from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6336, base_ring=CyclotomicField(240))
M = H._module
chi = DirichletCharacter(H, M([0,165,40,168]))
chi.galois_orbit()
[g,chi] = znchar(Mod(29,6336))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(6336\) | |
| Conductor: | \(6336\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(240\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | 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: | $\Q(\zeta_{240})$ |
| Fixed field: | Number field defined by a degree 240 polynomial (not computed) |
First 31 of 64 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(5\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{6336}(29,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{77}{240}\right)\) | \(e\left(\frac{53}{120}\right)\) | \(e\left(\frac{83}{240}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{73}{80}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{77}{120}\right)\) | \(e\left(\frac{151}{240}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{61}{80}\right)\) |
| \(\chi_{6336}(101,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{191}{240}\right)\) | \(e\left(\frac{119}{120}\right)\) | \(e\left(\frac{209}{240}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{19}{80}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{71}{120}\right)\) | \(e\left(\frac{13}{240}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{63}{80}\right)\) |
| \(\chi_{6336}(149,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{139}{240}\right)\) | \(e\left(\frac{91}{120}\right)\) | \(e\left(\frac{181}{240}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{31}{80}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{19}{120}\right)\) | \(e\left(\frac{17}{240}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{27}{80}\right)\) |
| \(\chi_{6336}(173,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{113}{240}\right)\) | \(e\left(\frac{17}{120}\right)\) | \(e\left(\frac{47}{240}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{77}{80}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{113}{120}\right)\) | \(e\left(\frac{19}{240}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{49}{80}\right)\) |
| \(\chi_{6336}(293,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{127}{240}\right)\) | \(e\left(\frac{103}{120}\right)\) | \(e\left(\frac{193}{240}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{3}{80}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{7}{120}\right)\) | \(e\left(\frac{221}{240}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{31}{80}\right)\) |
| \(\chi_{6336}(365,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{240}\right)\) | \(e\left(\frac{49}{120}\right)\) | \(e\left(\frac{79}{240}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{29}{80}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{1}{120}\right)\) | \(e\left(\frac{83}{240}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{33}{80}\right)\) |
| \(\chi_{6336}(437,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{163}{240}\right)\) | \(e\left(\frac{67}{120}\right)\) | \(e\left(\frac{157}{240}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{7}{80}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{43}{120}\right)\) | \(e\left(\frac{89}{240}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{19}{80}\right)\) |
| \(\chi_{6336}(677,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{239}{240}\right)\) | \(e\left(\frac{71}{120}\right)\) | \(e\left(\frac{161}{240}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{51}{80}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{119}{120}\right)\) | \(e\left(\frac{157}{240}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{47}{80}\right)\) |
| \(\chi_{6336}(821,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{227}{240}\right)\) | \(e\left(\frac{83}{120}\right)\) | \(e\left(\frac{173}{240}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{23}{80}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{107}{120}\right)\) | \(e\left(\frac{121}{240}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{51}{80}\right)\) |
| \(\chi_{6336}(893,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{101}{240}\right)\) | \(e\left(\frac{29}{120}\right)\) | \(e\left(\frac{59}{240}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{49}{80}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{101}{120}\right)\) | \(e\left(\frac{223}{240}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{53}{80}\right)\) |
| \(\chi_{6336}(941,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{49}{240}\right)\) | \(e\left(\frac{1}{120}\right)\) | \(e\left(\frac{31}{240}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{61}{80}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{49}{120}\right)\) | \(e\left(\frac{227}{240}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{17}{80}\right)\) |
| \(\chi_{6336}(965,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{240}\right)\) | \(e\left(\frac{47}{120}\right)\) | \(e\left(\frac{137}{240}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{27}{80}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{23}{120}\right)\) | \(e\left(\frac{229}{240}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{39}{80}\right)\) |
| \(\chi_{6336}(1085,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{240}\right)\) | \(e\left(\frac{13}{120}\right)\) | \(e\left(\frac{43}{240}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{33}{80}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{37}{120}\right)\) | \(e\left(\frac{191}{240}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{21}{80}\right)\) |
| \(\chi_{6336}(1157,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{151}{240}\right)\) | \(e\left(\frac{79}{120}\right)\) | \(e\left(\frac{169}{240}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{59}{80}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{31}{120}\right)\) | \(e\left(\frac{53}{240}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{23}{80}\right)\) |
| \(\chi_{6336}(1229,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{73}{240}\right)\) | \(e\left(\frac{97}{120}\right)\) | \(e\left(\frac{7}{240}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{37}{80}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{73}{120}\right)\) | \(e\left(\frac{59}{240}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{9}{80}\right)\) |
| \(\chi_{6336}(1469,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{149}{240}\right)\) | \(e\left(\frac{101}{120}\right)\) | \(e\left(\frac{11}{240}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{1}{80}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{29}{120}\right)\) | \(e\left(\frac{127}{240}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{37}{80}\right)\) |
| \(\chi_{6336}(1613,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{137}{240}\right)\) | \(e\left(\frac{113}{120}\right)\) | \(e\left(\frac{23}{240}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{53}{80}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{17}{120}\right)\) | \(e\left(\frac{91}{240}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{41}{80}\right)\) |
| \(\chi_{6336}(1685,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{240}\right)\) | \(e\left(\frac{59}{120}\right)\) | \(e\left(\frac{149}{240}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{79}{80}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{11}{120}\right)\) | \(e\left(\frac{193}{240}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{43}{80}\right)\) |
| \(\chi_{6336}(1733,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{199}{240}\right)\) | \(e\left(\frac{31}{120}\right)\) | \(e\left(\frac{121}{240}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{11}{80}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{79}{120}\right)\) | \(e\left(\frac{197}{240}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{7}{80}\right)\) |
| \(\chi_{6336}(1757,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{173}{240}\right)\) | \(e\left(\frac{77}{120}\right)\) | \(e\left(\frac{227}{240}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{57}{80}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{53}{120}\right)\) | \(e\left(\frac{199}{240}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{29}{80}\right)\) |
| \(\chi_{6336}(1877,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{187}{240}\right)\) | \(e\left(\frac{43}{120}\right)\) | \(e\left(\frac{133}{240}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{63}{80}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{67}{120}\right)\) | \(e\left(\frac{161}{240}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{11}{80}\right)\) |
| \(\chi_{6336}(1949,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{61}{240}\right)\) | \(e\left(\frac{109}{120}\right)\) | \(e\left(\frac{19}{240}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{9}{80}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{61}{120}\right)\) | \(e\left(\frac{23}{240}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{13}{80}\right)\) |
| \(\chi_{6336}(2021,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{223}{240}\right)\) | \(e\left(\frac{7}{120}\right)\) | \(e\left(\frac{97}{240}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{67}{80}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{103}{120}\right)\) | \(e\left(\frac{29}{240}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{79}{80}\right)\) |
| \(\chi_{6336}(2261,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{59}{240}\right)\) | \(e\left(\frac{11}{120}\right)\) | \(e\left(\frac{101}{240}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{31}{80}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{59}{120}\right)\) | \(e\left(\frac{97}{240}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{27}{80}\right)\) |
| \(\chi_{6336}(2405,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{47}{240}\right)\) | \(e\left(\frac{23}{120}\right)\) | \(e\left(\frac{113}{240}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{3}{80}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{47}{120}\right)\) | \(e\left(\frac{61}{240}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{31}{80}\right)\) |
| \(\chi_{6336}(2477,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{161}{240}\right)\) | \(e\left(\frac{89}{120}\right)\) | \(e\left(\frac{239}{240}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{29}{80}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{41}{120}\right)\) | \(e\left(\frac{163}{240}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{33}{80}\right)\) |
| \(\chi_{6336}(2525,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{109}{240}\right)\) | \(e\left(\frac{61}{120}\right)\) | \(e\left(\frac{211}{240}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{41}{80}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{109}{120}\right)\) | \(e\left(\frac{167}{240}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{77}{80}\right)\) |
| \(\chi_{6336}(2549,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{83}{240}\right)\) | \(e\left(\frac{107}{120}\right)\) | \(e\left(\frac{77}{240}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{7}{80}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{83}{120}\right)\) | \(e\left(\frac{169}{240}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{19}{80}\right)\) |
| \(\chi_{6336}(2669,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{97}{240}\right)\) | \(e\left(\frac{73}{120}\right)\) | \(e\left(\frac{223}{240}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{13}{80}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{97}{120}\right)\) | \(e\left(\frac{131}{240}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{1}{80}\right)\) |
| \(\chi_{6336}(2741,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{211}{240}\right)\) | \(e\left(\frac{19}{120}\right)\) | \(e\left(\frac{109}{240}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{39}{80}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{91}{120}\right)\) | \(e\left(\frac{233}{240}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{3}{80}\right)\) |
| \(\chi_{6336}(2813,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{133}{240}\right)\) | \(e\left(\frac{37}{120}\right)\) | \(e\left(\frac{187}{240}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{17}{80}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{13}{120}\right)\) | \(e\left(\frac{239}{240}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{69}{80}\right)\) |