sage: from sage.modular.dirichlet import DirichletCharacter
sage: H = DirichletGroup(950, base_ring=CyclotomicField(10))
sage: M = H._module
sage: chi = DirichletCharacter(H, M([2,0]))
pari: [g,chi] = znchar(Mod(191,950))
Basic properties
| Modulus: | \(950\) | |
| Conductor: | \(25\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(5\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{25}(16,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 950.h
\(\chi_{950}(191,\cdot)\) \(\chi_{950}(381,\cdot)\) \(\chi_{950}(571,\cdot)\) \(\chi_{950}(761,\cdot)\)
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
\((77,401)\) → \((e\left(\frac{1}{5}\right),1)\)
Values
| \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \(1\) | \(1\) | \(e\left(\frac{2}{5}\right)\) | \(1\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{2}{5}\right)\) |
Related number fields
| Field of values: | \(\Q(\zeta_{5})\) |
| Fixed field: | 5.5.390625.1 |
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{950}(191,\cdot)) = \sum_{r\in \Z/950\Z} \chi_{950}(191,r) e\left(\frac{r}{475}\right) = -3.6448431371+3.4227355296i \)
Jacobi sum
sage: chi.jacobi_sum(n)
\( \displaystyle J(\chi_{950}(191,\cdot),\chi_{950}(1,\cdot)) = \sum_{r\in \Z/950\Z} \chi_{950}(191,r) \chi_{950}(1,1-r) = 0 \)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{950}(191,·))
= \sum_{r \in \Z/950\Z}
\chi_{950}(191,r) e\left(\frac{1 r + 2 r^{-1}}{950}\right)
= 8.0748532232+5.8667242741i \)