from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5185, base_ring=CyclotomicField(240))
M = H._module
chi = DirichletCharacter(H, M([120,75,184]))
chi.galois_orbit()
[g,chi] = znchar(Mod(39,5185))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(5185\) | |
| Conductor: | \(5185\) | 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: | odd | 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 | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{5185}(39,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{77}{120}\right)\) | \(e\left(\frac{33}{80}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{13}{240}\right)\) | \(e\left(\frac{121}{240}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{167}{240}\right)\) | \(e\left(\frac{5}{12}\right)\) |
| \(\chi_{5185}(249,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{119}{120}\right)\) | \(e\left(\frac{11}{80}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{31}{240}\right)\) | \(e\left(\frac{67}{240}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{29}{240}\right)\) | \(e\left(\frac{11}{12}\right)\) |
| \(\chi_{5185}(309,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{49}{120}\right)\) | \(e\left(\frac{61}{80}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{41}{240}\right)\) | \(e\left(\frac{197}{240}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{139}{240}\right)\) | \(e\left(\frac{1}{12}\right)\) |
| \(\chi_{5185}(354,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{120}\right)\) | \(e\left(\frac{69}{80}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{209}{240}\right)\) | \(e\left(\frac{173}{240}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{211}{240}\right)\) | \(e\left(\frac{1}{12}\right)\) |
| \(\chi_{5185}(524,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{73}{120}\right)\) | \(e\left(\frac{37}{80}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{17}{240}\right)\) | \(e\left(\frac{29}{240}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{163}{240}\right)\) | \(e\left(\frac{1}{12}\right)\) |
| \(\chi_{5185}(534,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{120}\right)\) | \(e\left(\frac{79}{80}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{19}{240}\right)\) | \(e\left(\frac{103}{240}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{41}{240}\right)\) | \(e\left(\frac{11}{12}\right)\) |
| \(\chi_{5185}(554,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{120}\right)\) | \(e\left(\frac{71}{80}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{91}{240}\right)\) | \(e\left(\frac{127}{240}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{209}{240}\right)\) | \(e\left(\frac{11}{12}\right)\) |
| \(\chi_{5185}(649,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{120}\right)\) | \(e\left(\frac{13}{80}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{73}{240}\right)\) | \(e\left(\frac{181}{240}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{107}{240}\right)\) | \(e\left(\frac{5}{12}\right)\) |
| \(\chi_{5185}(839,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{71}{120}\right)\) | \(e\left(\frac{19}{80}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{199}{240}\right)\) | \(e\left(\frac{43}{240}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{101}{240}\right)\) | \(e\left(\frac{11}{12}\right)\) |
| \(\chi_{5185}(964,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{61}{120}\right)\) | \(e\left(\frac{9}{80}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{149}{240}\right)\) | \(e\left(\frac{113}{240}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{31}{240}\right)\) | \(e\left(\frac{1}{12}\right)\) |
| \(\chi_{5185}(1134,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{120}\right)\) | \(e\left(\frac{57}{80}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{197}{240}\right)\) | \(e\left(\frac{209}{240}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{223}{240}\right)\) | \(e\left(\frac{1}{12}\right)\) |
| \(\chi_{5185}(1144,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{101}{120}\right)\) | \(e\left(\frac{49}{80}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{109}{240}\right)\) | \(e\left(\frac{73}{240}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{71}{240}\right)\) | \(e\left(\frac{5}{12}\right)\) |
| \(\chi_{5185}(1204,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{113}{120}\right)\) | \(e\left(\frac{37}{80}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{97}{240}\right)\) | \(e\left(\frac{109}{240}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{83}{240}\right)\) | \(e\left(\frac{5}{12}\right)\) |
| \(\chi_{5185}(1269,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{120}\right)\) | \(e\left(\frac{59}{80}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{239}{240}\right)\) | \(e\left(\frac{83}{240}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{61}{240}\right)\) | \(e\left(\frac{7}{12}\right)\) |
| \(\chi_{5185}(1439,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{103}{120}\right)\) | \(e\left(\frac{27}{80}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{47}{240}\right)\) | \(e\left(\frac{179}{240}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{13}{240}\right)\) | \(e\left(\frac{7}{12}\right)\) |
| \(\chi_{5185}(1469,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{120}\right)\) | \(e\left(\frac{31}{80}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{211}{240}\right)\) | \(e\left(\frac{7}{240}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{89}{240}\right)\) | \(e\left(\frac{11}{12}\right)\) |
| \(\chi_{5185}(1544,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{97}{120}\right)\) | \(e\left(\frac{53}{80}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{113}{240}\right)\) | \(e\left(\frac{221}{240}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{67}{240}\right)\) | \(e\left(\frac{1}{12}\right)\) |
| \(\chi_{5185}(1574,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{91}{120}\right)\) | \(e\left(\frac{39}{80}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{59}{240}\right)\) | \(e\left(\frac{143}{240}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{1}{240}\right)\) | \(e\left(\frac{7}{12}\right)\) |
| \(\chi_{5185}(1744,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{120}\right)\) | \(e\left(\frac{7}{80}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{107}{240}\right)\) | \(e\left(\frac{239}{240}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{193}{240}\right)\) | \(e\left(\frac{7}{12}\right)\) |
| \(\chi_{5185}(1754,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{41}{120}\right)\) | \(e\left(\frac{29}{80}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{169}{240}\right)\) | \(e\left(\frac{133}{240}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{11}{240}\right)\) | \(e\left(\frac{5}{12}\right)\) |
| \(\chi_{5185}(1774,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{119}{120}\right)\) | \(e\left(\frac{51}{80}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{151}{240}\right)\) | \(e\left(\frac{187}{240}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{149}{240}\right)\) | \(e\left(\frac{11}{12}\right)\) |
| \(\chi_{5185}(1814,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{53}{120}\right)\) | \(e\left(\frac{57}{80}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{37}{240}\right)\) | \(e\left(\frac{49}{240}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{143}{240}\right)\) | \(e\left(\frac{5}{12}\right)\) |
| \(\chi_{5185}(2079,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{120}\right)\) | \(e\left(\frac{1}{80}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{61}{240}\right)\) | \(e\left(\frac{217}{240}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{119}{240}\right)\) | \(e\left(\frac{5}{12}\right)\) |
| \(\chi_{5185}(2119,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{120}\right)\) | \(e\left(\frac{27}{80}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{127}{240}\right)\) | \(e\left(\frac{19}{240}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{173}{240}\right)\) | \(e\left(\frac{11}{12}\right)\) |
| \(\chi_{5185}(2139,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{49}{120}\right)\) | \(e\left(\frac{21}{80}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{161}{240}\right)\) | \(e\left(\frac{77}{240}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{19}{240}\right)\) | \(e\left(\frac{1}{12}\right)\) |
| \(\chi_{5185}(2154,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{37}{120}\right)\) | \(e\left(\frac{73}{80}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{53}{240}\right)\) | \(e\left(\frac{161}{240}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{127}{240}\right)\) | \(e\left(\frac{1}{12}\right)\) |
| \(\chi_{5185}(2424,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{83}{120}\right)\) | \(e\left(\frac{7}{80}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{187}{240}\right)\) | \(e\left(\frac{79}{240}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{113}{240}\right)\) | \(e\left(\frac{11}{12}\right)\) |
| \(\chi_{5185}(2459,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{120}\right)\) | \(e\left(\frac{43}{80}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{143}{240}\right)\) | \(e\left(\frac{131}{240}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{157}{240}\right)\) | \(e\left(\frac{7}{12}\right)\) |
| \(\chi_{5185}(2479,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{120}\right)\) | \(e\left(\frac{53}{80}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{193}{240}\right)\) | \(e\left(\frac{61}{240}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{227}{240}\right)\) | \(e\left(\frac{5}{12}\right)\) |
| \(\chi_{5185}(2489,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{91}{120}\right)\) | \(e\left(\frac{79}{80}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{179}{240}\right)\) | \(e\left(\frac{23}{240}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{121}{240}\right)\) | \(e\left(\frac{7}{12}\right)\) |
| \(\chi_{5185}(2659,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{120}\right)\) | \(e\left(\frac{47}{80}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{227}{240}\right)\) | \(e\left(\frac{119}{240}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{73}{240}\right)\) | \(e\left(\frac{7}{12}\right)\) |