sage: from sage.modular.dirichlet import DirichletCharacter
sage: H = DirichletGroup(29, base_ring=CyclotomicField(28))
sage: M = H._module
sage: chi = DirichletCharacter(H, M([5]))
pari: [g,chi] = znchar(Mod(3,29))
Basic properties
| Modulus: | \(29\) | |
| Conductor: | \(29\) | 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: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 29.f
\(\chi_{29}(2,\cdot)\) \(\chi_{29}(3,\cdot)\) \(\chi_{29}(8,\cdot)\) \(\chi_{29}(10,\cdot)\) \(\chi_{29}(11,\cdot)\) \(\chi_{29}(14,\cdot)\) \(\chi_{29}(15,\cdot)\) \(\chi_{29}(18,\cdot)\) \(\chi_{29}(19,\cdot)\) \(\chi_{29}(21,\cdot)\) \(\chi_{29}(26,\cdot)\) \(\chi_{29}(27,\cdot)\)
sage: chi.galois_orbit()
pari: order = charorder(g,chi)
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Values on generators
\(2\) → \(e\left(\frac{5}{28}\right)\)
Values
| \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \(-1\) | \(1\) | \(e\left(\frac{5}{28}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{3}{28}\right)\) | \(e\left(\frac{13}{28}\right)\) |
Related number fields
| Field of values: | \(\Q(\zeta_{28})\) |
| Fixed field: | \(\Q(\zeta_{29})\) |
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{29}(3,\cdot)) = \sum_{r\in \Z/29\Z} \chi_{29}(3,r) e\left(\frac{2r}{29}\right) = 1.8410703114+5.0606778309i \)
Jacobi sum
sage: chi.jacobi_sum(n)
\( \displaystyle J(\chi_{29}(3,\cdot),\chi_{29}(1,\cdot)) = \sum_{r\in \Z/29\Z} \chi_{29}(3,r) \chi_{29}(1,1-r) = -1 \)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{29}(3,·))
= \sum_{r \in \Z/29\Z}
\chi_{29}(3,r) e\left(\frac{1 r + 2 r^{-1}}{29}\right)
= 0.4484787206+-0.713749795i \)