from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6422, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([9,22]))
pari: [g,chi] = znchar(Mod(775,6422))
Basic properties
| Modulus: | \(6422\) | |
| Conductor: | \(247\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{247}(34,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6422.bz
\(\chi_{6422}(775,\cdot)\) \(\chi_{6422}(915,\cdot)\) \(\chi_{6422}(1591,\cdot)\) \(\chi_{6422}(1789,\cdot)\) \(\chi_{6422}(1929,\cdot)\) \(\chi_{6422}(2465,\cdot)\) \(\chi_{6422}(2605,\cdot)\) \(\chi_{6422}(2803,\cdot)\) \(\chi_{6422}(3281,\cdot)\) \(\chi_{6422}(3479,\cdot)\) \(\chi_{6422}(4155,\cdot)\) \(\chi_{6422}(6323,\cdot)\)
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: | \(\Q(\zeta_{36})\) |
| Fixed field: | 36.36.35817308260975188347623162969257824484378877602003672125976639688418239357.1 |
Values on generators
\((4903,4733)\) → \((i,e\left(\frac{11}{18}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(15\) | \(17\) | \(21\) | \(23\) | \(25\) |
| \( \chi_{ 6422 }(775, a) \) | \(1\) | \(1\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{1}{18}\right)\) |
sage: chi.jacobi_sum(n)