from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(444, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([18,18,25]))
pari: [g,chi] = znchar(Mod(131,444))
Basic properties
| Modulus: | \(444\) | |
| Conductor: | \(444\) | 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 444.bi
\(\chi_{444}(35,\cdot)\) \(\chi_{444}(59,\cdot)\) \(\chi_{444}(131,\cdot)\) \(\chi_{444}(143,\cdot)\) \(\chi_{444}(167,\cdot)\) \(\chi_{444}(203,\cdot)\) \(\chi_{444}(227,\cdot)\) \(\chi_{444}(239,\cdot)\) \(\chi_{444}(311,\cdot)\) \(\chi_{444}(335,\cdot)\) \(\chi_{444}(383,\cdot)\) \(\chi_{444}(431,\cdot)\)
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: | 36.0.205268718591960451664107537071777407379898873740754655345797836302054326272.1 |
Values on generators
\((223,149,409)\) → \((-1,-1,e\left(\frac{25}{36}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 444 }(131, a) \) | \(-1\) | \(1\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(-i\) |
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)