sage: from sage.modular.dirichlet import DirichletCharacter
sage: H = DirichletGroup(133, base_ring=CyclotomicField(18))
sage: M = H._module
sage: chi = DirichletCharacter(H, M([12,8]))
pari: [g,chi] = znchar(Mod(123,133))
Basic properties
| Modulus: | \(133\) | |
| Conductor: | \(133\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(9\) | 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 133.w
\(\chi_{133}(4,\cdot)\) \(\chi_{133}(16,\cdot)\) \(\chi_{133}(25,\cdot)\) \(\chi_{133}(93,\cdot)\) \(\chi_{133}(100,\cdot)\) \(\chi_{133}(123,\cdot)\)
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
\((115,78)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{4}{9}\right))\)
Values
| \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \(1\) | \(1\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(1\) | \(1\) |
Related number fields
| Field of values: | \(\Q(\zeta_{9})\) |
| Fixed field: | 9.9.1998099208210609.1 |
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{133}(123,\cdot)) = \sum_{r\in \Z/133\Z} \chi_{133}(123,r) e\left(\frac{2r}{133}\right) = 0.5483922466+11.5195167409i \)
Jacobi sum
sage: chi.jacobi_sum(n)
\( \displaystyle J(\chi_{133}(123,\cdot),\chi_{133}(1,\cdot)) = \sum_{r\in \Z/133\Z} \chi_{133}(123,r) \chi_{133}(1,1-r) = 1 \)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{133}(123,·))
= \sum_{r \in \Z/133\Z}
\chi_{133}(123,r) e\left(\frac{1 r + 2 r^{-1}}{133}\right)
= 3.7219838887+-3.1231153083i \)