from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(592, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([0,9,1]))
pari: [g,chi] = znchar(Mod(261,592))
Basic properties
| Modulus: | \(592\) | |
| Conductor: | \(592\) | 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: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 592.cb
\(\chi_{592}(13,\cdot)\) \(\chi_{592}(69,\cdot)\) \(\chi_{592}(93,\cdot)\) \(\chi_{592}(109,\cdot)\) \(\chi_{592}(261,\cdot)\) \(\chi_{592}(277,\cdot)\) \(\chi_{592}(301,\cdot)\) \(\chi_{592}(357,\cdot)\) \(\chi_{592}(429,\cdot)\) \(\chi_{592}(461,\cdot)\) \(\chi_{592}(501,\cdot)\) \(\chi_{592}(533,\cdot)\)
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: | 36.0.4886860176107258124616704873602845327686728999915307588219200292503475176863258640384.1 |
Values on generators
\((223,149,113)\) → \((1,i,e\left(\frac{1}{36}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 592 }(261, a) \) | \(-1\) | \(1\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{31}{36}\right)\) |
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)