from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8032, base_ring=CyclotomicField(50))
M = H._module
chi = DirichletCharacter(H, M([25,0,6]))
pari: [g,chi] = znchar(Mod(63,8032))
Basic properties
| Modulus: | \(8032\) | |
| Conductor: | \(1004\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(50\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1004}(63,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8032.bj
\(\chi_{8032}(63,\cdot)\) \(\chi_{8032}(255,\cdot)\) \(\chi_{8032}(351,\cdot)\) \(\chi_{8032}(703,\cdot)\) \(\chi_{8032}(1055,\cdot)\) \(\chi_{8032}(1215,\cdot)\) \(\chi_{8032}(1247,\cdot)\) \(\chi_{8032}(1631,\cdot)\) \(\chi_{8032}(2463,\cdot)\) \(\chi_{8032}(2751,\cdot)\) \(\chi_{8032}(3103,\cdot)\) \(\chi_{8032}(3135,\cdot)\) \(\chi_{8032}(3327,\cdot)\) \(\chi_{8032}(3519,\cdot)\) \(\chi_{8032}(3583,\cdot)\) \(\chi_{8032}(4543,\cdot)\) \(\chi_{8032}(4767,\cdot)\) \(\chi_{8032}(4863,\cdot)\) \(\chi_{8032}(7295,\cdot)\) \(\chi_{8032}(7359,\cdot)\)
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
\((6527,3013,257)\) → \((-1,1,e\left(\frac{3}{25}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 8032 }(63, a) \) | \(-1\) | \(1\) | \(e\left(\frac{21}{50}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{13}{50}\right)\) | \(e\left(\frac{21}{25}\right)\) | \(e\left(\frac{41}{50}\right)\) | \(e\left(\frac{21}{25}\right)\) | \(e\left(\frac{1}{50}\right)\) | \(e\left(\frac{22}{25}\right)\) | \(e\left(\frac{3}{50}\right)\) | \(e\left(\frac{17}{25}\right)\) |
sage: chi.jacobi_sum(n)