from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(338130, base_ring=CyclotomicField(408))
M = H._module
chi = DirichletCharacter(H, M([340,204,374,351]))
chi.galois_orbit()
[g,chi] = znchar(Mod(59,338130))
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.bqt | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | 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 | \(-1\) | \(1\) | \(7\) | \(11\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) | \(47\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{338130}(59,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{107}{408}\right)\) | \(e\left(\frac{5}{136}\right)\) | \(e\left(\frac{32}{51}\right)\) | \(e\left(\frac{277}{408}\right)\) | \(e\left(\frac{5}{136}\right)\) | \(e\left(\frac{269}{408}\right)\) | \(e\left(\frac{365}{408}\right)\) | \(e\left(\frac{79}{408}\right)\) | \(e\left(\frac{19}{204}\right)\) | \(e\left(\frac{23}{204}\right)\) |
| \(\chi_{338130}(5159,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{355}{408}\right)\) | \(e\left(\frac{117}{136}\right)\) | \(e\left(\frac{28}{51}\right)\) | \(e\left(\frac{389}{408}\right)\) | \(e\left(\frac{117}{136}\right)\) | \(e\left(\frac{229}{408}\right)\) | \(e\left(\frac{109}{408}\right)\) | \(e\left(\frac{407}{408}\right)\) | \(e\left(\frac{23}{204}\right)\) | \(e\left(\frac{103}{204}\right)\) |
| \(\chi_{338130}(5969,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{49}{408}\right)\) | \(e\left(\frac{15}{136}\right)\) | \(e\left(\frac{28}{51}\right)\) | \(e\left(\frac{287}{408}\right)\) | \(e\left(\frac{15}{136}\right)\) | \(e\left(\frac{127}{408}\right)\) | \(e\left(\frac{7}{408}\right)\) | \(e\left(\frac{101}{408}\right)\) | \(e\left(\frac{125}{204}\right)\) | \(e\left(\frac{1}{204}\right)\) |
| \(\chi_{338130}(5999,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{5}{408}\right)\) | \(e\left(\frac{107}{136}\right)\) | \(e\left(\frac{32}{51}\right)\) | \(e\left(\frac{379}{408}\right)\) | \(e\left(\frac{107}{136}\right)\) | \(e\left(\frac{371}{408}\right)\) | \(e\left(\frac{59}{408}\right)\) | \(e\left(\frac{385}{408}\right)\) | \(e\left(\frac{121}{204}\right)\) | \(e\left(\frac{125}{204}\right)\) |
| \(\chi_{338130}(12929,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{383}{408}\right)\) | \(e\left(\frac{9}{136}\right)\) | \(e\left(\frac{44}{51}\right)\) | \(e\left(\frac{145}{408}\right)\) | \(e\left(\frac{9}{136}\right)\) | \(e\left(\frac{185}{408}\right)\) | \(e\left(\frac{113}{408}\right)\) | \(e\left(\frac{115}{408}\right)\) | \(e\left(\frac{7}{204}\right)\) | \(e\left(\frac{191}{204}\right)\) |
| \(\chi_{338130}(14159,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{349}{408}\right)\) | \(e\left(\frac{43}{136}\right)\) | \(e\left(\frac{10}{51}\right)\) | \(e\left(\frac{179}{408}\right)\) | \(e\left(\frac{43}{136}\right)\) | \(e\left(\frac{355}{408}\right)\) | \(e\left(\frac{283}{408}\right)\) | \(e\left(\frac{353}{408}\right)\) | \(e\left(\frac{41}{204}\right)\) | \(e\left(\frac{157}{204}\right)\) |
| \(\chi_{338130}(17699,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{209}{408}\right)\) | \(e\left(\frac{39}{136}\right)\) | \(e\left(\frac{32}{51}\right)\) | \(e\left(\frac{175}{408}\right)\) | \(e\left(\frac{39}{136}\right)\) | \(e\left(\frac{167}{408}\right)\) | \(e\left(\frac{263}{408}\right)\) | \(e\left(\frac{181}{408}\right)\) | \(e\left(\frac{121}{204}\right)\) | \(e\left(\frac{125}{204}\right)\) |
| \(\chi_{338130}(18029,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{271}{408}\right)\) | \(e\left(\frac{33}{136}\right)\) | \(e\left(\frac{31}{51}\right)\) | \(e\left(\frac{305}{408}\right)\) | \(e\left(\frac{33}{136}\right)\) | \(e\left(\frac{361}{408}\right)\) | \(e\left(\frac{97}{408}\right)\) | \(e\left(\frac{59}{408}\right)\) | \(e\left(\frac{71}{204}\right)\) | \(e\left(\frac{43}{204}\right)\) |
| \(\chi_{338130}(19949,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{299}{408}\right)\) | \(e\left(\frac{61}{136}\right)\) | \(e\left(\frac{47}{51}\right)\) | \(e\left(\frac{61}{408}\right)\) | \(e\left(\frac{61}{136}\right)\) | \(e\left(\frac{317}{408}\right)\) | \(e\left(\frac{101}{408}\right)\) | \(e\left(\frac{175}{408}\right)\) | \(e\left(\frac{55}{204}\right)\) | \(e\left(\frac{131}{204}\right)\) |
| \(\chi_{338130}(25049,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{139}{408}\right)\) | \(e\left(\frac{37}{136}\right)\) | \(e\left(\frac{43}{51}\right)\) | \(e\left(\frac{173}{408}\right)\) | \(e\left(\frac{37}{136}\right)\) | \(e\left(\frac{277}{408}\right)\) | \(e\left(\frac{253}{408}\right)\) | \(e\left(\frac{95}{408}\right)\) | \(e\left(\frac{59}{204}\right)\) | \(e\left(\frac{7}{204}\right)\) |
| \(\chi_{338130}(25859,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{408}\right)\) | \(e\left(\frac{103}{136}\right)\) | \(e\left(\frac{37}{51}\right)\) | \(e\left(\frac{239}{408}\right)\) | \(e\left(\frac{103}{136}\right)\) | \(e\left(\frac{319}{408}\right)\) | \(e\left(\frac{175}{408}\right)\) | \(e\left(\frac{77}{408}\right)\) | \(e\left(\frac{65}{204}\right)\) | \(e\left(\frac{25}{204}\right)\) |
| \(\chi_{338130}(25889,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{53}{408}\right)\) | \(e\left(\frac{19}{136}\right)\) | \(e\left(\frac{23}{51}\right)\) | \(e\left(\frac{19}{408}\right)\) | \(e\left(\frac{19}{136}\right)\) | \(e\left(\frac{179}{408}\right)\) | \(e\left(\frac{299}{408}\right)\) | \(e\left(\frac{1}{408}\right)\) | \(e\left(\frac{181}{204}\right)\) | \(e\left(\frac{101}{204}\right)\) |
| \(\chi_{338130}(32819,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{191}{408}\right)\) | \(e\left(\frac{89}{136}\right)\) | \(e\left(\frac{29}{51}\right)\) | \(e\left(\frac{361}{408}\right)\) | \(e\left(\frac{89}{136}\right)\) | \(e\left(\frac{137}{408}\right)\) | \(e\left(\frac{377}{408}\right)\) | \(e\left(\frac{19}{408}\right)\) | \(e\left(\frac{175}{204}\right)\) | \(e\left(\frac{83}{204}\right)\) |
| \(\chi_{338130}(34049,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{397}{408}\right)\) | \(e\left(\frac{91}{136}\right)\) | \(e\left(\frac{1}{51}\right)\) | \(e\left(\frac{227}{408}\right)\) | \(e\left(\frac{91}{136}\right)\) | \(e\left(\frac{163}{408}\right)\) | \(e\left(\frac{115}{408}\right)\) | \(e\left(\frac{377}{408}\right)\) | \(e\left(\frac{101}{204}\right)\) | \(e\left(\frac{133}{204}\right)\) |
| \(\chi_{338130}(37589,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{161}{408}\right)\) | \(e\left(\frac{127}{136}\right)\) | \(e\left(\frac{41}{51}\right)\) | \(e\left(\frac{127}{408}\right)\) | \(e\left(\frac{127}{136}\right)\) | \(e\left(\frac{359}{408}\right)\) | \(e\left(\frac{23}{408}\right)\) | \(e\left(\frac{157}{408}\right)\) | \(e\left(\frac{61}{204}\right)\) | \(e\left(\frac{149}{204}\right)\) |
| \(\chi_{338130}(37919,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{79}{408}\right)\) | \(e\left(\frac{113}{136}\right)\) | \(e\left(\frac{16}{51}\right)\) | \(e\left(\frac{113}{408}\right)\) | \(e\left(\frac{113}{136}\right)\) | \(e\left(\frac{313}{408}\right)\) | \(e\left(\frac{361}{408}\right)\) | \(e\left(\frac{371}{408}\right)\) | \(e\left(\frac{35}{204}\right)\) | \(e\left(\frac{139}{204}\right)\) |
| \(\chi_{338130}(39839,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{83}{408}\right)\) | \(e\left(\frac{117}{136}\right)\) | \(e\left(\frac{11}{51}\right)\) | \(e\left(\frac{253}{408}\right)\) | \(e\left(\frac{117}{136}\right)\) | \(e\left(\frac{365}{408}\right)\) | \(e\left(\frac{245}{408}\right)\) | \(e\left(\frac{271}{408}\right)\) | \(e\left(\frac{91}{204}\right)\) | \(e\left(\frac{35}{204}\right)\) |
| \(\chi_{338130}(44939,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{331}{408}\right)\) | \(e\left(\frac{93}{136}\right)\) | \(e\left(\frac{7}{51}\right)\) | \(e\left(\frac{365}{408}\right)\) | \(e\left(\frac{93}{136}\right)\) | \(e\left(\frac{325}{408}\right)\) | \(e\left(\frac{397}{408}\right)\) | \(e\left(\frac{191}{408}\right)\) | \(e\left(\frac{95}{204}\right)\) | \(e\left(\frac{115}{204}\right)\) |
| \(\chi_{338130}(45749,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{361}{408}\right)\) | \(e\left(\frac{55}{136}\right)\) | \(e\left(\frac{46}{51}\right)\) | \(e\left(\frac{191}{408}\right)\) | \(e\left(\frac{55}{136}\right)\) | \(e\left(\frac{103}{408}\right)\) | \(e\left(\frac{343}{408}\right)\) | \(e\left(\frac{53}{408}\right)\) | \(e\left(\frac{5}{204}\right)\) | \(e\left(\frac{49}{204}\right)\) |
| \(\chi_{338130}(45779,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{101}{408}\right)\) | \(e\left(\frac{67}{136}\right)\) | \(e\left(\frac{14}{51}\right)\) | \(e\left(\frac{67}{408}\right)\) | \(e\left(\frac{67}{136}\right)\) | \(e\left(\frac{395}{408}\right)\) | \(e\left(\frac{131}{408}\right)\) | \(e\left(\frac{25}{408}\right)\) | \(e\left(\frac{37}{204}\right)\) | \(e\left(\frac{77}{204}\right)\) |
| \(\chi_{338130}(52709,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{407}{408}\right)\) | \(e\left(\frac{33}{136}\right)\) | \(e\left(\frac{14}{51}\right)\) | \(e\left(\frac{169}{408}\right)\) | \(e\left(\frac{33}{136}\right)\) | \(e\left(\frac{89}{408}\right)\) | \(e\left(\frac{233}{408}\right)\) | \(e\left(\frac{331}{408}\right)\) | \(e\left(\frac{139}{204}\right)\) | \(e\left(\frac{179}{204}\right)\) |
| \(\chi_{338130}(53939,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{408}\right)\) | \(e\left(\frac{3}{136}\right)\) | \(e\left(\frac{43}{51}\right)\) | \(e\left(\frac{275}{408}\right)\) | \(e\left(\frac{3}{136}\right)\) | \(e\left(\frac{379}{408}\right)\) | \(e\left(\frac{355}{408}\right)\) | \(e\left(\frac{401}{408}\right)\) | \(e\left(\frac{161}{204}\right)\) | \(e\left(\frac{109}{204}\right)\) |
| \(\chi_{338130}(57479,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{113}{408}\right)\) | \(e\left(\frac{79}{136}\right)\) | \(e\left(\frac{50}{51}\right)\) | \(e\left(\frac{79}{408}\right)\) | \(e\left(\frac{79}{136}\right)\) | \(e\left(\frac{143}{408}\right)\) | \(e\left(\frac{191}{408}\right)\) | \(e\left(\frac{133}{408}\right)\) | \(e\left(\frac{1}{204}\right)\) | \(e\left(\frac{173}{204}\right)\) |
| \(\chi_{338130}(57809,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{295}{408}\right)\) | \(e\left(\frac{57}{136}\right)\) | \(e\left(\frac{1}{51}\right)\) | \(e\left(\frac{329}{408}\right)\) | \(e\left(\frac{57}{136}\right)\) | \(e\left(\frac{265}{408}\right)\) | \(e\left(\frac{217}{408}\right)\) | \(e\left(\frac{275}{408}\right)\) | \(e\left(\frac{203}{204}\right)\) | \(e\left(\frac{31}{204}\right)\) |
| \(\chi_{338130}(59729,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{275}{408}\right)\) | \(e\left(\frac{37}{136}\right)\) | \(e\left(\frac{26}{51}\right)\) | \(e\left(\frac{37}{408}\right)\) | \(e\left(\frac{37}{136}\right)\) | \(e\left(\frac{5}{408}\right)\) | \(e\left(\frac{389}{408}\right)\) | \(e\left(\frac{367}{408}\right)\) | \(e\left(\frac{127}{204}\right)\) | \(e\left(\frac{143}{204}\right)\) |
| \(\chi_{338130}(64829,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{115}{408}\right)\) | \(e\left(\frac{13}{136}\right)\) | \(e\left(\frac{22}{51}\right)\) | \(e\left(\frac{149}{408}\right)\) | \(e\left(\frac{13}{136}\right)\) | \(e\left(\frac{373}{408}\right)\) | \(e\left(\frac{133}{408}\right)\) | \(e\left(\frac{287}{408}\right)\) | \(e\left(\frac{131}{204}\right)\) | \(e\left(\frac{19}{204}\right)\) |
| \(\chi_{338130}(65639,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{313}{408}\right)\) | \(e\left(\frac{7}{136}\right)\) | \(e\left(\frac{4}{51}\right)\) | \(e\left(\frac{143}{408}\right)\) | \(e\left(\frac{7}{136}\right)\) | \(e\left(\frac{295}{408}\right)\) | \(e\left(\frac{103}{408}\right)\) | \(e\left(\frac{29}{408}\right)\) | \(e\left(\frac{149}{204}\right)\) | \(e\left(\frac{73}{204}\right)\) |
| \(\chi_{338130}(65669,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{149}{408}\right)\) | \(e\left(\frac{115}{136}\right)\) | \(e\left(\frac{5}{51}\right)\) | \(e\left(\frac{115}{408}\right)\) | \(e\left(\frac{115}{136}\right)\) | \(e\left(\frac{203}{408}\right)\) | \(e\left(\frac{371}{408}\right)\) | \(e\left(\frac{49}{408}\right)\) | \(e\left(\frac{97}{204}\right)\) | \(e\left(\frac{53}{204}\right)\) |
| \(\chi_{338130}(72599,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{215}{408}\right)\) | \(e\left(\frac{113}{136}\right)\) | \(e\left(\frac{50}{51}\right)\) | \(e\left(\frac{385}{408}\right)\) | \(e\left(\frac{113}{136}\right)\) | \(e\left(\frac{41}{408}\right)\) | \(e\left(\frac{89}{408}\right)\) | \(e\left(\frac{235}{408}\right)\) | \(e\left(\frac{103}{204}\right)\) | \(e\left(\frac{71}{204}\right)\) |
| \(\chi_{338130}(77369,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{65}{408}\right)\) | \(e\left(\frac{31}{136}\right)\) | \(e\left(\frac{8}{51}\right)\) | \(e\left(\frac{31}{408}\right)\) | \(e\left(\frac{31}{136}\right)\) | \(e\left(\frac{335}{408}\right)\) | \(e\left(\frac{359}{408}\right)\) | \(e\left(\frac{109}{408}\right)\) | \(e\left(\frac{145}{204}\right)\) | \(e\left(\frac{197}{204}\right)\) |
| \(\chi_{338130}(77699,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{103}{408}\right)\) | \(e\left(\frac{1}{136}\right)\) | \(e\left(\frac{37}{51}\right)\) | \(e\left(\frac{137}{408}\right)\) | \(e\left(\frac{1}{136}\right)\) | \(e\left(\frac{217}{408}\right)\) | \(e\left(\frac{73}{408}\right)\) | \(e\left(\frac{179}{408}\right)\) | \(e\left(\frac{167}{204}\right)\) | \(e\left(\frac{127}{204}\right)\) |