from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7488, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([24,45,40,8]))
pari: [g,chi] = znchar(Mod(563,7488))
Basic properties
| Modulus: | \(7488\) | |
| Conductor: | \(7488\) | 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 7488.ng
\(\chi_{7488}(563,\cdot)\) \(\chi_{7488}(803,\cdot)\) \(\chi_{7488}(1499,\cdot)\) \(\chi_{7488}(1739,\cdot)\) \(\chi_{7488}(2435,\cdot)\) \(\chi_{7488}(2675,\cdot)\) \(\chi_{7488}(3371,\cdot)\) \(\chi_{7488}(3611,\cdot)\) \(\chi_{7488}(4307,\cdot)\) \(\chi_{7488}(4547,\cdot)\) \(\chi_{7488}(5243,\cdot)\) \(\chi_{7488}(5483,\cdot)\) \(\chi_{7488}(6179,\cdot)\) \(\chi_{7488}(6419,\cdot)\) \(\chi_{7488}(7115,\cdot)\) \(\chi_{7488}(7355,\cdot)\)
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
\((703,6085,5825,5761)\) → \((-1,e\left(\frac{15}{16}\right),e\left(\frac{5}{6}\right),e\left(\frac{1}{6}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 7488 }(563, a) \) | \(1\) | \(1\) | \(e\left(\frac{29}{48}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{31}{48}\right)\) |
sage: chi.jacobi_sum(n)