from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2368, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([24,3,28]))
pari: [g,chi] = znchar(Mod(251,2368))
Basic properties
| Modulus: | \(2368\) | |
| Conductor: | \(2368\) | 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 2368.dh
\(\chi_{2368}(251,\cdot)\) \(\chi_{2368}(347,\cdot)\) \(\chi_{2368}(467,\cdot)\) \(\chi_{2368}(563,\cdot)\) \(\chi_{2368}(843,\cdot)\) \(\chi_{2368}(939,\cdot)\) \(\chi_{2368}(1059,\cdot)\) \(\chi_{2368}(1155,\cdot)\) \(\chi_{2368}(1435,\cdot)\) \(\chi_{2368}(1531,\cdot)\) \(\chi_{2368}(1651,\cdot)\) \(\chi_{2368}(1747,\cdot)\) \(\chi_{2368}(2027,\cdot)\) \(\chi_{2368}(2123,\cdot)\) \(\chi_{2368}(2243,\cdot)\) \(\chi_{2368}(2339,\cdot)\)
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
\((1407,1925,705)\) → \((-1,e\left(\frac{1}{16}\right),e\left(\frac{7}{12}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 2368 }(251, a) \) | \(1\) | \(1\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{23}{48}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{17}{48}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{17}{48}\right)\) | \(e\left(\frac{31}{48}\right)\) |
sage: chi.jacobi_sum(n)