from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3315, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([24,36,40,45]))
pari: [g,chi] = znchar(Mod(23,3315))
Basic properties
| Modulus: | \(3315\) | |
| Conductor: | \(3315\) | 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: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3315.kf
\(\chi_{3315}(23,\cdot)\) \(\chi_{3315}(368,\cdot)\) \(\chi_{3315}(758,\cdot)\) \(\chi_{3315}(1187,\cdot)\) \(\chi_{3315}(1382,\cdot)\) \(\chi_{3315}(1388,\cdot)\) \(\chi_{3315}(1778,\cdot)\) \(\chi_{3315}(1928,\cdot)\) \(\chi_{3315}(1967,\cdot)\) \(\chi_{3315}(2162,\cdot)\) \(\chi_{3315}(2207,\cdot)\) \(\chi_{3315}(2318,\cdot)\) \(\chi_{3315}(2402,\cdot)\) \(\chi_{3315}(2948,\cdot)\) \(\chi_{3315}(2987,\cdot)\) \(\chi_{3315}(3182,\cdot)\)
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
\((1106,1327,1276,2536)\) → \((-1,-i,e\left(\frac{5}{6}\right),e\left(\frac{15}{16}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(14\) | \(16\) | \(19\) | \(22\) | \(23\) |
| \( \chi_{ 3315 }(23, a) \) | \(-1\) | \(1\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{11}{48}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{5}{48}\right)\) | \(e\left(\frac{7}{48}\right)\) |
sage: chi.jacobi_sum(n)