from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(333, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([30,5]))
pari: [g,chi] = znchar(Mod(32,333))
Basic properties
| Modulus: | \(333\) | |
| Conductor: | \(333\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(36\) | 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 333.bv
\(\chi_{333}(2,\cdot)\) \(\chi_{333}(5,\cdot)\) \(\chi_{333}(20,\cdot)\) \(\chi_{333}(32,\cdot)\) \(\chi_{333}(50,\cdot)\) \(\chi_{333}(128,\cdot)\) \(\chi_{333}(146,\cdot)\) \(\chi_{333}(167,\cdot)\) \(\chi_{333}(200,\cdot)\) \(\chi_{333}(203,\cdot)\) \(\chi_{333}(281,\cdot)\) \(\chi_{333}(320,\cdot)\)
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: | \(\Q(\zeta_{36})\) |
| Fixed field: | Number field defined by a degree 36 polynomial |
Values on generators
\((38,298)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{5}{36}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 333 }(32, a) \) | \(1\) | \(1\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(1\) | \(e\left(\frac{7}{36}\right)\) | \(-i\) | \(e\left(\frac{8}{9}\right)\) |
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)