from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5185, base_ring=CyclotomicField(240))
M = H._module
chi = DirichletCharacter(H, M([0,225,28]))
chi.galois_orbit()
[g,chi] = znchar(Mod(6,5185))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(5185\) | |
| Conductor: | \(1037\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(240\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from 1037.cr | 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 | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{5185}(6,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{120}\right)\) | \(e\left(\frac{51}{80}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{211}{240}\right)\) | \(e\left(\frac{7}{240}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{29}{240}\right)\) | \(e\left(\frac{5}{12}\right)\) |
| \(\chi_{5185}(96,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{120}\right)\) | \(e\left(\frac{43}{80}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{203}{240}\right)\) | \(e\left(\frac{191}{240}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{37}{240}\right)\) | \(e\left(\frac{1}{12}\right)\) |
| \(\chi_{5185}(116,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{59}{120}\right)\) | \(e\left(\frac{21}{80}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{181}{240}\right)\) | \(e\left(\frac{97}{240}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{59}{240}\right)\) | \(e\left(\frac{11}{12}\right)\) |
| \(\chi_{5185}(176,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{53}{120}\right)\) | \(e\left(\frac{67}{80}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{67}{240}\right)\) | \(e\left(\frac{199}{240}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{173}{240}\right)\) | \(e\left(\frac{5}{12}\right)\) |
| \(\chi_{5185}(181,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{77}{120}\right)\) | \(e\left(\frac{43}{80}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{43}{240}\right)\) | \(e\left(\frac{31}{240}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{197}{240}\right)\) | \(e\left(\frac{5}{12}\right)\) |
| \(\chi_{5185}(201,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{120}\right)\) | \(e\left(\frac{69}{80}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{229}{240}\right)\) | \(e\left(\frac{193}{240}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{11}{240}\right)\) | \(e\left(\frac{11}{12}\right)\) |
| \(\chi_{5185}(261,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{109}{120}\right)\) | \(e\left(\frac{51}{80}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{131}{240}\right)\) | \(e\left(\frac{167}{240}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{109}{240}\right)\) | \(e\left(\frac{1}{12}\right)\) |
| \(\chi_{5185}(396,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{103}{120}\right)\) | \(e\left(\frac{17}{80}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{17}{240}\right)\) | \(e\left(\frac{29}{240}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{223}{240}\right)\) | \(e\left(\frac{7}{12}\right)\) |
| \(\chi_{5185}(401,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{97}{120}\right)\) | \(e\left(\frac{23}{80}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{23}{240}\right)\) | \(e\left(\frac{11}{240}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{217}{240}\right)\) | \(e\left(\frac{1}{12}\right)\) |
| \(\chi_{5185}(471,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{79}{120}\right)\) | \(e\left(\frac{41}{80}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{41}{240}\right)\) | \(e\left(\frac{197}{240}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{199}{240}\right)\) | \(e\left(\frac{7}{12}\right)\) |
| \(\chi_{5185}(486,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{120}\right)\) | \(e\left(\frac{23}{80}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{103}{240}\right)\) | \(e\left(\frac{91}{240}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{137}{240}\right)\) | \(e\left(\frac{5}{12}\right)\) |
| \(\chi_{5185}(556,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{120}\right)\) | \(e\left(\frac{57}{80}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{217}{240}\right)\) | \(e\left(\frac{229}{240}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{23}{240}\right)\) | \(e\left(\frac{11}{12}\right)\) |
| \(\chi_{5185}(726,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{119}{120}\right)\) | \(e\left(\frac{41}{80}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{121}{240}\right)\) | \(e\left(\frac{37}{240}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{119}{240}\right)\) | \(e\left(\frac{11}{12}\right)\) |
| \(\chi_{5185}(811,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{71}{120}\right)\) | \(e\left(\frac{9}{80}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{169}{240}\right)\) | \(e\left(\frac{133}{240}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{71}{240}\right)\) | \(e\left(\frac{11}{12}\right)\) |
| \(\chi_{5185}(946,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{120}\right)\) | \(e\left(\frac{27}{80}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{107}{240}\right)\) | \(e\left(\frac{239}{240}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{133}{240}\right)\) | \(e\left(\frac{1}{12}\right)\) |
| \(\chi_{5185}(966,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{91}{120}\right)\) | \(e\left(\frac{69}{80}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{149}{240}\right)\) | \(e\left(\frac{113}{240}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{91}{240}\right)\) | \(e\left(\frac{7}{12}\right)\) |
| \(\chi_{5185}(1006,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{120}\right)\) | \(e\left(\frac{77}{80}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{77}{240}\right)\) | \(e\left(\frac{89}{240}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{163}{240}\right)\) | \(e\left(\frac{7}{12}\right)\) |
| \(\chi_{5185}(1161,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{47}{120}\right)\) | \(e\left(\frac{33}{80}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{193}{240}\right)\) | \(e\left(\frac{61}{240}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{47}{240}\right)\) | \(e\left(\frac{11}{12}\right)\) |
| \(\chi_{5185}(1246,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{120}\right)\) | \(e\left(\frac{33}{80}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{113}{240}\right)\) | \(e\left(\frac{221}{240}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{127}{240}\right)\) | \(e\left(\frac{7}{12}\right)\) |
| \(\chi_{5185}(1251,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{73}{120}\right)\) | \(e\left(\frac{7}{80}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{167}{240}\right)\) | \(e\left(\frac{59}{240}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{73}{240}\right)\) | \(e\left(\frac{1}{12}\right)\) |
| \(\chi_{5185}(1316,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{97}{120}\right)\) | \(e\left(\frac{63}{80}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{143}{240}\right)\) | \(e\left(\frac{131}{240}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{97}{240}\right)\) | \(e\left(\frac{1}{12}\right)\) |
| \(\chi_{5185}(1401,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{120}\right)\) | \(e\left(\frac{63}{80}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{223}{240}\right)\) | \(e\left(\frac{211}{240}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{17}{240}\right)\) | \(e\left(\frac{5}{12}\right)\) |
| \(\chi_{5185}(1576,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{120}\right)\) | \(e\left(\frac{9}{80}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{89}{240}\right)\) | \(e\left(\frac{53}{240}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{151}{240}\right)\) | \(e\left(\frac{7}{12}\right)\) |
| \(\chi_{5185}(1621,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{120}\right)\) | \(e\left(\frac{3}{80}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{83}{240}\right)\) | \(e\left(\frac{71}{240}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{157}{240}\right)\) | \(e\left(\frac{1}{12}\right)\) |
| \(\chi_{5185}(1706,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{77}{120}\right)\) | \(e\left(\frac{3}{80}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{163}{240}\right)\) | \(e\left(\frac{151}{240}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{77}{240}\right)\) | \(e\left(\frac{5}{12}\right)\) |
| \(\chi_{5185}(1771,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{107}{120}\right)\) | \(e\left(\frac{13}{80}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{13}{240}\right)\) | \(e\left(\frac{121}{240}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{227}{240}\right)\) | \(e\left(\frac{11}{12}\right)\) |
| \(\chi_{5185}(1856,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{67}{120}\right)\) | \(e\left(\frac{13}{80}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{173}{240}\right)\) | \(e\left(\frac{41}{240}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{67}{240}\right)\) | \(e\left(\frac{7}{12}\right)\) |
| \(\chi_{5185}(2166,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{73}{120}\right)\) | \(e\left(\frac{47}{80}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{47}{240}\right)\) | \(e\left(\frac{179}{240}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{193}{240}\right)\) | \(e\left(\frac{1}{12}\right)\) |
| \(\chi_{5185}(2471,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{120}\right)\) | \(e\left(\frac{67}{80}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{227}{240}\right)\) | \(e\left(\frac{119}{240}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{13}{240}\right)\) | \(e\left(\frac{1}{12}\right)\) |
| \(\chi_{5185}(2606,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{79}{120}\right)\) | \(e\left(\frac{1}{80}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{161}{240}\right)\) | \(e\left(\frac{77}{240}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{79}{240}\right)\) | \(e\left(\frac{7}{12}\right)\) |
| \(\chi_{5185}(2691,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{120}\right)\) | \(e\left(\frac{17}{80}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{97}{240}\right)\) | \(e\left(\frac{109}{240}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{143}{240}\right)\) | \(e\left(\frac{11}{12}\right)\) |