from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(338130, base_ring=CyclotomicField(204))
M = H._module
chi = DirichletCharacter(H, M([34,51,51,75]))
chi.galois_orbit()
[g,chi] = znchar(Mod(47,338130))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(338130\) | |
| Conductor: | \(169065\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(204\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 169065.bib | 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: | $\Q(\zeta_{204})$ |
| Fixed field: | Number field defined by a degree 204 polynomial (not computed) |
First 31 of 64 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(7\) | \(11\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) | \(47\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{338130}(47,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{133}{204}\right)\) | \(e\left(\frac{19}{51}\right)\) | \(e\left(\frac{61}{68}\right)\) | \(e\left(\frac{7}{102}\right)\) | \(e\left(\frac{127}{204}\right)\) | \(e\left(\frac{91}{102}\right)\) | \(e\left(\frac{29}{68}\right)\) | \(e\left(\frac{97}{102}\right)\) | \(e\left(\frac{1}{204}\right)\) | \(e\left(\frac{95}{102}\right)\) |
| \(\chi_{338130}(11003,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{47}{204}\right)\) | \(e\left(\frac{14}{51}\right)\) | \(e\left(\frac{19}{68}\right)\) | \(e\left(\frac{83}{102}\right)\) | \(e\left(\frac{5}{204}\right)\) | \(e\left(\frac{59}{102}\right)\) | \(e\left(\frac{67}{68}\right)\) | \(e\left(\frac{101}{102}\right)\) | \(e\left(\frac{143}{204}\right)\) | \(e\left(\frac{19}{102}\right)\) |
| \(\chi_{338130}(13307,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{5}{204}\right)\) | \(e\left(\frac{8}{51}\right)\) | \(e\left(\frac{57}{68}\right)\) | \(e\left(\frac{11}{102}\right)\) | \(e\left(\frac{83}{204}\right)\) | \(e\left(\frac{41}{102}\right)\) | \(e\left(\frac{65}{68}\right)\) | \(e\left(\frac{65}{102}\right)\) | \(e\left(\frac{89}{204}\right)\) | \(e\left(\frac{91}{102}\right)\) |
| \(\chi_{338130}(17633,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{204}\right)\) | \(e\left(\frac{28}{51}\right)\) | \(e\left(\frac{55}{68}\right)\) | \(e\left(\frac{13}{102}\right)\) | \(e\left(\frac{61}{204}\right)\) | \(e\left(\frac{67}{102}\right)\) | \(e\left(\frac{15}{68}\right)\) | \(e\left(\frac{49}{102}\right)\) | \(e\left(\frac{31}{204}\right)\) | \(e\left(\frac{89}{102}\right)\) |
| \(\chi_{338130}(19937,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{145}{204}\right)\) | \(e\left(\frac{28}{51}\right)\) | \(e\left(\frac{21}{68}\right)\) | \(e\left(\frac{13}{102}\right)\) | \(e\left(\frac{163}{204}\right)\) | \(e\left(\frac{67}{102}\right)\) | \(e\left(\frac{49}{68}\right)\) | \(e\left(\frac{49}{102}\right)\) | \(e\left(\frac{133}{204}\right)\) | \(e\left(\frac{89}{102}\right)\) |
| \(\chi_{338130}(30893,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{35}{204}\right)\) | \(e\left(\frac{5}{51}\right)\) | \(e\left(\frac{59}{68}\right)\) | \(e\left(\frac{77}{102}\right)\) | \(e\left(\frac{173}{204}\right)\) | \(e\left(\frac{83}{102}\right)\) | \(e\left(\frac{47}{68}\right)\) | \(e\left(\frac{47}{102}\right)\) | \(e\left(\frac{11}{204}\right)\) | \(e\left(\frac{25}{102}\right)\) |
| \(\chi_{338130}(37523,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{204}\right)\) | \(e\left(\frac{19}{51}\right)\) | \(e\left(\frac{27}{68}\right)\) | \(e\left(\frac{7}{102}\right)\) | \(e\left(\frac{25}{204}\right)\) | \(e\left(\frac{91}{102}\right)\) | \(e\left(\frac{63}{68}\right)\) | \(e\left(\frac{97}{102}\right)\) | \(e\left(\frac{103}{204}\right)\) | \(e\left(\frac{95}{102}\right)\) |
| \(\chi_{338130}(39827,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{157}{204}\right)\) | \(e\left(\frac{37}{51}\right)\) | \(e\left(\frac{49}{68}\right)\) | \(e\left(\frac{19}{102}\right)\) | \(e\left(\frac{199}{204}\right)\) | \(e\left(\frac{43}{102}\right)\) | \(e\left(\frac{1}{68}\right)\) | \(e\left(\frac{1}{102}\right)\) | \(e\left(\frac{61}{204}\right)\) | \(e\left(\frac{83}{102}\right)\) |
| \(\chi_{338130}(50783,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{204}\right)\) | \(e\left(\frac{47}{51}\right)\) | \(e\left(\frac{31}{68}\right)\) | \(e\left(\frac{71}{102}\right)\) | \(e\left(\frac{137}{204}\right)\) | \(e\left(\frac{5}{102}\right)\) | \(e\left(\frac{27}{68}\right)\) | \(e\left(\frac{95}{102}\right)\) | \(e\left(\frac{83}{204}\right)\) | \(e\left(\frac{31}{102}\right)\) |
| \(\chi_{338130}(53087,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{204}\right)\) | \(e\left(\frac{26}{51}\right)\) | \(e\left(\frac{45}{68}\right)\) | \(e\left(\frac{23}{102}\right)\) | \(e\left(\frac{155}{204}\right)\) | \(e\left(\frac{95}{102}\right)\) | \(e\left(\frac{37}{68}\right)\) | \(e\left(\frac{71}{102}\right)\) | \(e\left(\frac{149}{204}\right)\) | \(e\left(\frac{79}{102}\right)\) |
| \(\chi_{338130}(57413,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{204}\right)\) | \(e\left(\frac{10}{51}\right)\) | \(e\left(\frac{67}{68}\right)\) | \(e\left(\frac{1}{102}\right)\) | \(e\left(\frac{193}{204}\right)\) | \(e\left(\frac{13}{102}\right)\) | \(e\left(\frac{43}{68}\right)\) | \(e\left(\frac{43}{102}\right)\) | \(e\left(\frac{175}{204}\right)\) | \(e\left(\frac{101}{102}\right)\) |
| \(\chi_{338130}(59717,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{169}{204}\right)\) | \(e\left(\frac{46}{51}\right)\) | \(e\left(\frac{9}{68}\right)\) | \(e\left(\frac{25}{102}\right)\) | \(e\left(\frac{31}{204}\right)\) | \(e\left(\frac{19}{102}\right)\) | \(e\left(\frac{21}{68}\right)\) | \(e\left(\frac{55}{102}\right)\) | \(e\left(\frac{193}{204}\right)\) | \(e\left(\frac{77}{102}\right)\) |
| \(\chi_{338130}(70673,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{204}\right)\) | \(e\left(\frac{38}{51}\right)\) | \(e\left(\frac{3}{68}\right)\) | \(e\left(\frac{65}{102}\right)\) | \(e\left(\frac{101}{204}\right)\) | \(e\left(\frac{29}{102}\right)\) | \(e\left(\frac{7}{68}\right)\) | \(e\left(\frac{41}{102}\right)\) | \(e\left(\frac{155}{204}\right)\) | \(e\left(\frac{37}{102}\right)\) |
| \(\chi_{338130}(72977,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{41}{204}\right)\) | \(e\left(\frac{35}{51}\right)\) | \(e\left(\frac{5}{68}\right)\) | \(e\left(\frac{29}{102}\right)\) | \(e\left(\frac{191}{204}\right)\) | \(e\left(\frac{71}{102}\right)\) | \(e\left(\frac{57}{68}\right)\) | \(e\left(\frac{23}{102}\right)\) | \(e\left(\frac{77}{204}\right)\) | \(e\left(\frac{73}{102}\right)\) |
| \(\chi_{338130}(77303,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{204}\right)\) | \(e\left(\frac{1}{51}\right)\) | \(e\left(\frac{39}{68}\right)\) | \(e\left(\frac{97}{102}\right)\) | \(e\left(\frac{157}{204}\right)\) | \(e\left(\frac{37}{102}\right)\) | \(e\left(\frac{23}{68}\right)\) | \(e\left(\frac{91}{102}\right)\) | \(e\left(\frac{43}{204}\right)\) | \(e\left(\frac{5}{102}\right)\) |
| \(\chi_{338130}(79607,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{181}{204}\right)\) | \(e\left(\frac{4}{51}\right)\) | \(e\left(\frac{37}{68}\right)\) | \(e\left(\frac{31}{102}\right)\) | \(e\left(\frac{67}{204}\right)\) | \(e\left(\frac{97}{102}\right)\) | \(e\left(\frac{41}{68}\right)\) | \(e\left(\frac{7}{102}\right)\) | \(e\left(\frac{121}{204}\right)\) | \(e\left(\frac{71}{102}\right)\) |
| \(\chi_{338130}(90563,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{203}{204}\right)\) | \(e\left(\frac{29}{51}\right)\) | \(e\left(\frac{43}{68}\right)\) | \(e\left(\frac{59}{102}\right)\) | \(e\left(\frac{65}{204}\right)\) | \(e\left(\frac{53}{102}\right)\) | \(e\left(\frac{55}{68}\right)\) | \(e\left(\frac{89}{102}\right)\) | \(e\left(\frac{23}{204}\right)\) | \(e\left(\frac{43}{102}\right)\) |
| \(\chi_{338130}(92867,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{53}{204}\right)\) | \(e\left(\frac{44}{51}\right)\) | \(e\left(\frac{33}{68}\right)\) | \(e\left(\frac{35}{102}\right)\) | \(e\left(\frac{23}{204}\right)\) | \(e\left(\frac{47}{102}\right)\) | \(e\left(\frac{9}{68}\right)\) | \(e\left(\frac{77}{102}\right)\) | \(e\left(\frac{5}{204}\right)\) | \(e\left(\frac{67}{102}\right)\) |
| \(\chi_{338130}(97193,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{199}{204}\right)\) | \(e\left(\frac{43}{51}\right)\) | \(e\left(\frac{11}{68}\right)\) | \(e\left(\frac{91}{102}\right)\) | \(e\left(\frac{121}{204}\right)\) | \(e\left(\frac{61}{102}\right)\) | \(e\left(\frac{3}{68}\right)\) | \(e\left(\frac{37}{102}\right)\) | \(e\left(\frac{115}{204}\right)\) | \(e\left(\frac{11}{102}\right)\) |
| \(\chi_{338130}(99497,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{193}{204}\right)\) | \(e\left(\frac{13}{51}\right)\) | \(e\left(\frac{65}{68}\right)\) | \(e\left(\frac{37}{102}\right)\) | \(e\left(\frac{103}{204}\right)\) | \(e\left(\frac{73}{102}\right)\) | \(e\left(\frac{61}{68}\right)\) | \(e\left(\frac{61}{102}\right)\) | \(e\left(\frac{49}{204}\right)\) | \(e\left(\frac{65}{102}\right)\) |
| \(\chi_{338130}(110453,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{191}{204}\right)\) | \(e\left(\frac{20}{51}\right)\) | \(e\left(\frac{15}{68}\right)\) | \(e\left(\frac{53}{102}\right)\) | \(e\left(\frac{29}{204}\right)\) | \(e\left(\frac{77}{102}\right)\) | \(e\left(\frac{35}{68}\right)\) | \(e\left(\frac{35}{102}\right)\) | \(e\left(\frac{95}{204}\right)\) | \(e\left(\frac{49}{102}\right)\) |
| \(\chi_{338130}(112757,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{65}{204}\right)\) | \(e\left(\frac{2}{51}\right)\) | \(e\left(\frac{61}{68}\right)\) | \(e\left(\frac{41}{102}\right)\) | \(e\left(\frac{59}{204}\right)\) | \(e\left(\frac{23}{102}\right)\) | \(e\left(\frac{29}{68}\right)\) | \(e\left(\frac{29}{102}\right)\) | \(e\left(\frac{137}{204}\right)\) | \(e\left(\frac{61}{102}\right)\) |
| \(\chi_{338130}(119387,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{204}\right)\) | \(e\left(\frac{22}{51}\right)\) | \(e\left(\frac{25}{68}\right)\) | \(e\left(\frac{43}{102}\right)\) | \(e\left(\frac{139}{204}\right)\) | \(e\left(\frac{49}{102}\right)\) | \(e\left(\frac{13}{68}\right)\) | \(e\left(\frac{13}{102}\right)\) | \(e\left(\frac{181}{204}\right)\) | \(e\left(\frac{59}{102}\right)\) |
| \(\chi_{338130}(130343,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{179}{204}\right)\) | \(e\left(\frac{11}{51}\right)\) | \(e\left(\frac{55}{68}\right)\) | \(e\left(\frac{47}{102}\right)\) | \(e\left(\frac{197}{204}\right)\) | \(e\left(\frac{101}{102}\right)\) | \(e\left(\frac{15}{68}\right)\) | \(e\left(\frac{83}{102}\right)\) | \(e\left(\frac{167}{204}\right)\) | \(e\left(\frac{55}{102}\right)\) |
| \(\chi_{338130}(132647,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{77}{204}\right)\) | \(e\left(\frac{11}{51}\right)\) | \(e\left(\frac{21}{68}\right)\) | \(e\left(\frac{47}{102}\right)\) | \(e\left(\frac{95}{204}\right)\) | \(e\left(\frac{101}{102}\right)\) | \(e\left(\frac{49}{68}\right)\) | \(e\left(\frac{83}{102}\right)\) | \(e\left(\frac{65}{204}\right)\) | \(e\left(\frac{55}{102}\right)\) |
| \(\chi_{338130}(136973,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{175}{204}\right)\) | \(e\left(\frac{25}{51}\right)\) | \(e\left(\frac{23}{68}\right)\) | \(e\left(\frac{79}{102}\right)\) | \(e\left(\frac{49}{204}\right)\) | \(e\left(\frac{7}{102}\right)\) | \(e\left(\frac{31}{68}\right)\) | \(e\left(\frac{31}{102}\right)\) | \(e\left(\frac{55}{204}\right)\) | \(e\left(\frac{23}{102}\right)\) |
| \(\chi_{338130}(139277,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{204}\right)\) | \(e\left(\frac{31}{51}\right)\) | \(e\left(\frac{53}{68}\right)\) | \(e\left(\frac{49}{102}\right)\) | \(e\left(\frac{175}{204}\right)\) | \(e\left(\frac{25}{102}\right)\) | \(e\left(\frac{33}{68}\right)\) | \(e\left(\frac{67}{102}\right)\) | \(e\left(\frac{109}{204}\right)\) | \(e\left(\frac{53}{102}\right)\) |
| \(\chi_{338130}(150233,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{167}{204}\right)\) | \(e\left(\frac{2}{51}\right)\) | \(e\left(\frac{27}{68}\right)\) | \(e\left(\frac{41}{102}\right)\) | \(e\left(\frac{161}{204}\right)\) | \(e\left(\frac{23}{102}\right)\) | \(e\left(\frac{63}{68}\right)\) | \(e\left(\frac{29}{102}\right)\) | \(e\left(\frac{35}{204}\right)\) | \(e\left(\frac{61}{102}\right)\) |
| \(\chi_{338130}(152537,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{89}{204}\right)\) | \(e\left(\frac{20}{51}\right)\) | \(e\left(\frac{49}{68}\right)\) | \(e\left(\frac{53}{102}\right)\) | \(e\left(\frac{131}{204}\right)\) | \(e\left(\frac{77}{102}\right)\) | \(e\left(\frac{1}{68}\right)\) | \(e\left(\frac{35}{102}\right)\) | \(e\left(\frac{197}{204}\right)\) | \(e\left(\frac{49}{102}\right)\) |
| \(\chi_{338130}(156863,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{163}{204}\right)\) | \(e\left(\frac{16}{51}\right)\) | \(e\left(\frac{63}{68}\right)\) | \(e\left(\frac{73}{102}\right)\) | \(e\left(\frac{13}{204}\right)\) | \(e\left(\frac{31}{102}\right)\) | \(e\left(\frac{11}{68}\right)\) | \(e\left(\frac{79}{102}\right)\) | \(e\left(\frac{127}{204}\right)\) | \(e\left(\frac{29}{102}\right)\) |
| \(\chi_{338130}(159167,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{25}{204}\right)\) | \(e\left(\frac{40}{51}\right)\) | \(e\left(\frac{13}{68}\right)\) | \(e\left(\frac{55}{102}\right)\) | \(e\left(\frac{7}{204}\right)\) | \(e\left(\frac{1}{102}\right)\) | \(e\left(\frac{53}{68}\right)\) | \(e\left(\frac{19}{102}\right)\) | \(e\left(\frac{37}{204}\right)\) | \(e\left(\frac{47}{102}\right)\) |