from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4719, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([22,32,11]))
pari: [g,chi] = znchar(Mod(320,4719))
Basic properties
| Modulus: | \(4719\) | |
| Conductor: | \(4719\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(44\) | 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)
|
Galois orbit 4719.ce
\(\chi_{4719}(320,\cdot)\) \(\chi_{4719}(551,\cdot)\) \(\chi_{4719}(749,\cdot)\) \(\chi_{4719}(980,\cdot)\) \(\chi_{4719}(1178,\cdot)\) \(\chi_{4719}(1409,\cdot)\) \(\chi_{4719}(1607,\cdot)\) \(\chi_{4719}(1838,\cdot)\) \(\chi_{4719}(2036,\cdot)\) \(\chi_{4719}(2267,\cdot)\) \(\chi_{4719}(2465,\cdot)\) \(\chi_{4719}(2696,\cdot)\) \(\chi_{4719}(2894,\cdot)\) \(\chi_{4719}(3125,\cdot)\) \(\chi_{4719}(3323,\cdot)\) \(\chi_{4719}(3554,\cdot)\) \(\chi_{4719}(3983,\cdot)\) \(\chi_{4719}(4181,\cdot)\) \(\chi_{4719}(4412,\cdot)\) \(\chi_{4719}(4610,\cdot)\)
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: | \(\Q(\zeta_{44})\) |
| Fixed field: | Number field defined by a degree 44 polynomial |
Values on generators
\((1574,3511,4357)\) → \((-1,e\left(\frac{8}{11}\right),i)\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 4719 }(320, a) \) | \(1\) | \(1\) | \(e\left(\frac{21}{44}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{25}{44}\right)\) | \(e\left(\frac{37}{44}\right)\) | \(e\left(\frac{19}{44}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{27}{44}\right)\) |
sage: chi.jacobi_sum(n)