from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(855, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([0,27,32]))
pari: [g,chi] = znchar(Mod(613,855))
Basic properties
Modulus: | \(855\) | |
Conductor: | \(95\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{95}(43,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 855.dj
\(\chi_{855}(28,\cdot)\) \(\chi_{855}(73,\cdot)\) \(\chi_{855}(82,\cdot)\) \(\chi_{855}(118,\cdot)\) \(\chi_{855}(253,\cdot)\) \(\chi_{855}(397,\cdot)\) \(\chi_{855}(442,\cdot)\) \(\chi_{855}(568,\cdot)\) \(\chi_{855}(613,\cdot)\) \(\chi_{855}(712,\cdot)\) \(\chi_{855}(757,\cdot)\) \(\chi_{855}(802,\cdot)\)
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: | 36.0.619876750267203693326033178758188478035934269428253173828125.1 |
Values on generators
\((191,172,496)\) → \((1,-i,e\left(\frac{8}{9}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(22\) |
\( \chi_{ 855 }(613, a) \) | \(-1\) | \(1\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{11}{36}\right)\) |
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)