sage: from sage.modular.dirichlet import DirichletCharacter
sage: H = DirichletGroup(61, base_ring=CyclotomicField(10))
sage: M = H._module
sage: chi = DirichletCharacter(H, M([9]))
pari: [g,chi] = znchar(Mod(41,61))
Basic properties
| Modulus: | \(61\) | |
| Conductor: | \(61\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(10\) | 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 61.g
\(\chi_{61}(3,\cdot)\) \(\chi_{61}(27,\cdot)\) \(\chi_{61}(41,\cdot)\) \(\chi_{61}(52,\cdot)\)
sage: chi.galois_orbit()
pari: order = charorder(g,chi)
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Related number fields
| Field of values: | \(\Q(\zeta_{5})\) |
| Fixed field: | 10.10.11694146092834141.1 |
Values on generators
\(2\) → \(e\left(\frac{9}{10}\right)\)
Values
| \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \(1\) | \(1\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(-1\) |
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{61}(41,\cdot)) = \sum_{r\in \Z/61\Z} \chi_{61}(41,r) e\left(\frac{2r}{61}\right) = 7.5076852062+-2.1528267103i \)
Jacobi sum
sage: chi.jacobi_sum(n)
\( \displaystyle J(\chi_{61}(41,\cdot),\chi_{61}(1,\cdot)) = \sum_{r\in \Z/61\Z} \chi_{61}(41,r) \chi_{61}(1,1-r) = -1 \)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{61}(41,·))
= \sum_{r \in \Z/61\Z}
\chi_{61}(41,r) e\left(\frac{1 r + 2 r^{-1}}{61}\right)
= 1.8120273001+-0.5887633599i \)