from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8003, base_ring=CyclotomicField(50))
M = H._module
chi = DirichletCharacter(H, M([0,48]))
pari: [g,chi] = znchar(Mod(3976,8003))
Basic properties
| Modulus: | \(8003\) | |
| Conductor: | \(151\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(25\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{151}(50,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8003.u
\(\chi_{8003}(160,\cdot)\) \(\chi_{8003}(425,\cdot)\) \(\chi_{8003}(531,\cdot)\) \(\chi_{8003}(690,\cdot)\) \(\chi_{8003}(849,\cdot)\) \(\chi_{8003}(1167,\cdot)\) \(\chi_{8003}(1379,\cdot)\) \(\chi_{8003}(1591,\cdot)\) \(\chi_{8003}(1856,\cdot)\) \(\chi_{8003}(2333,\cdot)\) \(\chi_{8003}(2651,\cdot)\) \(\chi_{8003}(3446,\cdot)\) \(\chi_{8003}(3923,\cdot)\) \(\chi_{8003}(3976,\cdot)\) \(\chi_{8003}(4506,\cdot)\) \(\chi_{8003}(4559,\cdot)\) \(\chi_{8003}(4930,\cdot)\) \(\chi_{8003}(6414,\cdot)\) \(\chi_{8003}(6467,\cdot)\) \(\chi_{8003}(7792,\cdot)\)
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 25 polynomial |
Values on generators
\((4984,7103)\) → \((1,e\left(\frac{24}{25}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 8003 }(3976, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{19}{25}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{13}{25}\right)\) | \(e\left(\frac{24}{25}\right)\) | \(e\left(\frac{8}{25}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{13}{25}\right)\) | \(e\left(\frac{18}{25}\right)\) | \(e\left(\frac{6}{25}\right)\) |
sage: chi.jacobi_sum(n)