from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(348, base_ring=CyclotomicField(28))
M = H._module
chi = DirichletCharacter(H, M([14,0,9]))
pari: [g,chi] = znchar(Mod(19,348))
Basic properties
| Modulus: | \(348\) | |
| Conductor: | \(116\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(28\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{116}(19,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 348.x
\(\chi_{348}(19,\cdot)\) \(\chi_{348}(31,\cdot)\) \(\chi_{348}(43,\cdot)\) \(\chi_{348}(55,\cdot)\) \(\chi_{348}(79,\cdot)\) \(\chi_{348}(127,\cdot)\) \(\chi_{348}(163,\cdot)\) \(\chi_{348}(211,\cdot)\) \(\chi_{348}(235,\cdot)\) \(\chi_{348}(247,\cdot)\) \(\chi_{348}(259,\cdot)\) \(\chi_{348}(271,\cdot)\)
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: | \(\Q(\zeta_{116})^+\) |
Values on generators
\((175,233,205)\) → \((-1,1,e\left(\frac{9}{28}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(31\) | \(35\) |
| \( \chi_{ 348 }(19, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(-i\) | \(e\left(\frac{11}{28}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(e\left(\frac{3}{7}\right)\) |
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)