sage: from sage.modular.dirichlet import DirichletCharacter
sage: H = DirichletGroup(143, base_ring=CyclotomicField(10))
sage: M = H._module
sage: chi = DirichletCharacter(H, M([7,0]))
pari: [g,chi] = znchar(Mod(40,143))
Basic properties
| Modulus: | \(143\) | |
| Conductor: | \(11\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(10\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{11}(7,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 143.m
\(\chi_{143}(40,\cdot)\) \(\chi_{143}(79,\cdot)\) \(\chi_{143}(105,\cdot)\) \(\chi_{143}(118,\cdot)\)
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: | \(\Q(\zeta_{5})\) |
| Fixed field: | \(\Q(\zeta_{11})\) |
Values on generators
\((79,67)\) → \((e\left(\frac{7}{10}\right),1)\)
Values
| \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
| \(-1\) | \(1\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(-1\) | \(1\) |
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{143}(40,\cdot)) = \sum_{r\in \Z/143\Z} \chi_{143}(40,r) e\left(\frac{2r}{143}\right) = -2.5412780247+2.1311747937i \)
Jacobi sum
sage: chi.jacobi_sum(n)
\( \displaystyle J(\chi_{143}(40,\cdot),\chi_{143}(1,\cdot)) = \sum_{r\in \Z/143\Z} \chi_{143}(40,r) \chi_{143}(1,1-r) = -11 \)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{143}(40,·))
= \sum_{r \in \Z/143\Z}
\chi_{143}(40,r) e\left(\frac{1 r + 2 r^{-1}}{143}\right)
= 2.006477495+1.4577912316i \)