from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(540, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([18,32,27]))
pari: [g,chi] = znchar(Mod(223,540))
Basic properties
| Modulus: | \(540\) | |
| Conductor: | \(540\) | 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 540.bh
\(\chi_{540}(7,\cdot)\) \(\chi_{540}(43,\cdot)\) \(\chi_{540}(67,\cdot)\) \(\chi_{540}(103,\cdot)\) \(\chi_{540}(187,\cdot)\) \(\chi_{540}(223,\cdot)\) \(\chi_{540}(247,\cdot)\) \(\chi_{540}(283,\cdot)\) \(\chi_{540}(367,\cdot)\) \(\chi_{540}(403,\cdot)\) \(\chi_{540}(427,\cdot)\) \(\chi_{540}(463,\cdot)\)
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
\((271,461,217)\) → \((-1,e\left(\frac{8}{9}\right),-i)\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 540 }(223, a) \) | \(1\) | \(1\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{1}{9}\right)\) |
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)