from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8281, base_ring=CyclotomicField(26))
M = H._module
chi = DirichletCharacter(H, M([0,9]))
pari: [g,chi] = znchar(Mod(246,8281))
Basic properties
| Modulus: | \(8281\) | |
| 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 \(\chi_{169}(77,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8281.bo
\(\chi_{8281}(246,\cdot)\) \(\chi_{8281}(883,\cdot)\) \(\chi_{8281}(2157,\cdot)\) \(\chi_{8281}(2794,\cdot)\) \(\chi_{8281}(3431,\cdot)\) \(\chi_{8281}(4068,\cdot)\) \(\chi_{8281}(4705,\cdot)\) \(\chi_{8281}(5342,\cdot)\) \(\chi_{8281}(5979,\cdot)\) \(\chi_{8281}(6616,\cdot)\) \(\chi_{8281}(7253,\cdot)\) \(\chi_{8281}(7890,\cdot)\)
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
\((1522,3382)\) → \((1,e\left(\frac{9}{26}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \( \chi_{ 8281 }(246, a) \) | \(1\) | \(1\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{3}{26}\right)\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{8}{13}\right)\) |
sage: chi.jacobi_sum(n)