from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(325, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([3,5]))
chi.galois_orbit()
[g,chi] = znchar(Mod(2,325))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
Modulus: | \(325\) | |
Conductor: | \(325\) | 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: | even | 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 | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(14\) |
---|---|---|---|---|---|---|---|---|---|---|---|---|
\(\chi_{325}(2,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{3}{10}\right)\) |
\(\chi_{325}(33,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{9}{10}\right)\) |
\(\chi_{325}(63,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{7}{10}\right)\) |
\(\chi_{325}(67,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{9}{10}\right)\) |
\(\chi_{325}(97,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) |
\(\chi_{325}(98,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{59}{60}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{3}{10}\right)\) |
\(\chi_{325}(128,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) |
\(\chi_{325}(162,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{41}{60}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{7}{10}\right)\) |
\(\chi_{325}(163,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{7}{10}\right)\) |
\(\chi_{325}(197,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) |
\(\chi_{325}(227,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{3}{10}\right)\) |
\(\chi_{325}(228,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) |
\(\chi_{325}(258,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{9}{10}\right)\) |
\(\chi_{325}(262,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{7}{10}\right)\) |
\(\chi_{325}(292,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{9}{10}\right)\) |
\(\chi_{325}(323,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{3}{10}\right)\) |