sage: from sage.modular.dirichlet import DirichletCharacter
sage: H = DirichletGroup(483, base_ring=CyclotomicField(6))
sage: M = H._module
sage: chi = DirichletCharacter(H, M([0,4,0]))
pari: [g,chi] = znchar(Mod(277,483))
Basic properties
| Modulus: | \(483\) | |
| Conductor: | \(7\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(3\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{7}(4,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 483.i
\(\chi_{483}(277,\cdot)\) \(\chi_{483}(415,\cdot)\)
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: | \(\Q(\sqrt{-3}) \) |
| Fixed field: | \(\Q(\zeta_{7})^+\) |
Values on generators
\((323,346,442)\) → \((1,e\left(\frac{2}{3}\right),1)\)
Values
| \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
| \(1\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(1\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) |
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{483}(277,\cdot)) = \sum_{r\in \Z/483\Z} \chi_{483}(277,r) e\left(\frac{2r}{483}\right) = -0.1675628019+-2.6404398701i \)
Jacobi sum
sage: chi.jacobi_sum(n)
\( \displaystyle J(\chi_{483}(277,\cdot),\chi_{483}(1,\cdot)) = \sum_{r\in \Z/483\Z} \chi_{483}(277,r) \chi_{483}(1,1-r) = -21 \)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{483}(277,·))
= \sum_{r \in \Z/483\Z}
\chi_{483}(277,r) e\left(\frac{1 r + 2 r^{-1}}{483}\right)
= 22.7243098758+39.3596592719i \)