from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(169065, base_ring=CyclotomicField(408))
M = H._module
chi = DirichletCharacter(H, M([68,306,306,177]))
chi.galois_orbit()
[g,chi] = znchar(Mod(83,169065))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(169065\) | |
| Conductor: | \(169065\) | 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 | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(14\) | \(16\) | \(19\) | \(22\) | \(23\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{169065}(83,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{204}\right)\) | \(e\left(\frac{19}{102}\right)\) | \(e\left(\frac{371}{408}\right)\) | \(e\left(\frac{19}{68}\right)\) | \(e\left(\frac{161}{408}\right)\) | \(e\left(\frac{1}{408}\right)\) | \(e\left(\frac{19}{51}\right)\) | \(e\left(\frac{11}{34}\right)\) | \(e\left(\frac{199}{408}\right)\) | \(e\left(\frac{133}{408}\right)\) |
| \(\chi_{169065}(1607,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{173}{204}\right)\) | \(e\left(\frac{71}{102}\right)\) | \(e\left(\frac{361}{408}\right)\) | \(e\left(\frac{37}{68}\right)\) | \(e\left(\frac{403}{408}\right)\) | \(e\left(\frac{299}{408}\right)\) | \(e\left(\frac{20}{51}\right)\) | \(e\left(\frac{25}{34}\right)\) | \(e\left(\frac{341}{408}\right)\) | \(e\left(\frac{191}{408}\right)\) |
| \(\chi_{169065}(1838,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{151}{204}\right)\) | \(e\left(\frac{49}{102}\right)\) | \(e\left(\frac{71}{408}\right)\) | \(e\left(\frac{15}{68}\right)\) | \(e\left(\frac{77}{408}\right)\) | \(e\left(\frac{373}{408}\right)\) | \(e\left(\frac{49}{51}\right)\) | \(e\left(\frac{23}{34}\right)\) | \(e\left(\frac{379}{408}\right)\) | \(e\left(\frac{241}{408}\right)\) |
| \(\chi_{169065}(3398,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{204}\right)\) | \(e\left(\frac{59}{102}\right)\) | \(e\left(\frac{379}{408}\right)\) | \(e\left(\frac{59}{68}\right)\) | \(e\left(\frac{49}{408}\right)\) | \(e\left(\frac{89}{408}\right)\) | \(e\left(\frac{8}{51}\right)\) | \(e\left(\frac{27}{34}\right)\) | \(e\left(\frac{167}{408}\right)\) | \(e\left(\frac{5}{408}\right)\) |
| \(\chi_{169065}(5153,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{179}{204}\right)\) | \(e\left(\frac{77}{102}\right)\) | \(e\left(\frac{199}{408}\right)\) | \(e\left(\frac{43}{68}\right)\) | \(e\left(\frac{325}{408}\right)\) | \(e\left(\frac{149}{408}\right)\) | \(e\left(\frac{26}{51}\right)\) | \(e\left(\frac{7}{34}\right)\) | \(e\left(\frac{275}{408}\right)\) | \(e\left(\frac{233}{408}\right)\) |
| \(\chi_{169065}(8237,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{193}{204}\right)\) | \(e\left(\frac{91}{102}\right)\) | \(e\left(\frac{161}{408}\right)\) | \(e\left(\frac{57}{68}\right)\) | \(e\left(\frac{347}{408}\right)\) | \(e\left(\frac{139}{408}\right)\) | \(e\left(\frac{40}{51}\right)\) | \(e\left(\frac{33}{34}\right)\) | \(e\left(\frac{325}{408}\right)\) | \(e\left(\frac{127}{408}\right)\) |
| \(\chi_{169065}(8627,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{41}{204}\right)\) | \(e\left(\frac{41}{102}\right)\) | \(e\left(\frac{253}{408}\right)\) | \(e\left(\frac{41}{68}\right)\) | \(e\left(\frac{79}{408}\right)\) | \(e\left(\frac{335}{408}\right)\) | \(e\left(\frac{41}{51}\right)\) | \(e\left(\frac{13}{34}\right)\) | \(e\left(\frac{161}{408}\right)\) | \(e\left(\frac{83}{408}\right)\) |
| \(\chi_{169065}(10028,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{139}{204}\right)\) | \(e\left(\frac{37}{102}\right)\) | \(e\left(\frac{395}{408}\right)\) | \(e\left(\frac{3}{68}\right)\) | \(e\left(\frac{233}{408}\right)\) | \(e\left(\frac{265}{408}\right)\) | \(e\left(\frac{37}{51}\right)\) | \(e\left(\frac{25}{34}\right)\) | \(e\left(\frac{103}{408}\right)\) | \(e\left(\frac{157}{408}\right)\) |
| \(\chi_{169065}(11552,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{101}{204}\right)\) | \(e\left(\frac{101}{102}\right)\) | \(e\left(\frac{265}{408}\right)\) | \(e\left(\frac{33}{68}\right)\) | \(e\left(\frac{115}{408}\right)\) | \(e\left(\frac{59}{408}\right)\) | \(e\left(\frac{50}{51}\right)\) | \(e\left(\frac{3}{34}\right)\) | \(e\left(\frac{317}{408}\right)\) | \(e\left(\frac{95}{408}\right)\) |
| \(\chi_{169065}(11783,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{204}\right)\) | \(e\left(\frac{31}{102}\right)\) | \(e\left(\frac{47}{408}\right)\) | \(e\left(\frac{31}{68}\right)\) | \(e\left(\frac{5}{408}\right)\) | \(e\left(\frac{109}{408}\right)\) | \(e\left(\frac{31}{51}\right)\) | \(e\left(\frac{9}{34}\right)\) | \(e\left(\frac{67}{408}\right)\) | \(e\left(\frac{217}{408}\right)\) |
| \(\chi_{169065}(13343,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{179}{204}\right)\) | \(e\left(\frac{77}{102}\right)\) | \(e\left(\frac{403}{408}\right)\) | \(e\left(\frac{43}{68}\right)\) | \(e\left(\frac{121}{408}\right)\) | \(e\left(\frac{353}{408}\right)\) | \(e\left(\frac{26}{51}\right)\) | \(e\left(\frac{7}{34}\right)\) | \(e\left(\frac{71}{408}\right)\) | \(e\left(\frac{29}{408}\right)\) |
| \(\chi_{169065}(15098,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{204}\right)\) | \(e\left(\frac{59}{102}\right)\) | \(e\left(\frac{175}{408}\right)\) | \(e\left(\frac{59}{68}\right)\) | \(e\left(\frac{253}{408}\right)\) | \(e\left(\frac{293}{408}\right)\) | \(e\left(\frac{8}{51}\right)\) | \(e\left(\frac{27}{34}\right)\) | \(e\left(\frac{371}{408}\right)\) | \(e\left(\frac{209}{408}\right)\) |
| \(\chi_{169065}(15257,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{157}{204}\right)\) | \(e\left(\frac{55}{102}\right)\) | \(e\left(\frac{317}{408}\right)\) | \(e\left(\frac{21}{68}\right)\) | \(e\left(\frac{407}{408}\right)\) | \(e\left(\frac{223}{408}\right)\) | \(e\left(\frac{4}{51}\right)\) | \(e\left(\frac{5}{34}\right)\) | \(e\left(\frac{313}{408}\right)\) | \(e\left(\frac{283}{408}\right)\) |
| \(\chi_{169065}(18182,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{121}{204}\right)\) | \(e\left(\frac{19}{102}\right)\) | \(e\left(\frac{65}{408}\right)\) | \(e\left(\frac{53}{68}\right)\) | \(e\left(\frac{59}{408}\right)\) | \(e\left(\frac{307}{408}\right)\) | \(e\left(\frac{19}{51}\right)\) | \(e\left(\frac{11}{34}\right)\) | \(e\left(\frac{301}{408}\right)\) | \(e\left(\frac{31}{408}\right)\) |
| \(\chi_{169065}(18572,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{113}{204}\right)\) | \(e\left(\frac{11}{102}\right)\) | \(e\left(\frac{349}{408}\right)\) | \(e\left(\frac{45}{68}\right)\) | \(e\left(\frac{367}{408}\right)\) | \(e\left(\frac{167}{408}\right)\) | \(e\left(\frac{11}{51}\right)\) | \(e\left(\frac{1}{34}\right)\) | \(e\left(\frac{185}{408}\right)\) | \(e\left(\frac{179}{408}\right)\) |
| \(\chi_{169065}(19973,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{55}{204}\right)\) | \(e\left(\frac{55}{102}\right)\) | \(e\left(\frac{11}{408}\right)\) | \(e\left(\frac{55}{68}\right)\) | \(e\left(\frac{305}{408}\right)\) | \(e\left(\frac{121}{408}\right)\) | \(e\left(\frac{4}{51}\right)\) | \(e\left(\frac{5}{34}\right)\) | \(e\left(\frac{7}{408}\right)\) | \(e\left(\frac{181}{408}\right)\) |
| \(\chi_{169065}(21497,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{204}\right)\) | \(e\left(\frac{29}{102}\right)\) | \(e\left(\frac{169}{408}\right)\) | \(e\left(\frac{29}{68}\right)\) | \(e\left(\frac{235}{408}\right)\) | \(e\left(\frac{227}{408}\right)\) | \(e\left(\frac{29}{51}\right)\) | \(e\left(\frac{15}{34}\right)\) | \(e\left(\frac{293}{408}\right)\) | \(e\left(\frac{407}{408}\right)\) |
| \(\chi_{169065}(21728,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{115}{204}\right)\) | \(e\left(\frac{13}{102}\right)\) | \(e\left(\frac{23}{408}\right)\) | \(e\left(\frac{47}{68}\right)\) | \(e\left(\frac{341}{408}\right)\) | \(e\left(\frac{253}{408}\right)\) | \(e\left(\frac{13}{51}\right)\) | \(e\left(\frac{29}{34}\right)\) | \(e\left(\frac{163}{408}\right)\) | \(e\left(\frac{193}{408}\right)\) |
| \(\chi_{169065}(23288,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{95}{204}\right)\) | \(e\left(\frac{95}{102}\right)\) | \(e\left(\frac{19}{408}\right)\) | \(e\left(\frac{27}{68}\right)\) | \(e\left(\frac{193}{408}\right)\) | \(e\left(\frac{209}{408}\right)\) | \(e\left(\frac{44}{51}\right)\) | \(e\left(\frac{21}{34}\right)\) | \(e\left(\frac{383}{408}\right)\) | \(e\left(\frac{53}{408}\right)\) |
| \(\chi_{169065}(25043,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{143}{204}\right)\) | \(e\left(\frac{41}{102}\right)\) | \(e\left(\frac{151}{408}\right)\) | \(e\left(\frac{7}{68}\right)\) | \(e\left(\frac{181}{408}\right)\) | \(e\left(\frac{29}{408}\right)\) | \(e\left(\frac{41}{51}\right)\) | \(e\left(\frac{13}{34}\right)\) | \(e\left(\frac{59}{408}\right)\) | \(e\left(\frac{185}{408}\right)\) |
| \(\chi_{169065}(25202,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{25}{204}\right)\) | \(e\left(\frac{25}{102}\right)\) | \(e\left(\frac{5}{408}\right)\) | \(e\left(\frac{25}{68}\right)\) | \(e\left(\frac{287}{408}\right)\) | \(e\left(\frac{55}{408}\right)\) | \(e\left(\frac{25}{51}\right)\) | \(e\left(\frac{27}{34}\right)\) | \(e\left(\frac{337}{408}\right)\) | \(e\left(\frac{379}{408}\right)\) |
| \(\chi_{169065}(28127,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{49}{204}\right)\) | \(e\left(\frac{49}{102}\right)\) | \(e\left(\frac{377}{408}\right)\) | \(e\left(\frac{49}{68}\right)\) | \(e\left(\frac{179}{408}\right)\) | \(e\left(\frac{67}{408}\right)\) | \(e\left(\frac{49}{51}\right)\) | \(e\left(\frac{23}{34}\right)\) | \(e\left(\frac{277}{408}\right)\) | \(e\left(\frac{343}{408}\right)\) |
| \(\chi_{169065}(28517,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{185}{204}\right)\) | \(e\left(\frac{83}{102}\right)\) | \(e\left(\frac{37}{408}\right)\) | \(e\left(\frac{49}{68}\right)\) | \(e\left(\frac{247}{408}\right)\) | \(e\left(\frac{407}{408}\right)\) | \(e\left(\frac{32}{51}\right)\) | \(e\left(\frac{23}{34}\right)\) | \(e\left(\frac{209}{408}\right)\) | \(e\left(\frac{275}{408}\right)\) |
| \(\chi_{169065}(29918,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{175}{204}\right)\) | \(e\left(\frac{73}{102}\right)\) | \(e\left(\frac{35}{408}\right)\) | \(e\left(\frac{39}{68}\right)\) | \(e\left(\frac{377}{408}\right)\) | \(e\left(\frac{385}{408}\right)\) | \(e\left(\frac{22}{51}\right)\) | \(e\left(\frac{19}{34}\right)\) | \(e\left(\frac{319}{408}\right)\) | \(e\left(\frac{205}{408}\right)\) |
| \(\chi_{169065}(31442,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{161}{204}\right)\) | \(e\left(\frac{59}{102}\right)\) | \(e\left(\frac{73}{408}\right)\) | \(e\left(\frac{25}{68}\right)\) | \(e\left(\frac{355}{408}\right)\) | \(e\left(\frac{395}{408}\right)\) | \(e\left(\frac{8}{51}\right)\) | \(e\left(\frac{27}{34}\right)\) | \(e\left(\frac{269}{408}\right)\) | \(e\left(\frac{311}{408}\right)\) |
| \(\chi_{169065}(31673,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{199}{204}\right)\) | \(e\left(\frac{97}{102}\right)\) | \(e\left(\frac{407}{408}\right)\) | \(e\left(\frac{63}{68}\right)\) | \(e\left(\frac{269}{408}\right)\) | \(e\left(\frac{397}{408}\right)\) | \(e\left(\frac{46}{51}\right)\) | \(e\left(\frac{15}{34}\right)\) | \(e\left(\frac{259}{408}\right)\) | \(e\left(\frac{169}{408}\right)\) |
| \(\chi_{169065}(33233,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{204}\right)\) | \(e\left(\frac{11}{102}\right)\) | \(e\left(\frac{43}{408}\right)\) | \(e\left(\frac{11}{68}\right)\) | \(e\left(\frac{265}{408}\right)\) | \(e\left(\frac{65}{408}\right)\) | \(e\left(\frac{11}{51}\right)\) | \(e\left(\frac{1}{34}\right)\) | \(e\left(\frac{287}{408}\right)\) | \(e\left(\frac{77}{408}\right)\) |
| \(\chi_{169065}(34988,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{204}\right)\) | \(e\left(\frac{23}{102}\right)\) | \(e\left(\frac{127}{408}\right)\) | \(e\left(\frac{23}{68}\right)\) | \(e\left(\frac{109}{408}\right)\) | \(e\left(\frac{173}{408}\right)\) | \(e\left(\frac{23}{51}\right)\) | \(e\left(\frac{33}{34}\right)\) | \(e\left(\frac{155}{408}\right)\) | \(e\left(\frac{161}{408}\right)\) |
| \(\chi_{169065}(35147,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{97}{204}\right)\) | \(e\left(\frac{97}{102}\right)\) | \(e\left(\frac{101}{408}\right)\) | \(e\left(\frac{29}{68}\right)\) | \(e\left(\frac{167}{408}\right)\) | \(e\left(\frac{295}{408}\right)\) | \(e\left(\frac{46}{51}\right)\) | \(e\left(\frac{15}{34}\right)\) | \(e\left(\frac{361}{408}\right)\) | \(e\left(\frac{67}{408}\right)\) |
| \(\chi_{169065}(38072,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{181}{204}\right)\) | \(e\left(\frac{79}{102}\right)\) | \(e\left(\frac{281}{408}\right)\) | \(e\left(\frac{45}{68}\right)\) | \(e\left(\frac{299}{408}\right)\) | \(e\left(\frac{235}{408}\right)\) | \(e\left(\frac{28}{51}\right)\) | \(e\left(\frac{1}{34}\right)\) | \(e\left(\frac{253}{408}\right)\) | \(e\left(\frac{247}{408}\right)\) |
| \(\chi_{169065}(38462,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{53}{204}\right)\) | \(e\left(\frac{53}{102}\right)\) | \(e\left(\frac{133}{408}\right)\) | \(e\left(\frac{53}{68}\right)\) | \(e\left(\frac{127}{408}\right)\) | \(e\left(\frac{239}{408}\right)\) | \(e\left(\frac{2}{51}\right)\) | \(e\left(\frac{11}{34}\right)\) | \(e\left(\frac{233}{408}\right)\) | \(e\left(\frac{371}{408}\right)\) |