from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(421, base_ring=CyclotomicField(70))
M = H._module
chi = DirichletCharacter(H, M([62]))
chi.galois_orbit()
[g,chi] = znchar(Mod(27,421))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(421\) | |
| Conductor: | \(421\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(35\) | 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: | $\Q(\zeta_{35})$ |
| Fixed field: | Number field defined by a degree 35 polynomial |
Characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{421}(27,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{35}\right)\) | \(e\left(\frac{29}{35}\right)\) | \(e\left(\frac{27}{35}\right)\) | \(e\left(\frac{8}{35}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{6}{35}\right)\) | \(e\left(\frac{23}{35}\right)\) | \(e\left(\frac{23}{35}\right)\) | \(e\left(\frac{4}{35}\right)\) | \(e\left(\frac{1}{35}\right)\) |
| \(\chi_{421}(48,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{34}{35}\right)\) | \(e\left(\frac{16}{35}\right)\) | \(e\left(\frac{33}{35}\right)\) | \(e\left(\frac{2}{35}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{19}{35}\right)\) | \(e\left(\frac{32}{35}\right)\) | \(e\left(\frac{32}{35}\right)\) | \(e\left(\frac{1}{35}\right)\) | \(e\left(\frac{9}{35}\right)\) |
| \(\chi_{421}(49,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{26}{35}\right)\) | \(e\left(\frac{4}{35}\right)\) | \(e\left(\frac{17}{35}\right)\) | \(e\left(\frac{18}{35}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{31}{35}\right)\) | \(e\left(\frac{8}{35}\right)\) | \(e\left(\frac{8}{35}\right)\) | \(e\left(\frac{9}{35}\right)\) | \(e\left(\frac{11}{35}\right)\) |
| \(\chi_{421}(60,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{22}{35}\right)\) | \(e\left(\frac{33}{35}\right)\) | \(e\left(\frac{9}{35}\right)\) | \(e\left(\frac{26}{35}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{2}{35}\right)\) | \(e\left(\frac{31}{35}\right)\) | \(e\left(\frac{31}{35}\right)\) | \(e\left(\frac{13}{35}\right)\) | \(e\left(\frac{12}{35}\right)\) |
| \(\chi_{421}(68,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{24}{35}\right)\) | \(e\left(\frac{1}{35}\right)\) | \(e\left(\frac{13}{35}\right)\) | \(e\left(\frac{22}{35}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{34}{35}\right)\) | \(e\left(\frac{2}{35}\right)\) | \(e\left(\frac{2}{35}\right)\) | \(e\left(\frac{11}{35}\right)\) | \(e\left(\frac{29}{35}\right)\) |
| \(\chi_{421}(78,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{35}\right)\) | \(e\left(\frac{6}{35}\right)\) | \(e\left(\frac{8}{35}\right)\) | \(e\left(\frac{27}{35}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{29}{35}\right)\) | \(e\left(\frac{12}{35}\right)\) | \(e\left(\frac{12}{35}\right)\) | \(e\left(\frac{31}{35}\right)\) | \(e\left(\frac{34}{35}\right)\) |
| \(\chi_{421}(85,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{12}{35}\right)\) | \(e\left(\frac{18}{35}\right)\) | \(e\left(\frac{24}{35}\right)\) | \(e\left(\frac{11}{35}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{17}{35}\right)\) | \(e\left(\frac{1}{35}\right)\) | \(e\left(\frac{1}{35}\right)\) | \(e\left(\frac{23}{35}\right)\) | \(e\left(\frac{32}{35}\right)\) |
| \(\chi_{421}(139,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{35}\right)\) | \(e\left(\frac{11}{35}\right)\) | \(e\left(\frac{3}{35}\right)\) | \(e\left(\frac{32}{35}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{24}{35}\right)\) | \(e\left(\frac{22}{35}\right)\) | \(e\left(\frac{22}{35}\right)\) | \(e\left(\frac{16}{35}\right)\) | \(e\left(\frac{4}{35}\right)\) |
| \(\chi_{421}(190,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{35}\right)\) | \(e\left(\frac{12}{35}\right)\) | \(e\left(\frac{16}{35}\right)\) | \(e\left(\frac{19}{35}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{23}{35}\right)\) | \(e\left(\frac{24}{35}\right)\) | \(e\left(\frac{24}{35}\right)\) | \(e\left(\frac{27}{35}\right)\) | \(e\left(\frac{33}{35}\right)\) |
| \(\chi_{421}(199,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{33}{35}\right)\) | \(e\left(\frac{32}{35}\right)\) | \(e\left(\frac{31}{35}\right)\) | \(e\left(\frac{4}{35}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{3}{35}\right)\) | \(e\left(\frac{29}{35}\right)\) | \(e\left(\frac{29}{35}\right)\) | \(e\left(\frac{2}{35}\right)\) | \(e\left(\frac{18}{35}\right)\) |
| \(\chi_{421}(232,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{35}\right)\) | \(e\left(\frac{31}{35}\right)\) | \(e\left(\frac{18}{35}\right)\) | \(e\left(\frac{17}{35}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{4}{35}\right)\) | \(e\left(\frac{27}{35}\right)\) | \(e\left(\frac{27}{35}\right)\) | \(e\left(\frac{26}{35}\right)\) | \(e\left(\frac{24}{35}\right)\) |
| \(\chi_{421}(290,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{32}{35}\right)\) | \(e\left(\frac{13}{35}\right)\) | \(e\left(\frac{29}{35}\right)\) | \(e\left(\frac{6}{35}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{22}{35}\right)\) | \(e\left(\frac{26}{35}\right)\) | \(e\left(\frac{26}{35}\right)\) | \(e\left(\frac{3}{35}\right)\) | \(e\left(\frac{27}{35}\right)\) |
| \(\chi_{421}(291,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{35}\right)\) | \(e\left(\frac{34}{35}\right)\) | \(e\left(\frac{22}{35}\right)\) | \(e\left(\frac{13}{35}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{1}{35}\right)\) | \(e\left(\frac{33}{35}\right)\) | \(e\left(\frac{33}{35}\right)\) | \(e\left(\frac{24}{35}\right)\) | \(e\left(\frac{6}{35}\right)\) |
| \(\chi_{421}(296,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{35}\right)\) | \(e\left(\frac{8}{35}\right)\) | \(e\left(\frac{34}{35}\right)\) | \(e\left(\frac{1}{35}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{27}{35}\right)\) | \(e\left(\frac{16}{35}\right)\) | \(e\left(\frac{16}{35}\right)\) | \(e\left(\frac{18}{35}\right)\) | \(e\left(\frac{22}{35}\right)\) |
| \(\chi_{421}(307,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{35}\right)\) | \(e\left(\frac{19}{35}\right)\) | \(e\left(\frac{2}{35}\right)\) | \(e\left(\frac{33}{35}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{16}{35}\right)\) | \(e\left(\frac{3}{35}\right)\) | \(e\left(\frac{3}{35}\right)\) | \(e\left(\frac{34}{35}\right)\) | \(e\left(\frac{26}{35}\right)\) |
| \(\chi_{421}(308,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{27}{35}\right)\) | \(e\left(\frac{23}{35}\right)\) | \(e\left(\frac{19}{35}\right)\) | \(e\left(\frac{16}{35}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{12}{35}\right)\) | \(e\left(\frac{11}{35}\right)\) | \(e\left(\frac{11}{35}\right)\) | \(e\left(\frac{8}{35}\right)\) | \(e\left(\frac{2}{35}\right)\) |
| \(\chi_{421}(315,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{16}{35}\right)\) | \(e\left(\frac{24}{35}\right)\) | \(e\left(\frac{32}{35}\right)\) | \(e\left(\frac{3}{35}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{11}{35}\right)\) | \(e\left(\frac{13}{35}\right)\) | \(e\left(\frac{13}{35}\right)\) | \(e\left(\frac{19}{35}\right)\) | \(e\left(\frac{31}{35}\right)\) |
| \(\chi_{421}(317,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{35}\right)\) | \(e\left(\frac{17}{35}\right)\) | \(e\left(\frac{11}{35}\right)\) | \(e\left(\frac{24}{35}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{18}{35}\right)\) | \(e\left(\frac{34}{35}\right)\) | \(e\left(\frac{34}{35}\right)\) | \(e\left(\frac{12}{35}\right)\) | \(e\left(\frac{3}{35}\right)\) |
| \(\chi_{421}(321,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{35}\right)\) | \(e\left(\frac{26}{35}\right)\) | \(e\left(\frac{23}{35}\right)\) | \(e\left(\frac{12}{35}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{9}{35}\right)\) | \(e\left(\frac{17}{35}\right)\) | \(e\left(\frac{17}{35}\right)\) | \(e\left(\frac{6}{35}\right)\) | \(e\left(\frac{19}{35}\right)\) |
| \(\chi_{421}(341,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{6}{35}\right)\) | \(e\left(\frac{9}{35}\right)\) | \(e\left(\frac{12}{35}\right)\) | \(e\left(\frac{23}{35}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{26}{35}\right)\) | \(e\left(\frac{18}{35}\right)\) | \(e\left(\frac{18}{35}\right)\) | \(e\left(\frac{29}{35}\right)\) | \(e\left(\frac{16}{35}\right)\) |
| \(\chi_{421}(357,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{18}{35}\right)\) | \(e\left(\frac{27}{35}\right)\) | \(e\left(\frac{1}{35}\right)\) | \(e\left(\frac{34}{35}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{8}{35}\right)\) | \(e\left(\frac{19}{35}\right)\) | \(e\left(\frac{19}{35}\right)\) | \(e\left(\frac{17}{35}\right)\) | \(e\left(\frac{13}{35}\right)\) |
| \(\chi_{421}(366,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{35}\right)\) | \(e\left(\frac{3}{35}\right)\) | \(e\left(\frac{4}{35}\right)\) | \(e\left(\frac{31}{35}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{32}{35}\right)\) | \(e\left(\frac{6}{35}\right)\) | \(e\left(\frac{6}{35}\right)\) | \(e\left(\frac{33}{35}\right)\) | \(e\left(\frac{17}{35}\right)\) |
| \(\chi_{421}(376,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{35}\right)\) | \(e\left(\frac{22}{35}\right)\) | \(e\left(\frac{6}{35}\right)\) | \(e\left(\frac{29}{35}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{13}{35}\right)\) | \(e\left(\frac{9}{35}\right)\) | \(e\left(\frac{9}{35}\right)\) | \(e\left(\frac{32}{35}\right)\) | \(e\left(\frac{8}{35}\right)\) |
| \(\chi_{421}(414,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{35}\right)\) | \(e\left(\frac{2}{35}\right)\) | \(e\left(\frac{26}{35}\right)\) | \(e\left(\frac{9}{35}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{33}{35}\right)\) | \(e\left(\frac{4}{35}\right)\) | \(e\left(\frac{4}{35}\right)\) | \(e\left(\frac{22}{35}\right)\) | \(e\left(\frac{23}{35}\right)\) |