sage: from sage.modular.dirichlet import DirichletCharacter
sage: H = DirichletGroup(966, base_ring=CyclotomicField(22))
sage: M = H._module
sage: chi = DirichletCharacter(H, M([0,0,8]))
pari: [g,chi] = znchar(Mod(85,966))
Basic properties
| Modulus: | \(966\) | |
| Conductor: | \(23\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(11\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{23}(16,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 966.q
\(\chi_{966}(85,\cdot)\) \(\chi_{966}(127,\cdot)\) \(\chi_{966}(169,\cdot)\) \(\chi_{966}(211,\cdot)\) \(\chi_{966}(463,\cdot)\) \(\chi_{966}(547,\cdot)\) \(\chi_{966}(673,\cdot)\) \(\chi_{966}(715,\cdot)\) \(\chi_{966}(841,\cdot)\) \(\chi_{966}(883,\cdot)\)
sage: chi.galois_orbit()
pari: order = charorder(g,chi)
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Values on generators
\((323,829,925)\) → \((1,1,e\left(\frac{4}{11}\right))\)
Values
| \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \(1\) | \(1\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{4}{11}\right)\) |
Related number fields
| Field of values: | \(\Q(\zeta_{11})\) |
| Fixed field: | \(\Q(\zeta_{23})^+\) |
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{966}(85,\cdot)) = \sum_{r\in \Z/966\Z} \chi_{966}(85,r) e\left(\frac{r}{483}\right) = -4.7386502643+0.7383723133i \)
Jacobi sum
sage: chi.jacobi_sum(n)
\( \displaystyle J(\chi_{966}(85,\cdot),\chi_{966}(1,\cdot)) = \sum_{r\in \Z/966\Z} \chi_{966}(85,r) \chi_{966}(1,1-r) = 0 \)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{966}(85,·))
= \sum_{r \in \Z/966\Z}
\chi_{966}(85,r) e\left(\frac{1 r + 2 r^{-1}}{966}\right)
= 18.1623995515+-20.9605264429i \)