left(\frac{65}{256}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{61}{128}\right)\) | \(e\left(\frac{139}{256}\right)\) | \(e\left(\frac{45}{64}\right)\) |
| \(\chi_{257}(48,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{193}{256}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{119}{256}\right)\) | \(e\left(\frac{241}{256}\right)\) | \(e\left(\frac{21}{256}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{65}{128}\right)\) | \(e\left(\frac{167}{256}\right)\) | \(e\left(\frac{49}{64}\right)\) |
| \(\chi_{257}(51,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{121}{256}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{255}{256}\right)\) | \(e\left(\frac{41}{256}\right)\) | \(e\left(\frac{45}{256}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{121}{128}\right)\) | \(e\left(\frac{175}{256}\right)\) | \(e\left(\frac{41}{64}\right)\) |
| \(\chi_{257}(53,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{89}{256}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{31}{256}\right)\) | \(e\left(\frac{9}{256}\right)\) | \(e\left(\frac{141}{256}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{89}{128}\right)\) | \(e\left(\frac{207}{256}\right)\) | \(e\left(\frac{9}{64}\right)\) |
| \(\chi_{257}(54,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{51}{256}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{245}{256}\right)\) | \(e\left(\frac{195}{256}\right)\) | \(e\left(\frac{239}{256}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{51}{128}\right)\) | \(e\left(\frac{133}{256}\right)\) | \(e\left(\frac{3}{64}\right)\) |
| \(\chi_{257}(55,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{251}{256}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{237}{256}\right)\) | \(e\left(\frac{11}{256}\right)\) | \(e\left(\frac{87}{256}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{123}{128}\right)\) | \(e\left(\frac{253}{256}\right)\) | \(e\left(\frac{11}{64}\right)\) |
| \(\chi_{257}(56,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{229}{256}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{51}{256}\right)\) | \(e\left(\frac{213}{256}\right)\) | \(e\left(\frac{9}{256}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{101}{128}\right)\) | \(e\left(\frac{35}{256}\right)\) | \(e\left(\frac{21}{64}\right)\) |
| \(\chi_{257}(63,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{87}{256}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{177}{256}\right)\) | \(e\left(\frac{167}{256}\right)\) | \(e\left(\frac{227}{256}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{87}{128}\right)\) | \(e\left(\frac{1}{256}\right)\) | \(e\left(\frac{39}{64}\right)\) |
| \(\chi_{257}(65,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{161}{256}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{151}{256}\right)\) | \(e\left(\frac{209}{256}\right)\) | \(e\left(\frac{117}{256}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{33}{128}\right)\) | \(e\left(\frac{199}{256}\right)\) | \(e\left(\frac{17}{64}\right)\) |
| \(\chi_{257}(66,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{245}{256}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{163}{256}\right)\) | \(e\left(\frac{229}{256}\right)\) | \(e\left(\frac{89}{256}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{117}{128}\right)\) | \(e\left(\frac{147}{256}\right)\) | \(e\left(\frac{37}{64}\right)\) |
| \(\chi_{257}(69,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{29}{256}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{59}{256}\right)\) | \(e\left(\frac{141}{256}\right)\) | \(e\left(\frac{161}{256}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{29}{128}\right)\) | \(e\left(\frac{171}{256}\right)\) | \(e\left(\frac{13}{64}\right)\) |