from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(760, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([18,18,27,8]))
pari: [g,chi] = znchar(Mod(643,760))
Basic properties
| Modulus: | \(760\) | |
| Conductor: | \(760\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(36\) | 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 760.cp
\(\chi_{760}(43,\cdot)\) \(\chi_{760}(123,\cdot)\) \(\chi_{760}(187,\cdot)\) \(\chi_{760}(283,\cdot)\) \(\chi_{760}(347,\cdot)\) \(\chi_{760}(403,\cdot)\) \(\chi_{760}(427,\cdot)\) \(\chi_{760}(443,\cdot)\) \(\chi_{760}(587,\cdot)\) \(\chi_{760}(643,\cdot)\) \(\chi_{760}(707,\cdot)\) \(\chi_{760}(747,\cdot)\)
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: | Number field defined by a degree 36 polynomial |
Values on generators
\((191,381,457,401)\) → \((-1,-1,-i,e\left(\frac{2}{9}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 760 }(643, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{7}{9}\right)\) |
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)