from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2304, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([24,3,8]))
pari: [g,chi] = znchar(Mod(911,2304))
Basic properties
| Modulus: | \(2304\) | |
| Conductor: | \(576\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(48\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{576}(443,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2304.bp
\(\chi_{2304}(47,\cdot)\) \(\chi_{2304}(239,\cdot)\) \(\chi_{2304}(335,\cdot)\) \(\chi_{2304}(527,\cdot)\) \(\chi_{2304}(623,\cdot)\) \(\chi_{2304}(815,\cdot)\) \(\chi_{2304}(911,\cdot)\) \(\chi_{2304}(1103,\cdot)\) \(\chi_{2304}(1199,\cdot)\) \(\chi_{2304}(1391,\cdot)\) \(\chi_{2304}(1487,\cdot)\) \(\chi_{2304}(1679,\cdot)\) \(\chi_{2304}(1775,\cdot)\) \(\chi_{2304}(1967,\cdot)\) \(\chi_{2304}(2063,\cdot)\) \(\chi_{2304}(2255,\cdot)\)
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
\((1279,2053,1793)\) → \((-1,e\left(\frac{1}{16}\right),e\left(\frac{1}{6}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 2304 }(911, a) \) | \(1\) | \(1\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{13}{48}\right)\) | \(i\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{1}{3}\right)\) |
sage: chi.jacobi_sum(n)