sage: from sage.modular.dirichlet import DirichletCharacter
sage: H = DirichletGroup(429, base_ring=CyclotomicField(4))
sage: M = H._module
sage: chi = DirichletCharacter(H, M([0,2,3]))
pari: [g,chi] = znchar(Mod(109,429))
Basic properties
| Modulus: | \(429\) | |
| Conductor: | \(143\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(4\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{143}(109,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 429.m
\(\chi_{429}(109,\cdot)\) \(\chi_{429}(307,\cdot)\)
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
\((287,79,67)\) → \((1,-1,-i)\)
Values
| \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \(1\) | \(1\) | \(i\) | \(-1\) | \(-i\) | \(-i\) | \(-i\) | \(1\) | \(1\) | \(1\) | \(1\) | \(i\) |
Related number fields
| Field of values: | \(\Q(\sqrt{-1}) \) |
| Fixed field: | 4.4.265837.1 |
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{429}(109,\cdot)) = \sum_{r\in \Z/429\Z} \chi_{429}(109,r) e\left(\frac{2r}{429}\right) = -3.4653144092+11.4451560079i \)
Jacobi sum
sage: chi.jacobi_sum(n)
\( \displaystyle J(\chi_{429}(109,\cdot),\chi_{429}(1,\cdot)) = \sum_{r\in \Z/429\Z} \chi_{429}(109,r) \chi_{429}(1,1-r) = 1 \)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{429}(109,·))
= \sum_{r \in \Z/429\Z}
\chi_{429}(109,r) e\left(\frac{1 r + 2 r^{-1}}{429}\right)
= 0.0 \)