from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2805, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([10,5,8,0]))
pari: [g,chi] = znchar(Mod(137,2805))
Basic properties
| Modulus: | \(2805\) | |
| Conductor: | \(165\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(20\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{165}(137,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2805.dv
\(\chi_{2805}(137,\cdot)\) \(\chi_{2805}(443,\cdot)\) \(\chi_{2805}(647,\cdot)\) \(\chi_{2805}(698,\cdot)\) \(\chi_{2805}(1208,\cdot)\) \(\chi_{2805}(1412,\cdot)\) \(\chi_{2805}(1973,\cdot)\) \(\chi_{2805}(2687,\cdot)\)
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: | \(\Q(\zeta_{20})\) |
| Fixed field: | Number field defined by a degree 20 polynomial |
Values on generators
\((1871,562,1531,496)\) → \((-1,i,e\left(\frac{2}{5}\right),1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(13\) | \(14\) | \(16\) | \(19\) | \(23\) | \(26\) |
| \( \chi_{ 2805 }(137, a) \) | \(1\) | \(1\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(i\) | \(e\left(\frac{3}{10}\right)\) |
sage: chi.jacobi_sum(n)