from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5185, base_ring=CyclotomicField(120))
M = H._module
chi = DirichletCharacter(H, M([30,75,112]))
chi.galois_orbit()
[g,chi] = znchar(Mod(42,5185))
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: | odd | 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 | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{5185}(42,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{109}{120}\right)\) | \(e\left(\frac{103}{120}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{101}{120}\right)\) | \(e\left(\frac{7}{12}\right)\) |
| \(\chi_{5185}(77,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{41}{120}\right)\) | \(e\left(\frac{107}{120}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{49}{120}\right)\) | \(e\left(\frac{11}{12}\right)\) |
| \(\chi_{5185}(83,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{59}{120}\right)\) | \(e\left(\frac{113}{120}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{91}{120}\right)\) | \(e\left(\frac{5}{12}\right)\) |
| \(\chi_{5185}(138,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{71}{120}\right)\) | \(e\left(\frac{77}{120}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{79}{120}\right)\) | \(e\left(\frac{5}{12}\right)\) |
| \(\chi_{5185}(382,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{101}{120}\right)\) | \(e\left(\frac{47}{120}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{109}{120}\right)\) | \(e\left(\frac{11}{12}\right)\) |
| \(\chi_{5185}(423,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{43}{120}\right)\) | \(e\left(\frac{1}{120}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{107}{120}\right)\) | \(e\left(\frac{1}{12}\right)\) |
| \(\chi_{5185}(757,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{1}{120}\right)\) | \(e\left(\frac{67}{120}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{89}{120}\right)\) | \(e\left(\frac{7}{12}\right)\) |
| \(\chi_{5185}(818,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{31}{120}\right)\) | \(e\left(\frac{37}{120}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{119}{120}\right)\) | \(e\left(\frac{1}{12}\right)\) |
| \(\chi_{5185}(927,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{97}{120}\right)\) | \(e\left(\frac{19}{120}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{113}{120}\right)\) | \(e\left(\frac{7}{12}\right)\) |
| \(\chi_{5185}(988,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{7}{120}\right)\) | \(e\left(\frac{109}{120}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{23}{120}\right)\) | \(e\left(\frac{1}{12}\right)\) |
| \(\chi_{5185}(1018,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{19}{120}\right)\) | \(e\left(\frac{73}{120}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{11}{120}\right)\) | \(e\left(\frac{1}{12}\right)\) |
| \(\chi_{5185}(1062,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{61}{120}\right)\) | \(e\left(\frac{7}{120}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{29}{120}\right)\) | \(e\left(\frac{7}{12}\right)\) |
| \(\chi_{5185}(1232,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{37}{120}\right)\) | \(e\left(\frac{79}{120}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{53}{120}\right)\) | \(e\left(\frac{7}{12}\right)\) |
| \(\chi_{5185}(1358,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{11}{120}\right)\) | \(e\left(\frac{17}{120}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{19}{120}\right)\) | \(e\left(\frac{5}{12}\right)\) |
| \(\chi_{5185}(1947,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{113}{120}\right)\) | \(e\left(\frac{11}{120}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{97}{120}\right)\) | \(e\left(\frac{11}{12}\right)\) |
| \(\chi_{5185}(2008,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{23}{120}\right)\) | \(e\left(\frac{101}{120}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{7}{120}\right)\) | \(e\left(\frac{5}{12}\right)\) |
| \(\chi_{5185}(2038,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{91}{120}\right)\) | \(e\left(\frac{97}{120}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{59}{120}\right)\) | \(e\left(\frac{1}{12}\right)\) |
| \(\chi_{5185}(2208,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{67}{120}\right)\) | \(e\left(\frac{49}{120}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{83}{120}\right)\) | \(e\left(\frac{1}{12}\right)\) |
| \(\chi_{5185}(2252,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{53}{120}\right)\) | \(e\left(\frac{71}{120}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{37}{120}\right)\) | \(e\left(\frac{11}{12}\right)\) |
| \(\chi_{5185}(2882,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{17}{120}\right)\) | \(e\left(\frac{59}{120}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{73}{120}\right)\) | \(e\left(\frac{11}{12}\right)\) |
| \(\chi_{5185}(2943,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{47}{120}\right)\) | \(e\left(\frac{29}{120}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{103}{120}\right)\) | \(e\left(\frac{5}{12}\right)\) |
| \(\chi_{5185}(3187,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{7}{40}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{77}{120}\right)\) | \(e\left(\frac{119}{120}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{13}{120}\right)\) | \(e\left(\frac{11}{12}\right)\) |
| \(\chi_{5185}(3228,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{33}{40}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{83}{120}\right)\) | \(e\left(\frac{41}{120}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{67}{120}\right)\) | \(e\left(\frac{5}{12}\right)\) |
| \(\chi_{5185}(3987,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{89}{120}\right)\) | \(e\left(\frac{83}{120}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{1}{120}\right)\) | \(e\left(\frac{11}{12}\right)\) |
| \(\chi_{5185}(4048,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{119}{120}\right)\) | \(e\left(\frac{53}{120}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{31}{120}\right)\) | \(e\left(\frac{5}{12}\right)\) |
| \(\chi_{5185}(4163,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{107}{120}\right)\) | \(e\left(\frac{89}{120}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{43}{120}\right)\) | \(e\left(\frac{5}{12}\right)\) |
| \(\chi_{5185}(4292,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{29}{120}\right)\) | \(e\left(\frac{23}{120}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{61}{120}\right)\) | \(e\left(\frac{11}{12}\right)\) |
| \(\chi_{5185}(4327,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{73}{120}\right)\) | \(e\left(\frac{91}{120}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{17}{120}\right)\) | \(e\left(\frac{7}{12}\right)\) |
| \(\chi_{5185}(4388,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{13}{40}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{103}{120}\right)\) | \(e\left(\frac{61}{120}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{47}{120}\right)\) | \(e\left(\frac{1}{12}\right)\) |
| \(\chi_{5185}(4632,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{13}{120}\right)\) | \(e\left(\frac{31}{120}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{77}{120}\right)\) | \(e\left(\frac{7}{12}\right)\) |
| \(\chi_{5185}(4922,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{49}{120}\right)\) | \(e\left(\frac{43}{120}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{41}{120}\right)\) | \(e\left(\frac{7}{12}\right)\) |