from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(241, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([31]))
pari: [g,chi] = znchar(Mod(9,241))
Basic properties
| Modulus: | \(241\) | |
| Conductor: | \(241\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | 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.q
\(\chi_{241}(9,\cdot)\) \(\chi_{241}(82,\cdot)\) \(\chi_{241}(83,\cdot)\) \(\chi_{241}(90,\cdot)\) \(\chi_{241}(96,\cdot)\) \(\chi_{241}(97,\cdot)\) \(\chi_{241}(107,\cdot)\) \(\chi_{241}(118,\cdot)\) \(\chi_{241}(123,\cdot)\) \(\chi_{241}(134,\cdot)\) \(\chi_{241}(144,\cdot)\) \(\chi_{241}(145,\cdot)\) \(\chi_{241}(151,\cdot)\) \(\chi_{241}(158,\cdot)\) \(\chi_{241}(159,\cdot)\) \(\chi_{241}(232,\cdot)\)
sage: chi.galois_orbit()
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Related number fields
| Field of values: | \(\Q(\zeta_{60})\) |
| Fixed field: | Number field defined by a degree 60 polynomial |
Values on generators
\(7\) → \(e\left(\frac{31}{60}\right)\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 241 }(9, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(-1\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{11}{12}\right)\) |
sage: chi.jacobi_sum(n)
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
Jacobi sum
sage: chi.jacobi_sum(n)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)