sage: from sage.modular.dirichlet import DirichletCharacter
sage: H = DirichletGroup(570, base_ring=CyclotomicField(6))
sage: M = H._module
sage: chi = DirichletCharacter(H, M([0,3,4]))
pari: [g,chi] = znchar(Mod(49,570))
Basic properties
| Modulus: | \(570\) | |
| Conductor: | \(95\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(6\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{95}(49,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 570.q
\(\chi_{570}(49,\cdot)\) \(\chi_{570}(349,\cdot)\)
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: | 6.6.16290125.1 |
Values on generators
\((191,457,211)\) → \((1,-1,e\left(\frac{2}{3}\right))\)
Values
| \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \(1\) | \(1\) | \(-1\) | \(1\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(1\) | \(-1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{6}\right)\) |
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{570}(49,\cdot)) = \sum_{r\in \Z/570\Z} \chi_{570}(49,r) e\left(\frac{r}{285}\right) = 9.5268301523+2.0590063741i \)
Jacobi sum
sage: chi.jacobi_sum(n)
\( \displaystyle J(\chi_{570}(49,\cdot),\chi_{570}(1,\cdot)) = \sum_{r\in \Z/570\Z} \chi_{570}(49,r) \chi_{570}(1,1-r) = 0 \)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{570}(49,·))
= \sum_{r \in \Z/570\Z}
\chi_{570}(49,r) e\left(\frac{1 r + 2 r^{-1}}{570}\right)
= 0.0 \)