from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(555, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([18,27,16]))
pari: [g,chi] = znchar(Mod(83,555))
Basic properties
Modulus: | \(555\) | |
Conductor: | \(555\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(36\) | 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 555.cc
\(\chi_{555}(53,\cdot)\) \(\chi_{555}(83,\cdot)\) \(\chi_{555}(107,\cdot)\) \(\chi_{555}(182,\cdot)\) \(\chi_{555}(197,\cdot)\) \(\chi_{555}(218,\cdot)\) \(\chi_{555}(293,\cdot)\) \(\chi_{555}(308,\cdot)\) \(\chi_{555}(377,\cdot)\) \(\chi_{555}(488,\cdot)\) \(\chi_{555}(497,\cdot)\) \(\chi_{555}(527,\cdot)\)
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: | \(\Q(\zeta_{36})\) |
Fixed field: | Number field defined by a degree 36 polynomial |
Values on generators
\((371,112,76)\) → \((-1,-i,e\left(\frac{4}{9}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
\( \chi_{ 555 }(83, a) \) | \(1\) | \(1\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{1}{18}\right)\) |
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)