from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(959, base_ring=CyclotomicField(408))
M = H._module
chi = DirichletCharacter(H, M([136,375]))
chi.galois_orbit()
[g,chi] = znchar(Mod(23,959))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(959\) | |
| Conductor: | \(959\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(408\) | 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: | $\Q(\zeta_{408})$ |
| Fixed field: | Number field defined by a degree 408 polynomial (not computed) |
First 31 of 128 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{959}(23,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{175}{204}\right)\) | \(e\left(\frac{103}{408}\right)\) | \(e\left(\frac{73}{102}\right)\) | \(e\left(\frac{245}{408}\right)\) | \(e\left(\frac{15}{136}\right)\) | \(e\left(\frac{39}{68}\right)\) | \(e\left(\frac{103}{204}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{95}{204}\right)\) | \(e\left(\frac{395}{408}\right)\) |
| \(\chi_{959}(46,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{53}{204}\right)\) | \(e\left(\frac{269}{408}\right)\) | \(e\left(\frac{53}{102}\right)\) | \(e\left(\frac{319}{408}\right)\) | \(e\left(\frac{125}{136}\right)\) | \(e\left(\frac{53}{68}\right)\) | \(e\left(\frac{65}{204}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{157}{204}\right)\) | \(e\left(\frac{73}{408}\right)\) |
| \(\chi_{959}(51,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{109}{204}\right)\) | \(e\left(\frac{253}{408}\right)\) | \(e\left(\frac{7}{102}\right)\) | \(e\left(\frac{71}{408}\right)\) | \(e\left(\frac{21}{136}\right)\) | \(e\left(\frac{41}{68}\right)\) | \(e\left(\frac{49}{204}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{65}{204}\right)\) | \(e\left(\frac{281}{408}\right)\) |
| \(\chi_{959}(53,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{197}{204}\right)\) | \(e\left(\frac{257}{408}\right)\) | \(e\left(\frac{95}{102}\right)\) | \(e\left(\frac{235}{408}\right)\) | \(e\left(\frac{81}{136}\right)\) | \(e\left(\frac{61}{68}\right)\) | \(e\left(\frac{53}{204}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{37}{204}\right)\) | \(e\left(\frac{229}{408}\right)\) |
| \(\chi_{959}(58,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{204}\right)\) | \(e\left(\frac{31}{408}\right)\) | \(e\left(\frac{19}{102}\right)\) | \(e\left(\frac{149}{408}\right)\) | \(e\left(\frac{23}{136}\right)\) | \(e\left(\frac{19}{68}\right)\) | \(e\left(\frac{31}{204}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{191}{204}\right)\) | \(e\left(\frac{107}{408}\right)\) |
| \(\chi_{959}(67,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{137}{204}\right)\) | \(e\left(\frac{41}{408}\right)\) | \(e\left(\frac{35}{102}\right)\) | \(e\left(\frac{355}{408}\right)\) | \(e\left(\frac{105}{136}\right)\) | \(e\left(\frac{1}{68}\right)\) | \(e\left(\frac{41}{204}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{121}{204}\right)\) | \(e\left(\frac{181}{408}\right)\) |
| \(\chi_{959}(79,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{204}\right)\) | \(e\left(\frac{235}{408}\right)\) | \(e\left(\frac{19}{102}\right)\) | \(e\left(\frac{353}{408}\right)\) | \(e\left(\frac{91}{136}\right)\) | \(e\left(\frac{19}{68}\right)\) | \(e\left(\frac{31}{204}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{191}{204}\right)\) | \(e\left(\frac{311}{408}\right)\) |
| \(\chi_{959}(86,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{109}{204}\right)\) | \(e\left(\frac{49}{408}\right)\) | \(e\left(\frac{7}{102}\right)\) | \(e\left(\frac{275}{408}\right)\) | \(e\left(\frac{89}{136}\right)\) | \(e\left(\frac{41}{68}\right)\) | \(e\left(\frac{49}{204}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{65}{204}\right)\) | \(e\left(\frac{77}{408}\right)\) |
| \(\chi_{959}(95,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{47}{204}\right)\) | \(e\left(\frac{227}{408}\right)\) | \(e\left(\frac{47}{102}\right)\) | \(e\left(\frac{25}{408}\right)\) | \(e\left(\frac{107}{136}\right)\) | \(e\left(\frac{47}{68}\right)\) | \(e\left(\frac{23}{204}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{43}{204}\right)\) | \(e\left(\frac{7}{408}\right)\) |
| \(\chi_{959}(102,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{191}{204}\right)\) | \(e\left(\frac{11}{408}\right)\) | \(e\left(\frac{89}{102}\right)\) | \(e\left(\frac{145}{408}\right)\) | \(e\left(\frac{131}{136}\right)\) | \(e\left(\frac{55}{68}\right)\) | \(e\left(\frac{11}{204}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{127}{204}\right)\) | \(e\left(\frac{367}{408}\right)\) |
| \(\chi_{959}(114,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{175}{204}\right)\) | \(e\left(\frac{307}{408}\right)\) | \(e\left(\frac{73}{102}\right)\) | \(e\left(\frac{41}{408}\right)\) | \(e\left(\frac{83}{136}\right)\) | \(e\left(\frac{39}{68}\right)\) | \(e\left(\frac{103}{204}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{95}{204}\right)\) | \(e\left(\frac{191}{408}\right)\) |
| \(\chi_{959}(116,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{101}{204}\right)\) | \(e\left(\frac{197}{408}\right)\) | \(e\left(\frac{101}{102}\right)\) | \(e\left(\frac{223}{408}\right)\) | \(e\left(\frac{133}{136}\right)\) | \(e\left(\frac{33}{68}\right)\) | \(e\left(\frac{197}{204}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{49}{204}\right)\) | \(e\left(\frac{193}{408}\right)\) |
| \(\chi_{959}(142,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{37}{204}\right)\) | \(e\left(\frac{361}{408}\right)\) | \(e\left(\frac{37}{102}\right)\) | \(e\left(\frac{11}{408}\right)\) | \(e\left(\frac{9}{136}\right)\) | \(e\left(\frac{37}{68}\right)\) | \(e\left(\frac{157}{204}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{125}{204}\right)\) | \(e\left(\frac{101}{408}\right)\) |
| \(\chi_{959}(149,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{204}\right)\) | \(e\left(\frac{199}{408}\right)\) | \(e\left(\frac{43}{102}\right)\) | \(e\left(\frac{101}{408}\right)\) | \(e\left(\frac{95}{136}\right)\) | \(e\left(\frac{43}{68}\right)\) | \(e\left(\frac{199}{204}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{35}{204}\right)\) | \(e\left(\frac{371}{408}\right)\) |
| \(\chi_{959}(158,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{101}{204}\right)\) | \(e\left(\frac{401}{408}\right)\) | \(e\left(\frac{101}{102}\right)\) | \(e\left(\frac{19}{408}\right)\) | \(e\left(\frac{65}{136}\right)\) | \(e\left(\frac{33}{68}\right)\) | \(e\left(\frac{197}{204}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{49}{204}\right)\) | \(e\left(\frac{397}{408}\right)\) |
| \(\chi_{959}(163,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{49}{204}\right)\) | \(e\left(\frac{241}{408}\right)\) | \(e\left(\frac{49}{102}\right)\) | \(e\left(\frac{395}{408}\right)\) | \(e\left(\frac{113}{136}\right)\) | \(e\left(\frac{49}{68}\right)\) | \(e\left(\frac{37}{204}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{149}{204}\right)\) | \(e\left(\frac{29}{408}\right)\) |
| \(\chi_{959}(170,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{145}{204}\right)\) | \(e\left(\frac{97}{408}\right)\) | \(e\left(\frac{43}{102}\right)\) | \(e\left(\frac{203}{408}\right)\) | \(e\left(\frac{129}{136}\right)\) | \(e\left(\frac{9}{68}\right)\) | \(e\left(\frac{97}{204}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{137}{204}\right)\) | \(e\left(\frac{269}{408}\right)\) |
| \(\chi_{959}(172,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{191}{204}\right)\) | \(e\left(\frac{215}{408}\right)\) | \(e\left(\frac{89}{102}\right)\) | \(e\left(\frac{349}{408}\right)\) | \(e\left(\frac{63}{136}\right)\) | \(e\left(\frac{55}{68}\right)\) | \(e\left(\frac{11}{204}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{127}{204}\right)\) | \(e\left(\frac{163}{408}\right)\) |
| \(\chi_{959}(177,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{79}{204}\right)\) | \(e\left(\frac{43}{408}\right)\) | \(e\left(\frac{79}{102}\right)\) | \(e\left(\frac{233}{408}\right)\) | \(e\left(\frac{67}{136}\right)\) | \(e\left(\frac{11}{68}\right)\) | \(e\left(\frac{43}{204}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{107}{204}\right)\) | \(e\left(\frac{359}{408}\right)\) |
| \(\chi_{959}(179,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{47}{204}\right)\) | \(e\left(\frac{23}{408}\right)\) | \(e\left(\frac{47}{102}\right)\) | \(e\left(\frac{229}{408}\right)\) | \(e\left(\frac{39}{136}\right)\) | \(e\left(\frac{47}{68}\right)\) | \(e\left(\frac{23}{204}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{43}{204}\right)\) | \(e\left(\frac{211}{408}\right)\) |
| \(\chi_{959}(184,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{204}\right)\) | \(e\left(\frac{193}{408}\right)\) | \(e\left(\frac{13}{102}\right)\) | \(e\left(\frac{59}{408}\right)\) | \(e\left(\frac{73}{136}\right)\) | \(e\left(\frac{13}{68}\right)\) | \(e\left(\frac{193}{204}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{77}{204}\right)\) | \(e\left(\frac{245}{408}\right)\) |
| \(\chi_{959}(191,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{127}{204}\right)\) | \(e\left(\frac{175}{408}\right)\) | \(e\left(\frac{25}{102}\right)\) | \(e\left(\frac{341}{408}\right)\) | \(e\left(\frac{7}{136}\right)\) | \(e\left(\frac{59}{68}\right)\) | \(e\left(\frac{175}{204}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{203}{204}\right)\) | \(e\left(\frac{275}{408}\right)\) |
| \(\chi_{959}(207,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{137}{204}\right)\) | \(e\left(\frac{245}{408}\right)\) | \(e\left(\frac{35}{102}\right)\) | \(e\left(\frac{151}{408}\right)\) | \(e\left(\frac{37}{136}\right)\) | \(e\left(\frac{1}{68}\right)\) | \(e\left(\frac{41}{204}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{121}{204}\right)\) | \(e\left(\frac{385}{408}\right)\) |
| \(\chi_{959}(212,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{157}{204}\right)\) | \(e\left(\frac{181}{408}\right)\) | \(e\left(\frac{55}{102}\right)\) | \(e\left(\frac{383}{408}\right)\) | \(e\left(\frac{29}{136}\right)\) | \(e\left(\frac{21}{68}\right)\) | \(e\left(\frac{181}{204}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{161}{204}\right)\) | \(e\left(\frac{401}{408}\right)\) |
| \(\chi_{959}(219,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{204}\right)\) | \(e\left(\frac{115}{408}\right)\) | \(e\left(\frac{31}{102}\right)\) | \(e\left(\frac{329}{408}\right)\) | \(e\left(\frac{59}{136}\right)\) | \(e\left(\frac{31}{68}\right)\) | \(e\left(\frac{115}{204}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{11}{204}\right)\) | \(e\left(\frac{239}{408}\right)\) |
| \(\chi_{959}(221,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{197}{204}\right)\) | \(e\left(\frac{53}{408}\right)\) | \(e\left(\frac{95}{102}\right)\) | \(e\left(\frac{31}{408}\right)\) | \(e\left(\frac{13}{136}\right)\) | \(e\left(\frac{61}{68}\right)\) | \(e\left(\frac{53}{204}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{37}{204}\right)\) | \(e\left(\frac{25}{408}\right)\) |
| \(\chi_{959}(226,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{139}{204}\right)\) | \(e\left(\frac{55}{408}\right)\) | \(e\left(\frac{37}{102}\right)\) | \(e\left(\frac{317}{408}\right)\) | \(e\left(\frac{111}{136}\right)\) | \(e\left(\frac{3}{68}\right)\) | \(e\left(\frac{55}{204}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{23}{204}\right)\) | \(e\left(\frac{203}{408}\right)\) |
| \(\chi_{959}(228,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{53}{204}\right)\) | \(e\left(\frac{65}{408}\right)\) | \(e\left(\frac{53}{102}\right)\) | \(e\left(\frac{115}{408}\right)\) | \(e\left(\frac{57}{136}\right)\) | \(e\left(\frac{53}{68}\right)\) | \(e\left(\frac{65}{204}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{157}{204}\right)\) | \(e\left(\frac{277}{408}\right)\) |
| \(\chi_{959}(247,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{181}{204}\right)\) | \(e\left(\frac{349}{408}\right)\) | \(e\left(\frac{79}{102}\right)\) | \(e\left(\frac{335}{408}\right)\) | \(e\left(\frac{101}{136}\right)\) | \(e\left(\frac{45}{68}\right)\) | \(e\left(\frac{145}{204}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{5}{204}\right)\) | \(e\left(\frac{257}{408}\right)\) |
| \(\chi_{959}(254,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{133}{204}\right)\) | \(e\left(\frac{217}{408}\right)\) | \(e\left(\frac{31}{102}\right)\) | \(e\left(\frac{227}{408}\right)\) | \(e\left(\frac{25}{136}\right)\) | \(e\left(\frac{65}{68}\right)\) | \(e\left(\frac{13}{204}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{113}{204}\right)\) | \(e\left(\frac{341}{408}\right)\) |
| \(\chi_{959}(261,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{103}{204}\right)\) | \(e\left(\frac{7}{408}\right)\) | \(e\left(\frac{1}{102}\right)\) | \(e\left(\frac{389}{408}\right)\) | \(e\left(\frac{71}{136}\right)\) | \(e\left(\frac{35}{68}\right)\) | \(e\left(\frac{7}{204}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{155}{204}\right)\) | \(e\left(\frac{11}{408}\right)\) |