from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(145, base_ring=CyclotomicField(28))
M = H._module
chi = DirichletCharacter(H, M([21,10]))
pari: [g,chi] = znchar(Mod(38,145))
Basic properties
Modulus: | \(145\) | |
Conductor: | \(145\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(28\) | 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 145.q
\(\chi_{145}(13,\cdot)\) \(\chi_{145}(22,\cdot)\) \(\chi_{145}(33,\cdot)\) \(\chi_{145}(38,\cdot)\) \(\chi_{145}(42,\cdot)\) \(\chi_{145}(62,\cdot)\) \(\chi_{145}(63,\cdot)\) \(\chi_{145}(67,\cdot)\) \(\chi_{145}(92,\cdot)\) \(\chi_{145}(93,\cdot)\) \(\chi_{145}(122,\cdot)\) \(\chi_{145}(138,\cdot)\)
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: | \(\Q(\zeta_{28})\) |
Fixed field: | 28.0.50201655190081835380839261671426578388690948486328125.1 |
Values on generators
\((117,31)\) → \((-i,e\left(\frac{5}{14}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
\( \chi_{ 145 }(38, a) \) | \(-1\) | \(1\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{1}{28}\right)\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(i\) | \(e\left(\frac{19}{28}\right)\) |
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)