from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2500, base_ring=CyclotomicField(50))
M = H._module
chi = DirichletCharacter(H, M([25,37]))
pari: [g,chi] = znchar(Mod(2299,2500))
Basic properties
| Modulus: | \(2500\) | |
| Conductor: | \(500\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(50\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{500}(159,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2500.n
\(\chi_{2500}(99,\cdot)\) \(\chi_{2500}(199,\cdot)\) \(\chi_{2500}(299,\cdot)\) \(\chi_{2500}(399,\cdot)\) \(\chi_{2500}(599,\cdot)\) \(\chi_{2500}(699,\cdot)\) \(\chi_{2500}(799,\cdot)\) \(\chi_{2500}(899,\cdot)\) \(\chi_{2500}(1099,\cdot)\) \(\chi_{2500}(1199,\cdot)\) \(\chi_{2500}(1299,\cdot)\) \(\chi_{2500}(1399,\cdot)\) \(\chi_{2500}(1599,\cdot)\) \(\chi_{2500}(1699,\cdot)\) \(\chi_{2500}(1799,\cdot)\) \(\chi_{2500}(1899,\cdot)\) \(\chi_{2500}(2099,\cdot)\) \(\chi_{2500}(2199,\cdot)\) \(\chi_{2500}(2299,\cdot)\) \(\chi_{2500}(2399,\cdot)\)
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
\((1251,1877)\) → \((-1,e\left(\frac{37}{50}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 2500 }(2299, a) \) | \(-1\) | \(1\) | \(e\left(\frac{17}{25}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{9}{25}\right)\) | \(e\left(\frac{37}{50}\right)\) | \(e\left(\frac{43}{50}\right)\) | \(e\left(\frac{1}{50}\right)\) | \(e\left(\frac{41}{50}\right)\) | \(e\left(\frac{2}{25}\right)\) | \(e\left(\frac{11}{25}\right)\) | \(e\left(\frac{1}{25}\right)\) |
sage: chi.jacobi_sum(n)