from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8041, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([24,9,16]))
pari: [g,chi] = znchar(Mod(7303,8041))
Basic properties
| Modulus: | \(8041\) | |
| Conductor: | \(8041\) | 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 8041.df
\(\chi_{8041}(1253,\cdot)\) \(\chi_{8041}(1627,\cdot)\) \(\chi_{8041}(2199,\cdot)\) \(\chi_{8041}(2573,\cdot)\) \(\chi_{8041}(2672,\cdot)\) \(\chi_{8041}(3046,\cdot)\) \(\chi_{8041}(3618,\cdot)\) \(\chi_{8041}(3992,\cdot)\) \(\chi_{8041}(4091,\cdot)\) \(\chi_{8041}(4465,\cdot)\) \(\chi_{8041}(5037,\cdot)\) \(\chi_{8041}(5411,\cdot)\) \(\chi_{8041}(6929,\cdot)\) \(\chi_{8041}(7303,\cdot)\) \(\chi_{8041}(7402,\cdot)\) \(\chi_{8041}(7776,\cdot)\)
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
\((6580,2366,562)\) → \((-1,e\left(\frac{3}{16}\right),e\left(\frac{1}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
| \( \chi_{ 8041 }(7303, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{25}{48}\right)\) | \(i\) | \(e\left(\frac{13}{48}\right)\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{11}{48}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{19}{48}\right)\) | \(e\left(\frac{37}{48}\right)\) |
sage: chi.jacobi_sum(n)