from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(185, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([27,2]))
pari: [g,chi] = znchar(Mod(78,185))
Basic properties
| Modulus: | \(185\) | |
| Conductor: | \(185\) | 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: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 185.y
\(\chi_{185}(3,\cdot)\) \(\chi_{185}(28,\cdot)\) \(\chi_{185}(58,\cdot)\) \(\chi_{185}(62,\cdot)\) \(\chi_{185}(67,\cdot)\) \(\chi_{185}(77,\cdot)\) \(\chi_{185}(78,\cdot)\) \(\chi_{185}(102,\cdot)\) \(\chi_{185}(132,\cdot)\) \(\chi_{185}(152,\cdot)\) \(\chi_{185}(173,\cdot)\) \(\chi_{185}(178,\cdot)\)
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: | 36.0.1552561224397970539319305607385300901659347120825982756912708282470703125.1 |
Values on generators
\((112,76)\) → \((-i,e\left(\frac{1}{18}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 185 }(78, a) \) | \(-1\) | \(1\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(-1\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{31}{36}\right)\) |
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)