sage: from sage.modular.dirichlet import DirichletCharacter
sage: H = DirichletGroup(360, base_ring=CyclotomicField(2))
sage: M = H._module
sage: chi = DirichletCharacter(H, M([1,1,1,1]))
pari: [g,chi] = znchar(Mod(179,360))
Basic properties
| Modulus: | \(360\) | |
| Conductor: | \(120\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(2\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | yes | |
| Primitive: | no, induced from \(\chi_{120}(59,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 360.m
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: | \(\Q\) |
| Fixed field: | \(\Q(\sqrt{30}) \) |
Values on generators
\((271,181,281,217)\) → \((-1,-1,-1,-1)\)
Values
| \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \(1\) | \(1\) | \(1\) | \(-1\) | \(1\) | \(1\) | \(1\) | \(-1\) | \(1\) | \(-1\) | \(1\) | \(-1\) |
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{360}(179,\cdot)) = \sum_{r\in \Z/360\Z} \chi_{360}(179,r) e\left(\frac{r}{180}\right) = 0.0 \)
Jacobi sum
sage: chi.jacobi_sum(n)
\( \displaystyle J(\chi_{360}(179,\cdot),\chi_{360}(1,\cdot)) = \sum_{r\in \Z/360\Z} \chi_{360}(179,r) \chi_{360}(1,1-r) = 0 \)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{360}(179,·))
= \sum_{r \in \Z/360\Z}
\chi_{360}(179,r) e\left(\frac{1 r + 2 r^{-1}}{360}\right)
= -0.0 \)