from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5577, base_ring=CyclotomicField(26))
M = H._module
chi = DirichletCharacter(H, M([0,0,15]))
pari: [g,chi] = znchar(Mod(298,5577))
Basic properties
| Modulus: | \(5577\) | |
| Conductor: | \(169\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(26\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{169}(129,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5577.bs
\(\chi_{5577}(298,\cdot)\) \(\chi_{5577}(727,\cdot)\) \(\chi_{5577}(1156,\cdot)\) \(\chi_{5577}(1585,\cdot)\) \(\chi_{5577}(2014,\cdot)\) \(\chi_{5577}(2443,\cdot)\) \(\chi_{5577}(3301,\cdot)\) \(\chi_{5577}(3730,\cdot)\) \(\chi_{5577}(4159,\cdot)\) \(\chi_{5577}(4588,\cdot)\) \(\chi_{5577}(5017,\cdot)\) \(\chi_{5577}(5446,\cdot)\)
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: | \(\Q(\zeta_{13})\) |
| Fixed field: | 26.26.3830224792147131369362629348887201408953937846517364173.1 |
Values on generators
\((3719,508,1354)\) → \((1,1,e\left(\frac{15}{26}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 5577 }(298, a) \) | \(1\) | \(1\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{5}{26}\right)\) | \(e\left(\frac{19}{26}\right)\) | \(e\left(\frac{19}{26}\right)\) | \(e\left(\frac{10}{13}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(-1\) |
sage: chi.jacobi_sum(n)