from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5775, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([0,13,0,10]))
pari: [g,chi] = znchar(Mod(967,5775))
Basic properties
| Modulus: | \(5775\) | |
| Conductor: | \(275\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(20\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{275}(142,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5775.gu
\(\chi_{5775}(967,\cdot)\) \(\chi_{5775}(1198,\cdot)\) \(\chi_{5775}(2122,\cdot)\) \(\chi_{5775}(2353,\cdot)\) \(\chi_{5775}(3277,\cdot)\) \(\chi_{5775}(3508,\cdot)\) \(\chi_{5775}(4663,\cdot)\) \(\chi_{5775}(5587,\cdot)\)
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: | 20.20.75487840807181783020496368408203125.1 |
Values on generators
\((3851,4852,4126,3676)\) → \((1,e\left(\frac{13}{20}\right),1,-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(13\) | \(16\) | \(17\) | \(19\) | \(23\) | \(26\) | \(29\) |
| \( \chi_{ 5775 }(967, a) \) | \(1\) | \(1\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(1\) | \(e\left(\frac{4}{5}\right)\) |
sage: chi.jacobi_sum(n)