sage: from sage.modular.dirichlet import DirichletCharacter
sage: H = DirichletGroup(950, base_ring=CyclotomicField(30))
sage: M = H._module
sage: chi = DirichletCharacter(H, M([24,20]))
pari: [g,chi] = znchar(Mod(11,950))
Basic properties
| Modulus: | \(950\) | |
| Conductor: | \(475\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(15\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{475}(11,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 950.r
\(\chi_{950}(11,\cdot)\) \(\chi_{950}(121,\cdot)\) \(\chi_{950}(311,\cdot)\) \(\chi_{950}(391,\cdot)\) \(\chi_{950}(581,\cdot)\) \(\chi_{950}(691,\cdot)\) \(\chi_{950}(771,\cdot)\) \(\chi_{950}(881,\cdot)\)
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
\((77,401)\) → \((e\left(\frac{4}{5}\right),e\left(\frac{2}{3}\right))\)
Values
| \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \(1\) | \(1\) | \(e\left(\frac{4}{15}\right)\) | \(1\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{14}{15}\right)\) |
Related number fields
| Field of values: | \(\Q(\zeta_{15})\) |
| Fixed field: | 15.15.365440026390612125396728515625.1 |
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{950}(11,\cdot)) = \sum_{r\in \Z/950\Z} \chi_{950}(11,r) e\left(\frac{r}{475}\right) = 12.3772476978+17.9388890243i \)
Jacobi sum
sage: chi.jacobi_sum(n)
\( \displaystyle J(\chi_{950}(11,\cdot),\chi_{950}(1,\cdot)) = \sum_{r\in \Z/950\Z} \chi_{950}(11,r) \chi_{950}(1,1-r) = 0 \)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{950}(11,·))
= \sum_{r \in \Z/950\Z}
\chi_{950}(11,r) e\left(\frac{1 r + 2 r^{-1}}{950}\right)
= 3.810567139+36.255124538i \)