from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2009, base_ring=CyclotomicField(70))
M = H._module
chi = DirichletCharacter(H, M([50,63]))
chi.galois_orbit()
[g,chi] = znchar(Mod(64,2009))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
Modulus: | \(2009\) | |
Conductor: | \(2009\) | 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: | 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 70 polynomial |
Characters in Galois orbit
Character | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
---|---|---|---|---|---|---|---|---|---|---|---|---|
\(\chi_{2009}(64,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{34}{35}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{33}{35}\right)\) | \(e\left(\frac{18}{35}\right)\) | \(e\left(\frac{13}{70}\right)\) | \(e\left(\frac{32}{35}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{17}{35}\right)\) | \(e\left(\frac{19}{70}\right)\) | \(e\left(\frac{11}{70}\right)\) |
\(\chi_{2009}(113,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{27}{35}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{19}{35}\right)\) | \(e\left(\frac{4}{35}\right)\) | \(e\left(\frac{69}{70}\right)\) | \(e\left(\frac{11}{35}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{31}{35}\right)\) | \(e\left(\frac{47}{70}\right)\) | \(e\left(\frac{53}{70}\right)\) |
\(\chi_{2009}(127,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{33}{35}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{31}{35}\right)\) | \(e\left(\frac{1}{35}\right)\) | \(e\left(\frac{61}{70}\right)\) | \(e\left(\frac{29}{35}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{34}{35}\right)\) | \(e\left(\frac{3}{70}\right)\) | \(e\left(\frac{57}{70}\right)\) |
\(\chi_{2009}(351,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{24}{35}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{13}{35}\right)\) | \(e\left(\frac{23}{35}\right)\) | \(e\left(\frac{3}{70}\right)\) | \(e\left(\frac{2}{35}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{12}{35}\right)\) | \(e\left(\frac{69}{70}\right)\) | \(e\left(\frac{51}{70}\right)\) |
\(\chi_{2009}(400,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{35}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{34}{35}\right)\) | \(e\left(\frac{9}{35}\right)\) | \(e\left(\frac{59}{70}\right)\) | \(e\left(\frac{16}{35}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{26}{35}\right)\) | \(e\left(\frac{27}{70}\right)\) | \(e\left(\frac{23}{70}\right)\) |
\(\chi_{2009}(414,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{35}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{11}{35}\right)\) | \(e\left(\frac{6}{35}\right)\) | \(e\left(\frac{51}{70}\right)\) | \(e\left(\frac{34}{35}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{29}{35}\right)\) | \(e\left(\frac{53}{70}\right)\) | \(e\left(\frac{27}{70}\right)\) |
\(\chi_{2009}(435,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{35}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{22}{35}\right)\) | \(e\left(\frac{12}{35}\right)\) | \(e\left(\frac{67}{70}\right)\) | \(e\left(\frac{33}{35}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{23}{35}\right)\) | \(e\left(\frac{1}{70}\right)\) | \(e\left(\frac{19}{70}\right)\) |
\(\chi_{2009}(701,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{35}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{26}{35}\right)\) | \(e\left(\frac{11}{35}\right)\) | \(e\left(\frac{41}{70}\right)\) | \(e\left(\frac{4}{35}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{24}{35}\right)\) | \(e\left(\frac{33}{70}\right)\) | \(e\left(\frac{67}{70}\right)\) |
\(\chi_{2009}(722,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{35}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{2}{35}\right)\) | \(e\left(\frac{17}{35}\right)\) | \(e\left(\frac{57}{70}\right)\) | \(e\left(\frac{3}{35}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{18}{35}\right)\) | \(e\left(\frac{51}{70}\right)\) | \(e\left(\frac{59}{70}\right)\) |
\(\chi_{2009}(925,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{35}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{8}{35}\right)\) | \(e\left(\frac{33}{35}\right)\) | \(e\left(\frac{53}{70}\right)\) | \(e\left(\frac{12}{35}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{2}{35}\right)\) | \(e\left(\frac{29}{70}\right)\) | \(e\left(\frac{61}{70}\right)\) |
\(\chi_{2009}(974,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{32}{35}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{29}{35}\right)\) | \(e\left(\frac{19}{35}\right)\) | \(e\left(\frac{39}{70}\right)\) | \(e\left(\frac{26}{35}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{16}{35}\right)\) | \(e\left(\frac{57}{70}\right)\) | \(e\left(\frac{33}{70}\right)\) |
\(\chi_{2009}(988,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{35}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{6}{35}\right)\) | \(e\left(\frac{16}{35}\right)\) | \(e\left(\frac{31}{70}\right)\) | \(e\left(\frac{9}{35}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{19}{35}\right)\) | \(e\left(\frac{13}{70}\right)\) | \(e\left(\frac{37}{70}\right)\) |
\(\chi_{2009}(1009,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{26}{35}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{17}{35}\right)\) | \(e\left(\frac{22}{35}\right)\) | \(e\left(\frac{47}{70}\right)\) | \(e\left(\frac{8}{35}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{13}{35}\right)\) | \(e\left(\frac{31}{70}\right)\) | \(e\left(\frac{29}{70}\right)\) |
\(\chi_{2009}(1212,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{35}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{23}{35}\right)\) | \(e\left(\frac{3}{35}\right)\) | \(e\left(\frac{43}{70}\right)\) | \(e\left(\frac{17}{35}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{32}{35}\right)\) | \(e\left(\frac{9}{70}\right)\) | \(e\left(\frac{31}{70}\right)\) |
\(\chi_{2009}(1261,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{22}{35}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{9}{35}\right)\) | \(e\left(\frac{24}{35}\right)\) | \(e\left(\frac{29}{70}\right)\) | \(e\left(\frac{31}{35}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{11}{35}\right)\) | \(e\left(\frac{37}{70}\right)\) | \(e\left(\frac{3}{70}\right)\) |
\(\chi_{2009}(1296,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{16}{35}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{32}{35}\right)\) | \(e\left(\frac{27}{35}\right)\) | \(e\left(\frac{37}{70}\right)\) | \(e\left(\frac{13}{35}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{8}{35}\right)\) | \(e\left(\frac{11}{70}\right)\) | \(e\left(\frac{69}{70}\right)\) |
\(\chi_{2009}(1499,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{35}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{3}{35}\right)\) | \(e\left(\frac{8}{35}\right)\) | \(e\left(\frac{33}{70}\right)\) | \(e\left(\frac{22}{35}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{27}{35}\right)\) | \(e\left(\frac{59}{70}\right)\) | \(e\left(\frac{1}{70}\right)\) |
\(\chi_{2009}(1548,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{12}{35}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{24}{35}\right)\) | \(e\left(\frac{29}{35}\right)\) | \(e\left(\frac{19}{70}\right)\) | \(e\left(\frac{1}{35}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{6}{35}\right)\) | \(e\left(\frac{17}{70}\right)\) | \(e\left(\frac{43}{70}\right)\) |
\(\chi_{2009}(1562,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{18}{35}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{1}{35}\right)\) | \(e\left(\frac{26}{35}\right)\) | \(e\left(\frac{11}{70}\right)\) | \(e\left(\frac{19}{35}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{9}{35}\right)\) | \(e\left(\frac{43}{70}\right)\) | \(e\left(\frac{47}{70}\right)\) |
\(\chi_{2009}(1583,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{6}{35}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{12}{35}\right)\) | \(e\left(\frac{32}{35}\right)\) | \(e\left(\frac{27}{70}\right)\) | \(e\left(\frac{18}{35}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{3}{35}\right)\) | \(e\left(\frac{61}{70}\right)\) | \(e\left(\frac{39}{70}\right)\) |
\(\chi_{2009}(1786,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{35}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{18}{35}\right)\) | \(e\left(\frac{13}{35}\right)\) | \(e\left(\frac{23}{70}\right)\) | \(e\left(\frac{27}{35}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{22}{35}\right)\) | \(e\left(\frac{39}{70}\right)\) | \(e\left(\frac{41}{70}\right)\) |
\(\chi_{2009}(1835,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{35}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{4}{35}\right)\) | \(e\left(\frac{34}{35}\right)\) | \(e\left(\frac{9}{70}\right)\) | \(e\left(\frac{6}{35}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{1}{35}\right)\) | \(e\left(\frac{67}{70}\right)\) | \(e\left(\frac{13}{70}\right)\) |
\(\chi_{2009}(1849,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{35}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{16}{35}\right)\) | \(e\left(\frac{31}{35}\right)\) | \(e\left(\frac{1}{70}\right)\) | \(e\left(\frac{24}{35}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{4}{35}\right)\) | \(e\left(\frac{23}{70}\right)\) | \(e\left(\frac{17}{70}\right)\) |
\(\chi_{2009}(1870,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{35}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{27}{35}\right)\) | \(e\left(\frac{2}{35}\right)\) | \(e\left(\frac{17}{70}\right)\) | \(e\left(\frac{23}{35}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{33}{35}\right)\) | \(e\left(\frac{41}{70}\right)\) | \(e\left(\frac{9}{70}\right)\) |