from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6025, base_ring=CyclotomicField(10))
M = H._module
chi = DirichletCharacter(H, M([0,2]))
pari: [g,chi] = znchar(Mod(2501,6025))
Basic properties
| Modulus: | \(6025\) | |
| Conductor: | \(241\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(5\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{241}(91,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6025.q
\(\chi_{6025}(1051,\cdot)\) \(\chi_{6025}(1651,\cdot)\) \(\chi_{6025}(2026,\cdot)\) \(\chi_{6025}(2501,\cdot)\)
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: | \(\Q(\zeta_{5})\) |
| Fixed field: | 5.5.3373402561.1 |
Values on generators
\((2652,2176)\) → \((1,e\left(\frac{1}{5}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 6025 }(2501, a) \) | \(1\) | \(1\) | \(1\) | \(e\left(\frac{2}{5}\right)\) | \(1\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(1\) | \(e\left(\frac{4}{5}\right)\) | \(1\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{2}{5}\right)\) |
sage: chi.jacobi_sum(n)