from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5185, base_ring=CyclotomicField(120))
M = H._module
chi = DirichletCharacter(H, M([30,105,2]))
chi.galois_orbit()
[g,chi] = znchar(Mod(2,5185))
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: | \(120\) | 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: | $\Q(\zeta_{120})$ |
| Fixed field: | Number field defined by a degree 120 polynomial (not computed) |
First 31 of 32 characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{5185}(2,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{29}{120}\right)\) | \(e\left(\frac{83}{120}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{91}{120}\right)\) | \(e\left(\frac{11}{12}\right)\) |
| \(\chi_{5185}(43,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{107}{120}\right)\) | \(e\left(\frac{29}{120}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{13}{120}\right)\) | \(e\left(\frac{5}{12}\right)\) |
| \(\chi_{5185}(87,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{109}{120}\right)\) | \(e\left(\frac{43}{120}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{11}{120}\right)\) | \(e\left(\frac{7}{12}\right)\) |
| \(\chi_{5185}(128,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{83}{120}\right)\) | \(e\left(\frac{101}{120}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{37}{120}\right)\) | \(e\left(\frac{5}{12}\right)\) |
| \(\chi_{5185}(298,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{11}{120}\right)\) | \(e\left(\frac{77}{120}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{109}{120}\right)\) | \(e\left(\frac{5}{12}\right)\) |
| \(\chi_{5185}(348,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{47}{120}\right)\) | \(e\left(\frac{89}{120}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{73}{120}\right)\) | \(e\left(\frac{5}{12}\right)\) |
| \(\chi_{5185}(383,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{43}{120}\right)\) | \(e\left(\frac{61}{120}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{77}{120}\right)\) | \(e\left(\frac{1}{12}\right)\) |
| \(\chi_{5185}(433,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{23}{120}\right)\) | \(e\left(\frac{41}{120}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{97}{120}\right)\) | \(e\left(\frac{5}{12}\right)\) |
| \(\chi_{5185}(457,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{1}{120}\right)\) | \(e\left(\frac{7}{120}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{119}{120}\right)\) | \(e\left(\frac{7}{12}\right)\) |
| \(\chi_{5185}(603,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{71}{120}\right)\) | \(e\left(\frac{17}{120}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{49}{120}\right)\) | \(e\left(\frac{5}{12}\right)\) |
| \(\chi_{5185}(688,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{103}{120}\right)\) | \(e\left(\frac{1}{120}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{17}{120}\right)\) | \(e\left(\frac{1}{12}\right)\) |
| \(\chi_{5185}(1222,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{89}{120}\right)\) | \(e\left(\frac{23}{120}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{31}{120}\right)\) | \(e\left(\frac{11}{12}\right)\) |
| \(\chi_{5185}(1307,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{49}{120}\right)\) | \(e\left(\frac{103}{120}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{71}{120}\right)\) | \(e\left(\frac{7}{12}\right)\) |
| \(\chi_{5185}(1447,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{13}{120}\right)\) | \(e\left(\frac{91}{120}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{107}{120}\right)\) | \(e\left(\frac{7}{12}\right)\) |
| \(\chi_{5185}(1532,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{101}{120}\right)\) | \(e\left(\frac{107}{120}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{19}{120}\right)\) | \(e\left(\frac{11}{12}\right)\) |
| \(\chi_{5185}(1702,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{53}{120}\right)\) | \(e\left(\frac{11}{120}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{67}{120}\right)\) | \(e\left(\frac{11}{12}\right)\) |
| \(\chi_{5185}(1743,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{19}{120}\right)\) | \(e\left(\frac{13}{120}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{101}{120}\right)\) | \(e\left(\frac{1}{12}\right)\) |
| \(\chi_{5185}(1787,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{77}{120}\right)\) | \(e\left(\frac{59}{120}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{43}{120}\right)\) | \(e\left(\frac{11}{12}\right)\) |
| \(\chi_{5185}(1828,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{59}{120}\right)\) | \(e\left(\frac{53}{120}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{61}{120}\right)\) | \(e\left(\frac{5}{12}\right)\) |
| \(\chi_{5185}(2048,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{79}{120}\right)\) | \(e\left(\frac{73}{120}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{41}{120}\right)\) | \(e\left(\frac{1}{12}\right)\) |
| \(\chi_{5185}(2133,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{119}{120}\right)\) | \(e\left(\frac{113}{120}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{1}{120}\right)\) | \(e\left(\frac{5}{12}\right)\) |
| \(\chi_{5185}(2552,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{37}{120}\right)\) | \(e\left(\frac{19}{120}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{83}{120}\right)\) | \(e\left(\frac{7}{12}\right)\) |
| \(\chi_{5185}(2593,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{91}{120}\right)\) | \(e\left(\frac{37}{120}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{29}{120}\right)\) | \(e\left(\frac{1}{12}\right)\) |
| \(\chi_{5185}(2667,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{73}{120}\right)\) | \(e\left(\frac{31}{120}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{47}{120}\right)\) | \(e\left(\frac{7}{12}\right)\) |
| \(\chi_{5185}(2752,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{41}{120}\right)\) | \(e\left(\frac{47}{120}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{79}{120}\right)\) | \(e\left(\frac{11}{12}\right)\) |
| \(\chi_{5185}(2898,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{31}{120}\right)\) | \(e\left(\frac{97}{120}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{89}{120}\right)\) | \(e\left(\frac{1}{12}\right)\) |
| \(\chi_{5185}(2922,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{113}{120}\right)\) | \(e\left(\frac{71}{120}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{7}{120}\right)\) | \(e\left(\frac{11}{12}\right)\) |
| \(\chi_{5185}(3007,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{17}{120}\right)\) | \(e\left(\frac{119}{120}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{103}{120}\right)\) | \(e\left(\frac{11}{12}\right)\) |
| \(\chi_{5185}(3772,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{97}{120}\right)\) | \(e\left(\frac{79}{120}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{23}{120}\right)\) | \(e\left(\frac{7}{12}\right)\) |
| \(\chi_{5185}(4422,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{61}{120}\right)\) | \(e\left(\frac{67}{120}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{59}{120}\right)\) | \(e\left(\frac{7}{12}\right)\) |
| \(\chi_{5185}(4463,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{67}{120}\right)\) | \(e\left(\frac{109}{120}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{53}{120}\right)\) | \(e\left(\frac{1}{12}\right)\) |