from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8005, base_ring=CyclotomicField(50))
M = H._module
chi = DirichletCharacter(H, M([25,36]))
pari: [g,chi] = znchar(Mod(19,8005))
Basic properties
| Modulus: | \(8005\) | |
| Conductor: | \(8005\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(50\) | 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 8005.bn
\(\chi_{8005}(19,\cdot)\) \(\chi_{8005}(519,\cdot)\) \(\chi_{8005}(1104,\cdot)\) \(\chi_{8005}(1264,\cdot)\) \(\chi_{8005}(1419,\cdot)\) \(\chi_{8005}(1499,\cdot)\) \(\chi_{8005}(1994,\cdot)\) \(\chi_{8005}(2354,\cdot)\) \(\chi_{8005}(2399,\cdot)\) \(\chi_{8005}(2664,\cdot)\) \(\chi_{8005}(4409,\cdot)\) \(\chi_{8005}(5164,\cdot)\) \(\chi_{8005}(5299,\cdot)\) \(\chi_{8005}(6299,\cdot)\) \(\chi_{8005}(6659,\cdot)\) \(\chi_{8005}(6859,\cdot)\) \(\chi_{8005}(7044,\cdot)\) \(\chi_{8005}(7389,\cdot)\) \(\chi_{8005}(7749,\cdot)\) \(\chi_{8005}(7944,\cdot)\)
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: | \(\Q(\zeta_{25})\) |
| Fixed field: | Number field defined by a degree 50 polynomial |
Values on generators
\((1602,4806)\) → \((-1,e\left(\frac{18}{25}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 8005 }(19, a) \) | \(1\) | \(1\) | \(e\left(\frac{47}{50}\right)\) | \(e\left(\frac{11}{50}\right)\) | \(e\left(\frac{22}{25}\right)\) | \(e\left(\frac{4}{25}\right)\) | \(e\left(\frac{7}{50}\right)\) | \(e\left(\frac{41}{50}\right)\) | \(e\left(\frac{11}{25}\right)\) | \(e\left(\frac{19}{25}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(-1\) |
sage: chi.jacobi_sum(n)