sage: from sage.modular.dirichlet import DirichletCharacter
sage: H = DirichletGroup(539, base_ring=CyclotomicField(14))
sage: M = H._module
sage: chi = DirichletCharacter(H, M([8,0]))
pari: [g,chi] = znchar(Mod(463,539))
Basic properties
| Modulus: | \(539\) | |
| Conductor: | \(49\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(7\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{49}(22,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 539.j
\(\chi_{539}(78,\cdot)\) \(\chi_{539}(155,\cdot)\) \(\chi_{539}(232,\cdot)\) \(\chi_{539}(309,\cdot)\) \(\chi_{539}(386,\cdot)\) \(\chi_{539}(463,\cdot)\)
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: | \(\Q(\zeta_{7})\) |
| Fixed field: | 7.7.13841287201.1 |
Values on generators
\((199,442)\) → \((e\left(\frac{4}{7}\right),1)\)
Values
| \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(12\) | \(13\) |
| \(1\) | \(1\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{6}{7}\right)\) |
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{539}(463,\cdot)) = \sum_{r\in \Z/539\Z} \chi_{539}(463,r) e\left(\frac{2r}{539}\right) = 4.7061062318+5.1819459795i \)
Jacobi sum
sage: chi.jacobi_sum(n)
\( \displaystyle J(\chi_{539}(463,\cdot),\chi_{539}(1,\cdot)) = \sum_{r\in \Z/539\Z} \chi_{539}(463,r) \chi_{539}(1,1-r) = 0 \)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{539}(463,·))
= \sum_{r \in \Z/539\Z}
\chi_{539}(463,r) e\left(\frac{1 r + 2 r^{-1}}{539}\right)
= 0.0 \)