sage: from sage.modular.dirichlet import DirichletCharacter
sage: H = DirichletGroup(116, base_ring=CyclotomicField(14))
sage: M = H._module
sage: chi = DirichletCharacter(H, M([0,11]))
pari: [g,chi] = znchar(Mod(5,116))
Basic properties
| Modulus: | \(116\) | |
| Conductor: | \(29\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(14\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{29}(5,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 116.i
\(\chi_{116}(5,\cdot)\) \(\chi_{116}(9,\cdot)\) \(\chi_{116}(13,\cdot)\) \(\chi_{116}(33,\cdot)\) \(\chi_{116}(93,\cdot)\) \(\chi_{116}(109,\cdot)\)
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: | \(\Q(\zeta_{29})^+\) |
Values on generators
\((59,89)\) → \((1,e\left(\frac{11}{14}\right))\)
Values
| \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \(1\) | \(1\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(-1\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{5}{14}\right)\) |
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{116}(5,\cdot)) = \sum_{r\in \Z/116\Z} \chi_{116}(5,r) e\left(\frac{r}{58}\right) = 8.9511484522+5.989736337i \)
Jacobi sum
sage: chi.jacobi_sum(n)
\( \displaystyle J(\chi_{116}(5,\cdot),\chi_{116}(1,\cdot)) = \sum_{r\in \Z/116\Z} \chi_{116}(5,r) \chi_{116}(1,1-r) = 0 \)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{116}(5,·))
= \sum_{r \in \Z/116\Z}
\chi_{116}(5,r) e\left(\frac{1 r + 2 r^{-1}}{116}\right)
= -0.0 \)