from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(338, base_ring=CyclotomicField(26))
M = H._module
chi = DirichletCharacter(H, M([3]))
pari: [g,chi] = znchar(Mod(25,338))
Basic properties
Modulus: | \(338\) | |
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 \(\chi_{169}(25,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 338.h
\(\chi_{338}(25,\cdot)\) \(\chi_{338}(51,\cdot)\) \(\chi_{338}(77,\cdot)\) \(\chi_{338}(103,\cdot)\) \(\chi_{338}(129,\cdot)\) \(\chi_{338}(155,\cdot)\) \(\chi_{338}(181,\cdot)\) \(\chi_{338}(207,\cdot)\) \(\chi_{338}(233,\cdot)\) \(\chi_{338}(259,\cdot)\) \(\chi_{338}(285,\cdot)\) \(\chi_{338}(311,\cdot)\)
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
\(171\) → \(e\left(\frac{3}{26}\right)\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(15\) | \(17\) | \(19\) | \(21\) | \(23\) |
\( \chi_{ 338 }(25, a) \) | \(1\) | \(1\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{23}{26}\right)\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(-1\) | \(e\left(\frac{17}{26}\right)\) | \(1\) |
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)