from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1859, base_ring=CyclotomicField(26))
M = H._module
chi = DirichletCharacter(H, M([0,21]))
pari: [g,chi] = znchar(Mod(12,1859))
Basic properties
Modulus: | \(1859\) | |
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 \(\chi_{169}(12,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1859.w
\(\chi_{1859}(12,\cdot)\) \(\chi_{1859}(155,\cdot)\) \(\chi_{1859}(298,\cdot)\) \(\chi_{1859}(441,\cdot)\) \(\chi_{1859}(584,\cdot)\) \(\chi_{1859}(727,\cdot)\) \(\chi_{1859}(870,\cdot)\) \(\chi_{1859}(1156,\cdot)\) \(\chi_{1859}(1299,\cdot)\) \(\chi_{1859}(1442,\cdot)\) \(\chi_{1859}(1585,\cdot)\) \(\chi_{1859}(1728,\cdot)\)
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
\((508,1354)\) → \((1,e\left(\frac{21}{26}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
\( \chi_{ 1859 }(12, a) \) | \(1\) | \(1\) | \(e\left(\frac{21}{26}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{25}{26}\right)\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{10}{13}\right)\) |
sage: chi.jacobi_sum(n)