from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2352, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,0,21,5]))
pari: [g,chi] = znchar(Mod(47,2352))
Basic properties
| Modulus: | \(2352\) | |
| Conductor: | \(588\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{588}(47,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2352.dj
\(\chi_{2352}(47,\cdot)\) \(\chi_{2352}(143,\cdot)\) \(\chi_{2352}(383,\cdot)\) \(\chi_{2352}(479,\cdot)\) \(\chi_{2352}(719,\cdot)\) \(\chi_{2352}(1055,\cdot)\) \(\chi_{2352}(1151,\cdot)\) \(\chi_{2352}(1487,\cdot)\) \(\chi_{2352}(1727,\cdot)\) \(\chi_{2352}(1823,\cdot)\) \(\chi_{2352}(2063,\cdot)\) \(\chi_{2352}(2159,\cdot)\)
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: | 42.0.5436948860695888782893198886016377149049148530413040928765325951335574011833955525853184.1 |
Values on generators
\((1471,1765,785,2257)\) → \((-1,1,-1,e\left(\frac{5}{42}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 2352 }(47, a) \) | \(-1\) | \(1\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{17}{21}\right)\) |
sage: chi.jacobi_sum(n)