from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2009, base_ring=CyclotomicField(70))
M = H._module
chi = DirichletCharacter(H, M([60,42]))
chi.galois_orbit()
[g,chi] = znchar(Mod(57,2009))
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: | \(35\) | 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 | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
---|---|---|---|---|---|---|---|---|---|---|---|---|
\(\chi_{2009}(57,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{31}{35}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{27}{35}\right)\) | \(e\left(\frac{2}{35}\right)\) | \(e\left(\frac{26}{35}\right)\) | \(e\left(\frac{23}{35}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{33}{35}\right)\) | \(e\left(\frac{3}{35}\right)\) | \(e\left(\frac{22}{35}\right)\) |
\(\chi_{2009}(78,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{33}{35}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{31}{35}\right)\) | \(e\left(\frac{1}{35}\right)\) | \(e\left(\frac{13}{35}\right)\) | \(e\left(\frac{29}{35}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{34}{35}\right)\) | \(e\left(\frac{19}{35}\right)\) | \(e\left(\frac{11}{35}\right)\) |
\(\chi_{2009}(92,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{32}{35}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{29}{35}\right)\) | \(e\left(\frac{19}{35}\right)\) | \(e\left(\frac{2}{35}\right)\) | \(e\left(\frac{26}{35}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{16}{35}\right)\) | \(e\left(\frac{11}{35}\right)\) | \(e\left(\frac{34}{35}\right)\) |
\(\chi_{2009}(141,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{35}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{8}{35}\right)\) | \(e\left(\frac{33}{35}\right)\) | \(e\left(\frac{9}{35}\right)\) | \(e\left(\frac{12}{35}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{2}{35}\right)\) | \(e\left(\frac{32}{35}\right)\) | \(e\left(\frac{13}{35}\right)\) |
\(\chi_{2009}(365,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{23}{35}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{11}{35}\right)\) | \(e\left(\frac{6}{35}\right)\) | \(e\left(\frac{8}{35}\right)\) | \(e\left(\frac{34}{35}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{29}{35}\right)\) | \(e\left(\frac{9}{35}\right)\) | \(e\left(\frac{31}{35}\right)\) |
\(\chi_{2009}(379,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{22}{35}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{9}{35}\right)\) | \(e\left(\frac{24}{35}\right)\) | \(e\left(\frac{32}{35}\right)\) | \(e\left(\frac{31}{35}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{11}{35}\right)\) | \(e\left(\frac{1}{35}\right)\) | \(e\left(\frac{19}{35}\right)\) |
\(\chi_{2009}(428,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{29}{35}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{23}{35}\right)\) | \(e\left(\frac{3}{35}\right)\) | \(e\left(\frac{4}{35}\right)\) | \(e\left(\frac{17}{35}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{32}{35}\right)\) | \(e\left(\frac{22}{35}\right)\) | \(e\left(\frac{33}{35}\right)\) |
\(\chi_{2009}(631,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{35}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{22}{35}\right)\) | \(e\left(\frac{12}{35}\right)\) | \(e\left(\frac{16}{35}\right)\) | \(e\left(\frac{33}{35}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{23}{35}\right)\) | \(e\left(\frac{18}{35}\right)\) | \(e\left(\frac{27}{35}\right)\) |
\(\chi_{2009}(652,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{35}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{26}{35}\right)\) | \(e\left(\frac{11}{35}\right)\) | \(e\left(\frac{3}{35}\right)\) | \(e\left(\frac{4}{35}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{24}{35}\right)\) | \(e\left(\frac{34}{35}\right)\) | \(e\left(\frac{16}{35}\right)\) |
\(\chi_{2009}(666,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{12}{35}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{24}{35}\right)\) | \(e\left(\frac{29}{35}\right)\) | \(e\left(\frac{27}{35}\right)\) | \(e\left(\frac{1}{35}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{6}{35}\right)\) | \(e\left(\frac{26}{35}\right)\) | \(e\left(\frac{4}{35}\right)\) |
\(\chi_{2009}(715,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{19}{35}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{3}{35}\right)\) | \(e\left(\frac{8}{35}\right)\) | \(e\left(\frac{34}{35}\right)\) | \(e\left(\frac{22}{35}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{27}{35}\right)\) | \(e\left(\frac{12}{35}\right)\) | \(e\left(\frac{18}{35}\right)\) |
\(\chi_{2009}(918,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{35}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{2}{35}\right)\) | \(e\left(\frac{17}{35}\right)\) | \(e\left(\frac{11}{35}\right)\) | \(e\left(\frac{3}{35}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{18}{35}\right)\) | \(e\left(\frac{8}{35}\right)\) | \(e\left(\frac{12}{35}\right)\) |
\(\chi_{2009}(939,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{35}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{6}{35}\right)\) | \(e\left(\frac{16}{35}\right)\) | \(e\left(\frac{33}{35}\right)\) | \(e\left(\frac{9}{35}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{19}{35}\right)\) | \(e\left(\frac{24}{35}\right)\) | \(e\left(\frac{1}{35}\right)\) |
\(\chi_{2009}(953,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{35}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{4}{35}\right)\) | \(e\left(\frac{34}{35}\right)\) | \(e\left(\frac{22}{35}\right)\) | \(e\left(\frac{6}{35}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{1}{35}\right)\) | \(e\left(\frac{16}{35}\right)\) | \(e\left(\frac{24}{35}\right)\) |
\(\chi_{2009}(1002,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{9}{35}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{18}{35}\right)\) | \(e\left(\frac{13}{35}\right)\) | \(e\left(\frac{29}{35}\right)\) | \(e\left(\frac{27}{35}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{22}{35}\right)\) | \(e\left(\frac{2}{35}\right)\) | \(e\left(\frac{3}{35}\right)\) |
\(\chi_{2009}(1205,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{26}{35}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{17}{35}\right)\) | \(e\left(\frac{22}{35}\right)\) | \(e\left(\frac{6}{35}\right)\) | \(e\left(\frac{8}{35}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{13}{35}\right)\) | \(e\left(\frac{33}{35}\right)\) | \(e\left(\frac{32}{35}\right)\) |
\(\chi_{2009}(1240,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{27}{35}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{19}{35}\right)\) | \(e\left(\frac{4}{35}\right)\) | \(e\left(\frac{17}{35}\right)\) | \(e\left(\frac{11}{35}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{31}{35}\right)\) | \(e\left(\frac{6}{35}\right)\) | \(e\left(\frac{9}{35}\right)\) |
\(\chi_{2009}(1289,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{34}{35}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{33}{35}\right)\) | \(e\left(\frac{18}{35}\right)\) | \(e\left(\frac{24}{35}\right)\) | \(e\left(\frac{32}{35}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{17}{35}\right)\) | \(e\left(\frac{27}{35}\right)\) | \(e\left(\frac{23}{35}\right)\) |
\(\chi_{2009}(1492,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{16}{35}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{32}{35}\right)\) | \(e\left(\frac{27}{35}\right)\) | \(e\left(\frac{1}{35}\right)\) | \(e\left(\frac{13}{35}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{8}{35}\right)\) | \(e\left(\frac{23}{35}\right)\) | \(e\left(\frac{17}{35}\right)\) |
\(\chi_{2009}(1513,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{18}{35}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{1}{35}\right)\) | \(e\left(\frac{26}{35}\right)\) | \(e\left(\frac{23}{35}\right)\) | \(e\left(\frac{19}{35}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{9}{35}\right)\) | \(e\left(\frac{4}{35}\right)\) | \(e\left(\frac{6}{35}\right)\) |
\(\chi_{2009}(1527,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{17}{35}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{34}{35}\right)\) | \(e\left(\frac{9}{35}\right)\) | \(e\left(\frac{12}{35}\right)\) | \(e\left(\frac{16}{35}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{26}{35}\right)\) | \(e\left(\frac{31}{35}\right)\) | \(e\left(\frac{29}{35}\right)\) |
\(\chi_{2009}(1576,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{24}{35}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{13}{35}\right)\) | \(e\left(\frac{23}{35}\right)\) | \(e\left(\frac{19}{35}\right)\) | \(e\left(\frac{2}{35}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{12}{35}\right)\) | \(e\left(\frac{17}{35}\right)\) | \(e\left(\frac{8}{35}\right)\) |
\(\chi_{2009}(1779,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{6}{35}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{12}{35}\right)\) | \(e\left(\frac{32}{35}\right)\) | \(e\left(\frac{31}{35}\right)\) | \(e\left(\frac{18}{35}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{3}{35}\right)\) | \(e\left(\frac{13}{35}\right)\) | \(e\left(\frac{2}{35}\right)\) |
\(\chi_{2009}(1800,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{35}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{16}{35}\right)\) | \(e\left(\frac{31}{35}\right)\) | \(e\left(\frac{18}{35}\right)\) | \(e\left(\frac{24}{35}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{4}{35}\right)\) | \(e\left(\frac{29}{35}\right)\) | \(e\left(\frac{26}{35}\right)\) |