from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(319, base_ring=CyclotomicField(70))
M = H._module
chi = DirichletCharacter(H, M([63,15]))
chi.galois_orbit()
[g,chi] = znchar(Mod(6,319))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
| Modulus: | \(319\) | |
| Conductor: | \(319\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(70\) | 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: | $\Q(\zeta_{35})$ |
| Fixed field: | Number field defined by a degree 70 polynomial |
Characters in Galois orbit
| Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| \(\chi_{319}(6,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{4}{35}\right)\) | \(e\left(\frac{19}{70}\right)\) | \(e\left(\frac{8}{35}\right)\) | \(e\left(\frac{11}{35}\right)\) | \(e\left(\frac{27}{70}\right)\) | \(e\left(\frac{61}{70}\right)\) | \(e\left(\frac{12}{35}\right)\) | \(e\left(\frac{19}{35}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(-1\) |
| \(\chi_{319}(13,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{26}{35}\right)\) | \(e\left(\frac{1}{70}\right)\) | \(e\left(\frac{17}{35}\right)\) | \(e\left(\frac{19}{35}\right)\) | \(e\left(\frac{53}{70}\right)\) | \(e\left(\frac{29}{70}\right)\) | \(e\left(\frac{8}{35}\right)\) | \(e\left(\frac{1}{35}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(-1\) |
| \(\chi_{319}(35,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{11}{35}\right)\) | \(e\left(\frac{61}{70}\right)\) | \(e\left(\frac{22}{35}\right)\) | \(e\left(\frac{4}{35}\right)\) | \(e\left(\frac{13}{70}\right)\) | \(e\left(\frac{19}{70}\right)\) | \(e\left(\frac{33}{35}\right)\) | \(e\left(\frac{26}{35}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(-1\) |
| \(\chi_{319}(51,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{22}{35}\right)\) | \(e\left(\frac{17}{70}\right)\) | \(e\left(\frac{9}{35}\right)\) | \(e\left(\frac{8}{35}\right)\) | \(e\left(\frac{61}{70}\right)\) | \(e\left(\frac{3}{70}\right)\) | \(e\left(\frac{31}{35}\right)\) | \(e\left(\frac{17}{35}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(-1\) |
| \(\chi_{319}(62,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{27}{35}\right)\) | \(e\left(\frac{67}{70}\right)\) | \(e\left(\frac{19}{35}\right)\) | \(e\left(\frac{13}{35}\right)\) | \(e\left(\frac{51}{70}\right)\) | \(e\left(\frac{53}{70}\right)\) | \(e\left(\frac{11}{35}\right)\) | \(e\left(\frac{32}{35}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(-1\) |
| \(\chi_{319}(63,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{3}{35}\right)\) | \(e\left(\frac{23}{70}\right)\) | \(e\left(\frac{6}{35}\right)\) | \(e\left(\frac{17}{35}\right)\) | \(e\left(\frac{29}{70}\right)\) | \(e\left(\frac{37}{70}\right)\) | \(e\left(\frac{9}{35}\right)\) | \(e\left(\frac{23}{35}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(-1\) |
| \(\chi_{319}(96,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{23}{35}\right)\) | \(e\left(\frac{13}{70}\right)\) | \(e\left(\frac{11}{35}\right)\) | \(e\left(\frac{2}{35}\right)\) | \(e\left(\frac{59}{70}\right)\) | \(e\left(\frac{27}{70}\right)\) | \(e\left(\frac{34}{35}\right)\) | \(e\left(\frac{13}{35}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(-1\) |
| \(\chi_{319}(129,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{33}{35}\right)\) | \(e\left(\frac{43}{70}\right)\) | \(e\left(\frac{31}{35}\right)\) | \(e\left(\frac{12}{35}\right)\) | \(e\left(\frac{39}{70}\right)\) | \(e\left(\frac{57}{70}\right)\) | \(e\left(\frac{29}{35}\right)\) | \(e\left(\frac{8}{35}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(-1\) |
| \(\chi_{319}(138,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{29}{35}\right)\) | \(e\left(\frac{59}{70}\right)\) | \(e\left(\frac{23}{35}\right)\) | \(e\left(\frac{1}{35}\right)\) | \(e\left(\frac{47}{70}\right)\) | \(e\left(\frac{31}{70}\right)\) | \(e\left(\frac{17}{35}\right)\) | \(e\left(\frac{24}{35}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(-1\) |
| \(\chi_{319}(149,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{34}{35}\right)\) | \(e\left(\frac{39}{70}\right)\) | \(e\left(\frac{33}{35}\right)\) | \(e\left(\frac{6}{35}\right)\) | \(e\left(\frac{37}{70}\right)\) | \(e\left(\frac{11}{70}\right)\) | \(e\left(\frac{32}{35}\right)\) | \(e\left(\frac{4}{35}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(-1\) |
| \(\chi_{319}(150,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{17}{35}\right)\) | \(e\left(\frac{37}{70}\right)\) | \(e\left(\frac{34}{35}\right)\) | \(e\left(\frac{3}{35}\right)\) | \(e\left(\frac{1}{70}\right)\) | \(e\left(\frac{23}{70}\right)\) | \(e\left(\frac{16}{35}\right)\) | \(e\left(\frac{2}{35}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(-1\) |
| \(\chi_{319}(151,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{18}{35}\right)\) | \(e\left(\frac{33}{70}\right)\) | \(e\left(\frac{1}{35}\right)\) | \(e\left(\frac{32}{35}\right)\) | \(e\left(\frac{69}{70}\right)\) | \(e\left(\frac{47}{70}\right)\) | \(e\left(\frac{19}{35}\right)\) | \(e\left(\frac{33}{35}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(-1\) |
| \(\chi_{319}(167,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{1}{35}\right)\) | \(e\left(\frac{31}{70}\right)\) | \(e\left(\frac{2}{35}\right)\) | \(e\left(\frac{29}{35}\right)\) | \(e\left(\frac{33}{70}\right)\) | \(e\left(\frac{59}{70}\right)\) | \(e\left(\frac{3}{35}\right)\) | \(e\left(\frac{31}{35}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(-1\) |
| \(\chi_{319}(178,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{6}{35}\right)\) | \(e\left(\frac{11}{70}\right)\) | \(e\left(\frac{12}{35}\right)\) | \(e\left(\frac{34}{35}\right)\) | \(e\left(\frac{23}{70}\right)\) | \(e\left(\frac{39}{70}\right)\) | \(e\left(\frac{18}{35}\right)\) | \(e\left(\frac{11}{35}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(-1\) |
| \(\chi_{319}(183,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{2}{35}\right)\) | \(e\left(\frac{27}{70}\right)\) | \(e\left(\frac{4}{35}\right)\) | \(e\left(\frac{23}{35}\right)\) | \(e\left(\frac{31}{70}\right)\) | \(e\left(\frac{13}{70}\right)\) | \(e\left(\frac{6}{35}\right)\) | \(e\left(\frac{27}{35}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(-1\) |
| \(\chi_{319}(216,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{12}{35}\right)\) | \(e\left(\frac{57}{70}\right)\) | \(e\left(\frac{24}{35}\right)\) | \(e\left(\frac{33}{35}\right)\) | \(e\left(\frac{11}{70}\right)\) | \(e\left(\frac{43}{70}\right)\) | \(e\left(\frac{1}{35}\right)\) | \(e\left(\frac{22}{35}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(-1\) |
| \(\chi_{319}(237,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{24}{35}\right)\) | \(e\left(\frac{9}{70}\right)\) | \(e\left(\frac{13}{35}\right)\) | \(e\left(\frac{31}{35}\right)\) | \(e\left(\frac{57}{70}\right)\) | \(e\left(\frac{51}{70}\right)\) | \(e\left(\frac{2}{35}\right)\) | \(e\left(\frac{9}{35}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(-1\) |
| \(\chi_{319}(238,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{32}{35}\right)\) | \(e\left(\frac{47}{70}\right)\) | \(e\left(\frac{29}{35}\right)\) | \(e\left(\frac{18}{35}\right)\) | \(e\left(\frac{41}{70}\right)\) | \(e\left(\frac{33}{70}\right)\) | \(e\left(\frac{26}{35}\right)\) | \(e\left(\frac{12}{35}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(-1\) |
| \(\chi_{319}(266,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{31}{35}\right)\) | \(e\left(\frac{51}{70}\right)\) | \(e\left(\frac{27}{35}\right)\) | \(e\left(\frac{24}{35}\right)\) | \(e\left(\frac{43}{70}\right)\) | \(e\left(\frac{9}{70}\right)\) | \(e\left(\frac{23}{35}\right)\) | \(e\left(\frac{16}{35}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(-1\) |
| \(\chi_{319}(270,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{9}{35}\right)\) | \(e\left(\frac{69}{70}\right)\) | \(e\left(\frac{18}{35}\right)\) | \(e\left(\frac{16}{35}\right)\) | \(e\left(\frac{17}{70}\right)\) | \(e\left(\frac{41}{70}\right)\) | \(e\left(\frac{27}{35}\right)\) | \(e\left(\frac{34}{35}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(-1\) |
| \(\chi_{319}(283,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{8}{35}\right)\) | \(e\left(\frac{3}{70}\right)\) | \(e\left(\frac{16}{35}\right)\) | \(e\left(\frac{22}{35}\right)\) | \(e\left(\frac{19}{70}\right)\) | \(e\left(\frac{17}{70}\right)\) | \(e\left(\frac{24}{35}\right)\) | \(e\left(\frac{3}{35}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(-1\) |
| \(\chi_{319}(294,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{13}{35}\right)\) | \(e\left(\frac{53}{70}\right)\) | \(e\left(\frac{26}{35}\right)\) | \(e\left(\frac{27}{35}\right)\) | \(e\left(\frac{9}{70}\right)\) | \(e\left(\frac{67}{70}\right)\) | \(e\left(\frac{4}{35}\right)\) | \(e\left(\frac{18}{35}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(-1\) |
| \(\chi_{319}(299,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{16}{35}\right)\) | \(e\left(\frac{41}{70}\right)\) | \(e\left(\frac{32}{35}\right)\) | \(e\left(\frac{9}{35}\right)\) | \(e\left(\frac{3}{70}\right)\) | \(e\left(\frac{69}{70}\right)\) | \(e\left(\frac{13}{35}\right)\) | \(e\left(\frac{6}{35}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(-1\) |
| \(\chi_{319}(303,\cdot)\) | \(-1\) | \(1\) | \(e\left(\frac{19}{35}\right)\) | \(e\left(\frac{29}{70}\right)\) | \(e\left(\frac{3}{35}\right)\) | \(e\left(\frac{26}{35}\right)\) | \(e\left(\frac{67}{70}\right)\) | \(e\left(\frac{1}{70}\right)\) | \(e\left(\frac{22}{35}\right)\) | \(e\left(\frac{29}{35}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(-1\) |