from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(353, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([7]))
pari: [g,chi] = znchar(Mod(35,353))
Basic properties
| Modulus: | \(353\) | |
| Conductor: | \(353\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(44\) | 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 353.i
\(\chi_{353}(4,\cdot)\) \(\chi_{353}(34,\cdot)\) \(\chi_{353}(35,\cdot)\) \(\chi_{353}(64,\cdot)\) \(\chi_{353}(88,\cdot)\) \(\chi_{353}(121,\cdot)\) \(\chi_{353}(135,\cdot)\) \(\chi_{353}(146,\cdot)\) \(\chi_{353}(162,\cdot)\) \(\chi_{353}(171,\cdot)\) \(\chi_{353}(182,\cdot)\) \(\chi_{353}(191,\cdot)\) \(\chi_{353}(207,\cdot)\) \(\chi_{353}(218,\cdot)\) \(\chi_{353}(232,\cdot)\) \(\chi_{353}(265,\cdot)\) \(\chi_{353}(289,\cdot)\) \(\chi_{353}(318,\cdot)\) \(\chi_{353}(319,\cdot)\) \(\chi_{353}(349,\cdot)\)
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: | \(\Q(\zeta_{44})\) |
| Fixed field: | Number field defined by a degree 44 polynomial |
Values on generators
\(3\) → \(e\left(\frac{7}{44}\right)\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 353 }(35, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{7}{44}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{7}{44}\right)\) | \(i\) | \(-i\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(i\) | \(e\left(\frac{6}{11}\right)\) |
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)