from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2352, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,21,21,38]))
pari: [g,chi] = znchar(Mod(23,2352))
Basic properties
| Modulus: | \(2352\) | |
| Conductor: | \(1176\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1176}(611,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2352.df
\(\chi_{2352}(23,\cdot)\) \(\chi_{2352}(359,\cdot)\) \(\chi_{2352}(599,\cdot)\) \(\chi_{2352}(695,\cdot)\) \(\chi_{2352}(935,\cdot)\) \(\chi_{2352}(1031,\cdot)\) \(\chi_{2352}(1271,\cdot)\) \(\chi_{2352}(1367,\cdot)\) \(\chi_{2352}(1607,\cdot)\) \(\chi_{2352}(1703,\cdot)\) \(\chi_{2352}(1943,\cdot)\) \(\chi_{2352}(2279,\cdot)\)
sage: chi.galois_orbit()
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Related number fields
| Field of values: | \(\Q(\zeta_{21})\) |
| Fixed field: | Number field defined by a degree 42 polynomial |
Values on generators
\((1471,1765,785,2257)\) → \((-1,-1,-1,e\left(\frac{19}{21}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 2352 }(23, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{19}{42}\right)\) |
sage: chi.jacobi_sum(n)