sage: from sage.modular.dirichlet import DirichletCharacter
sage: H = DirichletGroup(241, base_ring=CyclotomicField(20))
sage: M = H._module
sage: chi = DirichletCharacter(H, M([1]))
pari: [g,chi] = znchar(Mod(235,241))
Basic properties
| Modulus: | \(241\) | |
| Conductor: | \(241\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(20\) | 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 241.l
\(\chi_{241}(6,\cdot)\) \(\chi_{241}(25,\cdot)\) \(\chi_{241}(40,\cdot)\) \(\chi_{241}(106,\cdot)\) \(\chi_{241}(135,\cdot)\) \(\chi_{241}(201,\cdot)\) \(\chi_{241}(216,\cdot)\) \(\chi_{241}(235,\cdot)\)
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: | \(\Q(\zeta_{20})\) |
| Fixed field: | 20.20.1812691127043107566808878608684122677382270161.1 |
Values on generators
\(7\) → \(e\left(\frac{1}{20}\right)\)
Values
| \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \(1\) | \(1\) | \(-1\) | \(e\left(\frac{1}{10}\right)\) | \(1\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(-1\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(i\) |
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
\(\displaystyle \tau_{2}(\chi_{241}(235,\cdot)) = \sum_{r\in \Z/241\Z} \chi_{241}(235,r) e\left(\frac{2r}{241}\right) = -10.4722826988+-11.4599866962i \)
Jacobi sum
sage: chi.jacobi_sum(n)
\( \displaystyle J(\chi_{241}(235,\cdot),\chi_{241}(1,\cdot)) = \sum_{r\in \Z/241\Z} \chi_{241}(235,r) \chi_{241}(1,1-r) = -1 \)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)
\( \displaystyle K(1,2,\chi_{241}(235,·))
= \sum_{r \in \Z/241\Z}
\chi_{241}(235,r) e\left(\frac{1 r + 2 r^{-1}}{241}\right)
= -20.9471623697i \)