sage: from sage.modular.dirichlet import DirichletCharacter
sage: H = DirichletGroup(175, base_ring=CyclotomicField(60))
sage: M = H._module
sage: chi = DirichletCharacter(H, M([3,20]))
sage: chi.galois_orbit()
pari: [g,chi] = znchar(Mod(2,175))
pari: order = charorder(g,chi)
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(175\) | |
| Conductor: | \(175\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | 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: | \(\Q(\zeta_{60})\) |
| Fixed field: | Number field defined by a degree 60 polynomial |
Characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) | \(16\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{175}(2,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{13}{15}\right)\) |
| \(\chi_{175}(23,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{13}{15}\right)\) |
| \(\chi_{175}(37,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{7}{15}\right)\) |
| \(\chi_{175}(53,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{11}{15}\right)\) |
| \(\chi_{175}(58,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{4}{15}\right)\) |
| \(\chi_{175}(67,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{14}{15}\right)\) |
| \(\chi_{175}(72,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{1}{15}\right)\) |
| \(\chi_{175}(88,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{2}{15}\right)\) |
| \(\chi_{175}(102,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{8}{15}\right)\) |
| \(\chi_{175}(123,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{8}{15}\right)\) |
| \(\chi_{175}(128,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{1}{15}\right)\) |
| \(\chi_{175}(137,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{2}{15}\right)\) |
| \(\chi_{175}(142,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{4}{15}\right)\) |
| \(\chi_{175}(158,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{14}{15}\right)\) |
| \(\chi_{175}(163,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{7}{15}\right)\) |
| \(\chi_{175}(172,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{11}{15}\right)\) |