sage: from sage.modular.dirichlet import DirichletCharacter
sage: H = DirichletGroup(119, base_ring=CyclotomicField(16))
sage: M = H._module
sage: chi = DirichletCharacter(H, M([0,7]))
pari: [g,chi] = znchar(Mod(113,119))
Basic properties
| Modulus: | \(119\) | |
| Conductor: | \(17\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(16\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{17}(11,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 119.o
\(\chi_{119}(22,\cdot)\) \(\chi_{119}(29,\cdot)\) \(\chi_{119}(57,\cdot)\) \(\chi_{119}(71,\cdot)\) \(\chi_{119}(78,\cdot)\) \(\chi_{119}(92,\cdot)\) \(\chi_{119}(99,\cdot)\) \(\chi_{119}(113,\cdot)\)
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: | \(\Q(\zeta_{16})\) |
| Fixed field: | \(\Q(\zeta_{17})\) |
Values on generators
\((52,71)\) → \((1,e\left(\frac{7}{16}\right))\)
Values
| \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
| \(-1\) | \(1\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(i\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{11}{16}\right)\) |
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{119}(113,\cdot)) = \sum_{r\in \Z/119\Z} \chi_{119}(113,r) e\left(\frac{2r}{119}\right) = -1.0320212957+-3.9918582196i \)
Jacobi sum
sage: chi.jacobi_sum(n)
\( \displaystyle J(\chi_{119}(113,\cdot),\chi_{119}(1,\cdot)) = \sum_{r\in \Z/119\Z} \chi_{119}(113,r) \chi_{119}(1,1-r) = -5 \)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{119}(113,·))
= \sum_{r \in \Z/119\Z}
\chi_{119}(113,r) e\left(\frac{1 r + 2 r^{-1}}{119}\right)
= 3.9058394587+-9.4295305936i \)