sage: from sage.modular.dirichlet import DirichletCharacter
sage: H = DirichletGroup(48, base_ring=CyclotomicField(4))
sage: M = H._module
sage: chi = DirichletCharacter(H, M([0,1,0]))
pari: [g,chi] = znchar(Mod(37,48))
Basic properties
| Modulus: | \(48\) | |
| Conductor: | \(16\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(4\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{16}(5,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 48.j
\(\chi_{48}(13,\cdot)\) \(\chi_{48}(37,\cdot)\)
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: | \(\Q(\sqrt{-1}) \) |
| Fixed field: | \(\Q(\zeta_{16})^+\) |
Values on generators
\((31,37,17)\) → \((1,i,1)\)
Values
| \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \(1\) | \(1\) | \(i\) | \(-1\) | \(i\) | \(-i\) | \(1\) | \(-i\) | \(-1\) | \(-1\) | \(-i\) | \(1\) |
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{48}(37,\cdot)) = \sum_{r\in \Z/48\Z} \chi_{48}(37,r) e\left(\frac{r}{24}\right) = 0.0 \)
Jacobi sum
sage: chi.jacobi_sum(n)
\( \displaystyle J(\chi_{48}(37,\cdot),\chi_{48}(1,\cdot)) = \sum_{r\in \Z/48\Z} \chi_{48}(37,r) \chi_{48}(1,1-r) = 0 \)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{48}(37,·))
= \sum_{r \in \Z/48\Z}
\chi_{48}(37,r) e\left(\frac{1 r + 2 r^{-1}}{48}\right)
= 7.3910362601+-3.0614674589i \)