from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(145, base_ring=CyclotomicField(28))
M = H._module
chi = DirichletCharacter(H, M([7,15]))
pari: [g,chi] = znchar(Mod(27,145))
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: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 145.t
\(\chi_{145}(3,\cdot)\) \(\chi_{145}(27,\cdot)\) \(\chi_{145}(37,\cdot)\) \(\chi_{145}(43,\cdot)\) \(\chi_{145}(47,\cdot)\) \(\chi_{145}(48,\cdot)\) \(\chi_{145}(97,\cdot)\) \(\chi_{145}(98,\cdot)\) \(\chi_{145}(102,\cdot)\) \(\chi_{145}(108,\cdot)\) \(\chi_{145}(118,\cdot)\) \(\chi_{145}(142,\cdot)\)
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: | 28.28.1455848000512373226044338588471370773272037506103515625.2 |
Values on generators
\((117,31)\) → \((i,e\left(\frac{15}{28}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
\( \chi_{ 145 }(27, a) \) | \(1\) | \(1\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{11}{28}\right)\) | \(1\) | \(e\left(\frac{11}{28}\right)\) |
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)