from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(237, base_ring=CyclotomicField(26))
M = H._module
chi = DirichletCharacter(H, M([13,11]))
pari: [g,chi] = znchar(Mod(215,237))
Basic properties
Modulus: | \(237\) | |
Conductor: | \(237\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(26\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 237.l
\(\chi_{237}(14,\cdot)\) \(\chi_{237}(17,\cdot)\) \(\chi_{237}(41,\cdot)\) \(\chi_{237}(71,\cdot)\) \(\chi_{237}(137,\cdot)\) \(\chi_{237}(140,\cdot)\) \(\chi_{237}(170,\cdot)\) \(\chi_{237}(173,\cdot)\) \(\chi_{237}(185,\cdot)\) \(\chi_{237}(191,\cdot)\) \(\chi_{237}(215,\cdot)\) \(\chi_{237}(227,\cdot)\)
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: | 26.26.439798347913742167434960513305495870460482985171691677.1 |
Values on generators
\((80,82)\) → \((-1,e\left(\frac{11}{26}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
\( \chi_{ 237 }(215, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{26}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{19}{26}\right)\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{15}{26}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{10}{13}\right)\) |
sage: chi.jacobi_sum(n)
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
Jacobi sum
sage: chi.jacobi_sum(n)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)