from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(760, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([0,18,9,16]))
pari: [g,chi] = znchar(Mod(237,760))
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: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 760.cq
\(\chi_{760}(93,\cdot)\) \(\chi_{760}(157,\cdot)\) \(\chi_{760}(213,\cdot)\) \(\chi_{760}(237,\cdot)\) \(\chi_{760}(253,\cdot)\) \(\chi_{760}(397,\cdot)\) \(\chi_{760}(453,\cdot)\) \(\chi_{760}(517,\cdot)\) \(\chi_{760}(557,\cdot)\) \(\chi_{760}(613,\cdot)\) \(\chi_{760}(693,\cdot)\) \(\chi_{760}(757,\cdot)\)
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
\((191,381,457,401)\) → \((1,-1,i,e\left(\frac{4}{9}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 760 }(237, a) \) | \(-1\) | \(1\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{5}{9}\right)\) |
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)