sage: from sage.modular.dirichlet import DirichletCharacter
sage: H = DirichletGroup(690, base_ring=CyclotomicField(22))
sage: M = H._module
sage: chi = DirichletCharacter(H, M([11,11,5]))
pari: [g,chi] = znchar(Mod(89,690))
Basic properties
| Modulus: | \(690\) | |
| Conductor: | \(345\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(22\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{345}(89,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 690.n
\(\chi_{690}(89,\cdot)\) \(\chi_{690}(149,\cdot)\) \(\chi_{690}(329,\cdot)\) \(\chi_{690}(359,\cdot)\) \(\chi_{690}(389,\cdot)\) \(\chi_{690}(419,\cdot)\) \(\chi_{690}(479,\cdot)\) \(\chi_{690}(539,\cdot)\) \(\chi_{690}(569,\cdot)\) \(\chi_{690}(659,\cdot)\)
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
\((461,277,511)\) → \((-1,-1,e\left(\frac{5}{22}\right))\)
Values
| \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \(1\) | \(1\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{7}{11}\right)\) |
Related number fields
| Field of values: | \(\Q(\zeta_{11})\) |
| Fixed field: | 22.22.341419566026798986253349758444608447265625.1 |
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{690}(89,\cdot)) = \sum_{r\in \Z/690\Z} \chi_{690}(89,r) e\left(\frac{r}{345}\right) = -13.8694039523+-12.3547413574i \)
Jacobi sum
sage: chi.jacobi_sum(n)
\( \displaystyle J(\chi_{690}(89,\cdot),\chi_{690}(1,\cdot)) = \sum_{r\in \Z/690\Z} \chi_{690}(89,r) \chi_{690}(1,1-r) = 0 \)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{690}(89,·))
= \sum_{r \in \Z/690\Z}
\chi_{690}(89,r) e\left(\frac{1 r + 2 r^{-1}}{690}\right)
= -0.0 \)