from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2496, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([24,45,24,40]))
pari: [g,chi] = znchar(Mod(179,2496))
Basic properties
| Modulus: | \(2496\) | |
| Conductor: | \(2496\) | 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 2496.fa
\(\chi_{2496}(179,\cdot)\) \(\chi_{2496}(251,\cdot)\) \(\chi_{2496}(491,\cdot)\) \(\chi_{2496}(563,\cdot)\) \(\chi_{2496}(803,\cdot)\) \(\chi_{2496}(875,\cdot)\) \(\chi_{2496}(1115,\cdot)\) \(\chi_{2496}(1187,\cdot)\) \(\chi_{2496}(1427,\cdot)\) \(\chi_{2496}(1499,\cdot)\) \(\chi_{2496}(1739,\cdot)\) \(\chi_{2496}(1811,\cdot)\) \(\chi_{2496}(2051,\cdot)\) \(\chi_{2496}(2123,\cdot)\) \(\chi_{2496}(2363,\cdot)\) \(\chi_{2496}(2435,\cdot)\)
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
\((703,1093,833,769)\) → \((-1,e\left(\frac{15}{16}\right),-1,e\left(\frac{5}{6}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 2496 }(179, a) \) | \(1\) | \(1\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{11}{48}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(-1\) | \(e\left(\frac{47}{48}\right)\) |
sage: chi.jacobi_sum(n)