from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(211, base_ring=CyclotomicField(70))
M = H._module
chi = DirichletCharacter(H, M([39]))
pari: [g,chi] = znchar(Mod(124,211))
Basic properties
| Modulus: | \(211\) | |
| Conductor: | \(211\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(70\) | 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 211.n
\(\chi_{211}(8,\cdot)\) \(\chi_{211}(18,\cdot)\) \(\chi_{211}(27,\cdot)\) \(\chi_{211}(28,\cdot)\) \(\chi_{211}(42,\cdot)\) \(\chi_{211}(60,\cdot)\) \(\chi_{211}(68,\cdot)\) \(\chi_{211}(86,\cdot)\) \(\chi_{211}(89,\cdot)\) \(\chi_{211}(90,\cdot)\) \(\chi_{211}(97,\cdot)\) \(\chi_{211}(98,\cdot)\) \(\chi_{211}(102,\cdot)\) \(\chi_{211}(115,\cdot)\) \(\chi_{211}(124,\cdot)\) \(\chi_{211}(129,\cdot)\) \(\chi_{211}(132,\cdot)\) \(\chi_{211}(135,\cdot)\) \(\chi_{211}(146,\cdot)\) \(\chi_{211}(147,\cdot)\) \(\chi_{211}(186,\cdot)\) \(\chi_{211}(198,\cdot)\) \(\chi_{211}(200,\cdot)\) \(\chi_{211}(206,\cdot)\)
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: | $\Q(\zeta_{35})$ |
| Fixed field: | Number field defined by a degree 70 polynomial |
Values on generators
\(2\) → \(e\left(\frac{39}{70}\right)\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 211 }(124, a) \) | \(-1\) | \(1\) | \(e\left(\frac{39}{70}\right)\) | \(e\left(\frac{67}{70}\right)\) | \(e\left(\frac{4}{35}\right)\) | \(e\left(\frac{19}{35}\right)\) | \(e\left(\frac{18}{35}\right)\) | \(e\left(\frac{31}{70}\right)\) | \(e\left(\frac{47}{70}\right)\) | \(e\left(\frac{32}{35}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{9}{35}\right)\) |
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)