from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(338130, base_ring=CyclotomicField(408))
M = H._module
chi = DirichletCharacter(H, M([68,306,306,177]))
chi.galois_orbit()
[g,chi] = znchar(Mod(83,338130))
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: | \(408\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 169065.btl | 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 | \(-1\) | \(1\) | \(7\) | \(11\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) | \(47\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{338130}(83,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{371}{408}\right)\) | \(e\left(\frac{161}{408}\right)\) | \(e\left(\frac{11}{34}\right)\) | \(e\left(\frac{133}{408}\right)\) | \(e\left(\frac{365}{408}\right)\) | \(e\left(\frac{403}{408}\right)\) | \(e\left(\frac{131}{136}\right)\) | \(e\left(\frac{313}{408}\right)\) | \(e\left(\frac{47}{102}\right)\) | \(e\left(\frac{28}{51}\right)\) |
| \(\chi_{338130}(1607,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{361}{408}\right)\) | \(e\left(\frac{403}{408}\right)\) | \(e\left(\frac{25}{34}\right)\) | \(e\left(\frac{191}{408}\right)\) | \(e\left(\frac{199}{408}\right)\) | \(e\left(\frac{137}{408}\right)\) | \(e\left(\frac{1}{136}\right)\) | \(e\left(\frac{155}{408}\right)\) | \(e\left(\frac{79}{102}\right)\) | \(e\left(\frac{8}{51}\right)\) |
| \(\chi_{338130}(5153,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{199}{408}\right)\) | \(e\left(\frac{325}{408}\right)\) | \(e\left(\frac{7}{34}\right)\) | \(e\left(\frac{233}{408}\right)\) | \(e\left(\frac{121}{408}\right)\) | \(e\left(\frac{71}{408}\right)\) | \(e\left(\frac{71}{136}\right)\) | \(e\left(\frac{125}{408}\right)\) | \(e\left(\frac{67}{102}\right)\) | \(e\left(\frac{41}{51}\right)\) |
| \(\chi_{338130}(8237,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{161}{408}\right)\) | \(e\left(\frac{347}{408}\right)\) | \(e\left(\frac{33}{34}\right)\) | \(e\left(\frac{127}{408}\right)\) | \(e\left(\frac{143}{408}\right)\) | \(e\left(\frac{121}{408}\right)\) | \(e\left(\frac{121}{136}\right)\) | \(e\left(\frac{259}{408}\right)\) | \(e\left(\frac{5}{102}\right)\) | \(e\left(\frac{16}{51}\right)\) |
| \(\chi_{338130}(8627,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{253}{408}\right)\) | \(e\left(\frac{79}{408}\right)\) | \(e\left(\frac{13}{34}\right)\) | \(e\left(\frac{83}{408}\right)\) | \(e\left(\frac{283}{408}\right)\) | \(e\left(\frac{365}{408}\right)\) | \(e\left(\frac{93}{136}\right)\) | \(e\left(\frac{407}{408}\right)\) | \(e\left(\frac{37}{102}\right)\) | \(e\left(\frac{47}{51}\right)\) |
| \(\chi_{338130}(11783,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{47}{408}\right)\) | \(e\left(\frac{5}{408}\right)\) | \(e\left(\frac{9}{34}\right)\) | \(e\left(\frac{217}{408}\right)\) | \(e\left(\frac{209}{408}\right)\) | \(e\left(\frac{271}{408}\right)\) | \(e\left(\frac{135}{136}\right)\) | \(e\left(\frac{253}{408}\right)\) | \(e\left(\frac{23}{102}\right)\) | \(e\left(\frac{43}{51}\right)\) |
| \(\chi_{338130}(13343,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{403}{408}\right)\) | \(e\left(\frac{121}{408}\right)\) | \(e\left(\frac{7}{34}\right)\) | \(e\left(\frac{29}{408}\right)\) | \(e\left(\frac{325}{408}\right)\) | \(e\left(\frac{275}{408}\right)\) | \(e\left(\frac{3}{136}\right)\) | \(e\left(\frac{329}{408}\right)\) | \(e\left(\frac{67}{102}\right)\) | \(e\left(\frac{41}{51}\right)\) |
| \(\chi_{338130}(15257,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{317}{408}\right)\) | \(e\left(\frac{407}{408}\right)\) | \(e\left(\frac{5}{34}\right)\) | \(e\left(\frac{283}{408}\right)\) | \(e\left(\frac{203}{408}\right)\) | \(e\left(\frac{109}{408}\right)\) | \(e\left(\frac{109}{136}\right)\) | \(e\left(\frac{31}{408}\right)\) | \(e\left(\frac{77}{102}\right)\) | \(e\left(\frac{22}{51}\right)\) |
| \(\chi_{338130}(19973,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{408}\right)\) | \(e\left(\frac{305}{408}\right)\) | \(e\left(\frac{5}{34}\right)\) | \(e\left(\frac{181}{408}\right)\) | \(e\left(\frac{101}{408}\right)\) | \(e\left(\frac{211}{408}\right)\) | \(e\left(\frac{75}{136}\right)\) | \(e\left(\frac{337}{408}\right)\) | \(e\left(\frac{77}{102}\right)\) | \(e\left(\frac{22}{51}\right)\) |
| \(\chi_{338130}(21497,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{169}{408}\right)\) | \(e\left(\frac{235}{408}\right)\) | \(e\left(\frac{15}{34}\right)\) | \(e\left(\frac{407}{408}\right)\) | \(e\left(\frac{31}{408}\right)\) | \(e\left(\frac{89}{408}\right)\) | \(e\left(\frac{89}{136}\right)\) | \(e\left(\frac{59}{408}\right)\) | \(e\left(\frac{61}{102}\right)\) | \(e\left(\frac{32}{51}\right)\) |
| \(\chi_{338130}(25043,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{151}{408}\right)\) | \(e\left(\frac{181}{408}\right)\) | \(e\left(\frac{13}{34}\right)\) | \(e\left(\frac{185}{408}\right)\) | \(e\left(\frac{385}{408}\right)\) | \(e\left(\frac{263}{408}\right)\) | \(e\left(\frac{127}{136}\right)\) | \(e\left(\frac{101}{408}\right)\) | \(e\left(\frac{37}{102}\right)\) | \(e\left(\frac{47}{51}\right)\) |
| \(\chi_{338130}(28127,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{377}{408}\right)\) | \(e\left(\frac{179}{408}\right)\) | \(e\left(\frac{23}{34}\right)\) | \(e\left(\frac{343}{408}\right)\) | \(e\left(\frac{383}{408}\right)\) | \(e\left(\frac{73}{408}\right)\) | \(e\left(\frac{73}{136}\right)\) | \(e\left(\frac{163}{408}\right)\) | \(e\left(\frac{89}{102}\right)\) | \(e\left(\frac{40}{51}\right)\) |
| \(\chi_{338130}(28517,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{37}{408}\right)\) | \(e\left(\frac{247}{408}\right)\) | \(e\left(\frac{23}{34}\right)\) | \(e\left(\frac{275}{408}\right)\) | \(e\left(\frac{43}{408}\right)\) | \(e\left(\frac{5}{408}\right)\) | \(e\left(\frac{5}{136}\right)\) | \(e\left(\frac{95}{408}\right)\) | \(e\left(\frac{55}{102}\right)\) | \(e\left(\frac{23}{51}\right)\) |
| \(\chi_{338130}(31673,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{407}{408}\right)\) | \(e\left(\frac{269}{408}\right)\) | \(e\left(\frac{15}{34}\right)\) | \(e\left(\frac{169}{408}\right)\) | \(e\left(\frac{65}{408}\right)\) | \(e\left(\frac{55}{408}\right)\) | \(e\left(\frac{55}{136}\right)\) | \(e\left(\frac{229}{408}\right)\) | \(e\left(\frac{95}{102}\right)\) | \(e\left(\frac{49}{51}\right)\) |
| \(\chi_{338130}(33233,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{408}\right)\) | \(e\left(\frac{265}{408}\right)\) | \(e\left(\frac{1}{34}\right)\) | \(e\left(\frac{77}{408}\right)\) | \(e\left(\frac{61}{408}\right)\) | \(e\left(\frac{83}{408}\right)\) | \(e\left(\frac{83}{136}\right)\) | \(e\left(\frac{353}{408}\right)\) | \(e\left(\frac{97}{102}\right)\) | \(e\left(\frac{35}{51}\right)\) |
| \(\chi_{338130}(35147,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{101}{408}\right)\) | \(e\left(\frac{167}{408}\right)\) | \(e\left(\frac{15}{34}\right)\) | \(e\left(\frac{67}{408}\right)\) | \(e\left(\frac{371}{408}\right)\) | \(e\left(\frac{157}{408}\right)\) | \(e\left(\frac{21}{136}\right)\) | \(e\left(\frac{127}{408}\right)\) | \(e\left(\frac{95}{102}\right)\) | \(e\left(\frac{49}{51}\right)\) |
| \(\chi_{338130}(39863,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{408}\right)\) | \(e\left(\frac{41}{408}\right)\) | \(e\left(\frac{33}{34}\right)\) | \(e\left(\frac{229}{408}\right)\) | \(e\left(\frac{245}{408}\right)\) | \(e\left(\frac{19}{408}\right)\) | \(e\left(\frac{19}{136}\right)\) | \(e\left(\frac{361}{408}\right)\) | \(e\left(\frac{5}{102}\right)\) | \(e\left(\frac{16}{51}\right)\) |
| \(\chi_{338130}(41387,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{385}{408}\right)\) | \(e\left(\frac{67}{408}\right)\) | \(e\left(\frac{5}{34}\right)\) | \(e\left(\frac{215}{408}\right)\) | \(e\left(\frac{271}{408}\right)\) | \(e\left(\frac{41}{408}\right)\) | \(e\left(\frac{41}{136}\right)\) | \(e\left(\frac{371}{408}\right)\) | \(e\left(\frac{43}{102}\right)\) | \(e\left(\frac{5}{51}\right)\) |
| \(\chi_{338130}(44933,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{103}{408}\right)\) | \(e\left(\frac{37}{408}\right)\) | \(e\left(\frac{19}{34}\right)\) | \(e\left(\frac{137}{408}\right)\) | \(e\left(\frac{241}{408}\right)\) | \(e\left(\frac{47}{408}\right)\) | \(e\left(\frac{47}{136}\right)\) | \(e\left(\frac{77}{408}\right)\) | \(e\left(\frac{7}{102}\right)\) | \(e\left(\frac{2}{51}\right)\) |
| \(\chi_{338130}(48017,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{185}{408}\right)\) | \(e\left(\frac{11}{408}\right)\) | \(e\left(\frac{13}{34}\right)\) | \(e\left(\frac{151}{408}\right)\) | \(e\left(\frac{215}{408}\right)\) | \(e\left(\frac{25}{408}\right)\) | \(e\left(\frac{25}{136}\right)\) | \(e\left(\frac{67}{408}\right)\) | \(e\left(\frac{71}{102}\right)\) | \(e\left(\frac{13}{51}\right)\) |
| \(\chi_{338130}(48407,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{229}{408}\right)\) | \(e\left(\frac{7}{408}\right)\) | \(e\left(\frac{33}{34}\right)\) | \(e\left(\frac{59}{408}\right)\) | \(e\left(\frac{211}{408}\right)\) | \(e\left(\frac{53}{408}\right)\) | \(e\left(\frac{53}{136}\right)\) | \(e\left(\frac{191}{408}\right)\) | \(e\left(\frac{73}{102}\right)\) | \(e\left(\frac{50}{51}\right)\) |
| \(\chi_{338130}(51563,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{359}{408}\right)\) | \(e\left(\frac{125}{408}\right)\) | \(e\left(\frac{21}{34}\right)\) | \(e\left(\frac{121}{408}\right)\) | \(e\left(\frac{329}{408}\right)\) | \(e\left(\frac{247}{408}\right)\) | \(e\left(\frac{111}{136}\right)\) | \(e\left(\frac{205}{408}\right)\) | \(e\left(\frac{65}{102}\right)\) | \(e\left(\frac{4}{51}\right)\) |
| \(\chi_{338130}(53123,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{91}{408}\right)\) | \(e\left(\frac{1}{408}\right)\) | \(e\left(\frac{29}{34}\right)\) | \(e\left(\frac{125}{408}\right)\) | \(e\left(\frac{205}{408}\right)\) | \(e\left(\frac{299}{408}\right)\) | \(e\left(\frac{27}{136}\right)\) | \(e\left(\frac{377}{408}\right)\) | \(e\left(\frac{25}{102}\right)\) | \(e\left(\frac{29}{51}\right)\) |
| \(\chi_{338130}(55037,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{293}{408}\right)\) | \(e\left(\frac{335}{408}\right)\) | \(e\left(\frac{25}{34}\right)\) | \(e\left(\frac{259}{408}\right)\) | \(e\left(\frac{131}{408}\right)\) | \(e\left(\frac{205}{408}\right)\) | \(e\left(\frac{69}{136}\right)\) | \(e\left(\frac{223}{408}\right)\) | \(e\left(\frac{11}{102}\right)\) | \(e\left(\frac{25}{51}\right)\) |
| \(\chi_{338130}(59753,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{107}{408}\right)\) | \(e\left(\frac{185}{408}\right)\) | \(e\left(\frac{27}{34}\right)\) | \(e\left(\frac{277}{408}\right)\) | \(e\left(\frac{389}{408}\right)\) | \(e\left(\frac{235}{408}\right)\) | \(e\left(\frac{99}{136}\right)\) | \(e\left(\frac{385}{408}\right)\) | \(e\left(\frac{35}{102}\right)\) | \(e\left(\frac{10}{51}\right)\) |
| \(\chi_{338130}(61277,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{193}{408}\right)\) | \(e\left(\frac{307}{408}\right)\) | \(e\left(\frac{29}{34}\right)\) | \(e\left(\frac{23}{408}\right)\) | \(e\left(\frac{103}{408}\right)\) | \(e\left(\frac{401}{408}\right)\) | \(e\left(\frac{129}{136}\right)\) | \(e\left(\frac{275}{408}\right)\) | \(e\left(\frac{25}{102}\right)\) | \(e\left(\frac{29}{51}\right)\) |
| \(\chi_{338130}(64823,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{55}{408}\right)\) | \(e\left(\frac{301}{408}\right)\) | \(e\left(\frac{25}{34}\right)\) | \(e\left(\frac{89}{408}\right)\) | \(e\left(\frac{97}{408}\right)\) | \(e\left(\frac{239}{408}\right)\) | \(e\left(\frac{103}{136}\right)\) | \(e\left(\frac{53}{408}\right)\) | \(e\left(\frac{79}{102}\right)\) | \(e\left(\frac{8}{51}\right)\) |
| \(\chi_{338130}(67907,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{401}{408}\right)\) | \(e\left(\frac{251}{408}\right)\) | \(e\left(\frac{3}{34}\right)\) | \(e\left(\frac{367}{408}\right)\) | \(e\left(\frac{47}{408}\right)\) | \(e\left(\frac{385}{408}\right)\) | \(e\left(\frac{113}{136}\right)\) | \(e\left(\frac{379}{408}\right)\) | \(e\left(\frac{53}{102}\right)\) | \(e\left(\frac{37}{51}\right)\) |
| \(\chi_{338130}(68297,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{408}\right)\) | \(e\left(\frac{175}{408}\right)\) | \(e\left(\frac{9}{34}\right)\) | \(e\left(\frac{251}{408}\right)\) | \(e\left(\frac{379}{408}\right)\) | \(e\left(\frac{101}{408}\right)\) | \(e\left(\frac{101}{136}\right)\) | \(e\left(\frac{287}{408}\right)\) | \(e\left(\frac{91}{102}\right)\) | \(e\left(\frac{26}{51}\right)\) |
| \(\chi_{338130}(71453,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{311}{408}\right)\) | \(e\left(\frac{389}{408}\right)\) | \(e\left(\frac{27}{34}\right)\) | \(e\left(\frac{73}{408}\right)\) | \(e\left(\frac{185}{408}\right)\) | \(e\left(\frac{31}{408}\right)\) | \(e\left(\frac{31}{136}\right)\) | \(e\left(\frac{181}{408}\right)\) | \(e\left(\frac{35}{102}\right)\) | \(e\left(\frac{10}{51}\right)\) |
| \(\chi_{338130}(73013,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{139}{408}\right)\) | \(e\left(\frac{145}{408}\right)\) | \(e\left(\frac{23}{34}\right)\) | \(e\left(\frac{173}{408}\right)\) | \(e\left(\frac{349}{408}\right)\) | \(e\left(\frac{107}{408}\right)\) | \(e\left(\frac{107}{136}\right)\) | \(e\left(\frac{401}{408}\right)\) | \(e\left(\frac{55}{102}\right)\) | \(e\left(\frac{23}{51}\right)\) |