sage: from sage.modular.dirichlet import DirichletCharacter
sage: H = DirichletGroup(247, base_ring=CyclotomicField(18))
sage: M = H._module
sage: chi = DirichletCharacter(H, M([6,8]))
pari: [g,chi] = znchar(Mod(237,247))
Basic properties
Modulus: | \(247\) | |
Conductor: | \(247\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(9\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 247.w
\(\chi_{247}(35,\cdot)\) \(\chi_{247}(42,\cdot)\) \(\chi_{247}(74,\cdot)\) \(\chi_{247}(100,\cdot)\) \(\chi_{247}(120,\cdot)\) \(\chi_{247}(237,\cdot)\)
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
\((210,40)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{4}{9}\right))\)
Values
\(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
\(1\) | \(1\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{2}{3}\right)\) |
Related number fields
Field of values: | \(\Q(\zeta_{9})\) |
Fixed field: | 9.9.81976414938366169.2 |
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{247}(237,\cdot)) = \sum_{r\in \Z/247\Z} \chi_{247}(237,r) e\left(\frac{2r}{247}\right) = -1.7172394554+15.6221345741i \)
Jacobi sum
sage: chi.jacobi_sum(n)
\( \displaystyle J(\chi_{247}(237,\cdot),\chi_{247}(1,\cdot)) = \sum_{r\in \Z/247\Z} \chi_{247}(237,r) \chi_{247}(1,1-r) = 1 \)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{247}(237,·))
= \sum_{r \in \Z/247\Z}
\chi_{247}(237,r) e\left(\frac{1 r + 2 r^{-1}}{247}\right)
= 1.5202811183+-1.2756673256i \)