sage: from sage.modular.dirichlet import DirichletCharacter
sage: H = DirichletGroup(416, base_ring=CyclotomicField(24))
sage: M = H._module
sage: chi = DirichletCharacter(H, M([12,21,20]))
pari: [g,chi] = znchar(Mod(179,416))
Basic properties
| Modulus: | \(416\) | |
| Conductor: | \(416\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(24\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 416.cb
\(\chi_{416}(43,\cdot)\) \(\chi_{416}(75,\cdot)\) \(\chi_{416}(147,\cdot)\) \(\chi_{416}(179,\cdot)\) \(\chi_{416}(251,\cdot)\) \(\chi_{416}(283,\cdot)\) \(\chi_{416}(355,\cdot)\) \(\chi_{416}(387,\cdot)\)
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
\((287,261,353)\) → \((-1,e\left(\frac{7}{8}\right),e\left(\frac{5}{6}\right))\)
Values
| \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(15\) | \(17\) | \(19\) | \(21\) | \(23\) |
| \(-1\) | \(1\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{1}{12}\right)\) |
Related number fields
| Field of values: | \(\Q(\zeta_{24})\) |
| Fixed field: | 24.0.188216044816745326913150945765287080795084676923392.1 |
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{416}(179,\cdot)) = \sum_{r\in \Z/416\Z} \chi_{416}(179,r) e\left(\frac{r}{208}\right) = 0.0 \)
Jacobi sum
sage: chi.jacobi_sum(n)
\( \displaystyle J(\chi_{416}(179,\cdot),\chi_{416}(1,\cdot)) = \sum_{r\in \Z/416\Z} \chi_{416}(179,r) \chi_{416}(1,1-r) = 0 \)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{416}(179,·))
= \sum_{r \in \Z/416\Z}
\chi_{416}(179,r) e\left(\frac{1 r + 2 r^{-1}}{416}\right)
= 21.3427286066+1.3988763388i \)