from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(380, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([0,27,8]))
pari: [g,chi] = znchar(Mod(73,380))
Basic properties
Modulus: | \(380\) | |
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 \(\chi_{95}(73,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 380.bg
\(\chi_{380}(17,\cdot)\) \(\chi_{380}(73,\cdot)\) \(\chi_{380}(93,\cdot)\) \(\chi_{380}(137,\cdot)\) \(\chi_{380}(157,\cdot)\) \(\chi_{380}(177,\cdot)\) \(\chi_{380}(213,\cdot)\) \(\chi_{380}(233,\cdot)\) \(\chi_{380}(237,\cdot)\) \(\chi_{380}(253,\cdot)\) \(\chi_{380}(313,\cdot)\) \(\chi_{380}(377,\cdot)\)
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
\((191,77,21)\) → \((1,-i,e\left(\frac{2}{9}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(21\) | \(23\) | \(27\) | \(29\) |
\( \chi_{ 380 }(73, a) \) | \(-1\) | \(1\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{5}{18}\right)\) |
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)