from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(750, base_ring=CyclotomicField(50))
M = H._module
chi = DirichletCharacter(H, M([0,24]))
chi.galois_orbit()
[g,chi] = znchar(Mod(31,750))
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Basic properties
Modulus: | \(750\) | |
Conductor: | \(125\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(25\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from 125.g | 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: | \(\Q(\zeta_{25})\) |
Fixed field: | Number field defined by a degree 25 polynomial |
Characters in Galois orbit
Character | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
---|---|---|---|---|---|---|---|---|---|---|---|---|
\(\chi_{750}(31,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{12}{25}\right)\) | \(e\left(\frac{18}{25}\right)\) | \(e\left(\frac{1}{25}\right)\) | \(e\left(\frac{16}{25}\right)\) | \(e\left(\frac{22}{25}\right)\) | \(e\left(\frac{19}{25}\right)\) | \(e\left(\frac{1}{25}\right)\) | \(e\left(\frac{23}{25}\right)\) | \(e\left(\frac{3}{25}\right)\) |
\(\chi_{750}(61,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{14}{25}\right)\) | \(e\left(\frac{21}{25}\right)\) | \(e\left(\frac{22}{25}\right)\) | \(e\left(\frac{2}{25}\right)\) | \(e\left(\frac{9}{25}\right)\) | \(e\left(\frac{18}{25}\right)\) | \(e\left(\frac{22}{25}\right)\) | \(e\left(\frac{6}{25}\right)\) | \(e\left(\frac{16}{25}\right)\) |
\(\chi_{750}(91,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{6}{25}\right)\) | \(e\left(\frac{9}{25}\right)\) | \(e\left(\frac{13}{25}\right)\) | \(e\left(\frac{8}{25}\right)\) | \(e\left(\frac{11}{25}\right)\) | \(e\left(\frac{22}{25}\right)\) | \(e\left(\frac{13}{25}\right)\) | \(e\left(\frac{24}{25}\right)\) | \(e\left(\frac{14}{25}\right)\) |
\(\chi_{750}(121,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{13}{25}\right)\) | \(e\left(\frac{7}{25}\right)\) | \(e\left(\frac{24}{25}\right)\) | \(e\left(\frac{9}{25}\right)\) | \(e\left(\frac{3}{25}\right)\) | \(e\left(\frac{6}{25}\right)\) | \(e\left(\frac{24}{25}\right)\) | \(e\left(\frac{2}{25}\right)\) | \(e\left(\frac{22}{25}\right)\) |
\(\chi_{750}(181,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{22}{25}\right)\) | \(e\left(\frac{8}{25}\right)\) | \(e\left(\frac{6}{25}\right)\) | \(e\left(\frac{21}{25}\right)\) | \(e\left(\frac{7}{25}\right)\) | \(e\left(\frac{14}{25}\right)\) | \(e\left(\frac{6}{25}\right)\) | \(e\left(\frac{13}{25}\right)\) | \(e\left(\frac{18}{25}\right)\) |
\(\chi_{750}(211,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{24}{25}\right)\) | \(e\left(\frac{11}{25}\right)\) | \(e\left(\frac{2}{25}\right)\) | \(e\left(\frac{7}{25}\right)\) | \(e\left(\frac{19}{25}\right)\) | \(e\left(\frac{13}{25}\right)\) | \(e\left(\frac{2}{25}\right)\) | \(e\left(\frac{21}{25}\right)\) | \(e\left(\frac{6}{25}\right)\) |
\(\chi_{750}(241,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{16}{25}\right)\) | \(e\left(\frac{24}{25}\right)\) | \(e\left(\frac{18}{25}\right)\) | \(e\left(\frac{13}{25}\right)\) | \(e\left(\frac{21}{25}\right)\) | \(e\left(\frac{17}{25}\right)\) | \(e\left(\frac{18}{25}\right)\) | \(e\left(\frac{14}{25}\right)\) | \(e\left(\frac{4}{25}\right)\) |
\(\chi_{750}(271,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{23}{25}\right)\) | \(e\left(\frac{22}{25}\right)\) | \(e\left(\frac{4}{25}\right)\) | \(e\left(\frac{14}{25}\right)\) | \(e\left(\frac{13}{25}\right)\) | \(e\left(\frac{1}{25}\right)\) | \(e\left(\frac{4}{25}\right)\) | \(e\left(\frac{17}{25}\right)\) | \(e\left(\frac{12}{25}\right)\) |
\(\chi_{750}(331,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{7}{25}\right)\) | \(e\left(\frac{23}{25}\right)\) | \(e\left(\frac{11}{25}\right)\) | \(e\left(\frac{1}{25}\right)\) | \(e\left(\frac{17}{25}\right)\) | \(e\left(\frac{9}{25}\right)\) | \(e\left(\frac{11}{25}\right)\) | \(e\left(\frac{3}{25}\right)\) | \(e\left(\frac{8}{25}\right)\) |
\(\chi_{750}(361,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{9}{25}\right)\) | \(e\left(\frac{1}{25}\right)\) | \(e\left(\frac{7}{25}\right)\) | \(e\left(\frac{12}{25}\right)\) | \(e\left(\frac{4}{25}\right)\) | \(e\left(\frac{8}{25}\right)\) | \(e\left(\frac{7}{25}\right)\) | \(e\left(\frac{11}{25}\right)\) | \(e\left(\frac{21}{25}\right)\) |
\(\chi_{750}(391,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{1}{25}\right)\) | \(e\left(\frac{14}{25}\right)\) | \(e\left(\frac{23}{25}\right)\) | \(e\left(\frac{18}{25}\right)\) | \(e\left(\frac{6}{25}\right)\) | \(e\left(\frac{12}{25}\right)\) | \(e\left(\frac{23}{25}\right)\) | \(e\left(\frac{4}{25}\right)\) | \(e\left(\frac{19}{25}\right)\) |
\(\chi_{750}(421,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{8}{25}\right)\) | \(e\left(\frac{12}{25}\right)\) | \(e\left(\frac{9}{25}\right)\) | \(e\left(\frac{19}{25}\right)\) | \(e\left(\frac{23}{25}\right)\) | \(e\left(\frac{21}{25}\right)\) | \(e\left(\frac{9}{25}\right)\) | \(e\left(\frac{7}{25}\right)\) | \(e\left(\frac{2}{25}\right)\) |
\(\chi_{750}(481,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{17}{25}\right)\) | \(e\left(\frac{13}{25}\right)\) | \(e\left(\frac{16}{25}\right)\) | \(e\left(\frac{6}{25}\right)\) | \(e\left(\frac{2}{25}\right)\) | \(e\left(\frac{4}{25}\right)\) | \(e\left(\frac{16}{25}\right)\) | \(e\left(\frac{18}{25}\right)\) | \(e\left(\frac{23}{25}\right)\) |
\(\chi_{750}(511,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{19}{25}\right)\) | \(e\left(\frac{16}{25}\right)\) | \(e\left(\frac{12}{25}\right)\) | \(e\left(\frac{17}{25}\right)\) | \(e\left(\frac{14}{25}\right)\) | \(e\left(\frac{3}{25}\right)\) | \(e\left(\frac{12}{25}\right)\) | \(e\left(\frac{1}{25}\right)\) | \(e\left(\frac{11}{25}\right)\) |
\(\chi_{750}(541,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{11}{25}\right)\) | \(e\left(\frac{4}{25}\right)\) | \(e\left(\frac{3}{25}\right)\) | \(e\left(\frac{23}{25}\right)\) | \(e\left(\frac{16}{25}\right)\) | \(e\left(\frac{7}{25}\right)\) | \(e\left(\frac{3}{25}\right)\) | \(e\left(\frac{19}{25}\right)\) | \(e\left(\frac{9}{25}\right)\) |
\(\chi_{750}(571,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{18}{25}\right)\) | \(e\left(\frac{2}{25}\right)\) | \(e\left(\frac{14}{25}\right)\) | \(e\left(\frac{24}{25}\right)\) | \(e\left(\frac{8}{25}\right)\) | \(e\left(\frac{16}{25}\right)\) | \(e\left(\frac{14}{25}\right)\) | \(e\left(\frac{22}{25}\right)\) | \(e\left(\frac{17}{25}\right)\) |
\(\chi_{750}(631,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{2}{25}\right)\) | \(e\left(\frac{3}{25}\right)\) | \(e\left(\frac{21}{25}\right)\) | \(e\left(\frac{11}{25}\right)\) | \(e\left(\frac{12}{25}\right)\) | \(e\left(\frac{24}{25}\right)\) | \(e\left(\frac{21}{25}\right)\) | \(e\left(\frac{8}{25}\right)\) | \(e\left(\frac{13}{25}\right)\) |
\(\chi_{750}(661,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{4}{25}\right)\) | \(e\left(\frac{6}{25}\right)\) | \(e\left(\frac{17}{25}\right)\) | \(e\left(\frac{22}{25}\right)\) | \(e\left(\frac{24}{25}\right)\) | \(e\left(\frac{23}{25}\right)\) | \(e\left(\frac{17}{25}\right)\) | \(e\left(\frac{16}{25}\right)\) | \(e\left(\frac{1}{25}\right)\) |
\(\chi_{750}(691,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{21}{25}\right)\) | \(e\left(\frac{19}{25}\right)\) | \(e\left(\frac{8}{25}\right)\) | \(e\left(\frac{3}{25}\right)\) | \(e\left(\frac{1}{25}\right)\) | \(e\left(\frac{2}{25}\right)\) | \(e\left(\frac{8}{25}\right)\) | \(e\left(\frac{9}{25}\right)\) | \(e\left(\frac{24}{25}\right)\) |
\(\chi_{750}(721,\cdot)\) | \(1\) | \(1\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{3}{25}\right)\) | \(e\left(\frac{17}{25}\right)\) | \(e\left(\frac{19}{25}\right)\) | \(e\left(\frac{4}{25}\right)\) | \(e\left(\frac{18}{25}\right)\) | \(e\left(\frac{11}{25}\right)\) | \(e\left(\frac{19}{25}\right)\) | \(e\left(\frac{12}{25}\right)\) | \(e\left(\frac{7}{25}\right)\) |