from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6369, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([24,24,29]))
pari: [g,chi] = znchar(Mod(1220,6369))
Basic properties
| Modulus: | \(6369\) | |
| Conductor: | \(6369\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(48\) | 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 6369.cn
\(\chi_{6369}(131,\cdot)\) \(\chi_{6369}(197,\cdot)\) \(\chi_{6369}(1154,\cdot)\) \(\chi_{6369}(1220,\cdot)\) \(\chi_{6369}(2144,\cdot)\) \(\chi_{6369}(2375,\cdot)\) \(\chi_{6369}(2738,\cdot)\) \(\chi_{6369}(3134,\cdot)\) \(\chi_{6369}(3233,\cdot)\) \(\chi_{6369}(3695,\cdot)\) \(\chi_{6369}(4025,\cdot)\) \(\chi_{6369}(4487,\cdot)\) \(\chi_{6369}(4586,\cdot)\) \(\chi_{6369}(4982,\cdot)\) \(\chi_{6369}(5345,\cdot)\) \(\chi_{6369}(5576,\cdot)\)
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: | \(\Q(\zeta_{48})\) |
| Fixed field: | Number field defined by a degree 48 polynomial |
Values on generators
\((4247,4633,2707)\) → \((-1,-1,e\left(\frac{29}{48}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(13\) | \(14\) | \(16\) | \(17\) |
| \( \chi_{ 6369 }(1220, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{5}{48}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{35}{48}\right)\) |
sage: chi.jacobi_sum(n)