from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(900, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,10,33]))
pari: [g,chi] = znchar(Mod(623,900))
Basic properties
| Modulus: | \(900\) | |
| Conductor: | \(900\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | 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 900.bu
\(\chi_{900}(23,\cdot)\) \(\chi_{900}(47,\cdot)\) \(\chi_{900}(83,\cdot)\) \(\chi_{900}(167,\cdot)\) \(\chi_{900}(203,\cdot)\) \(\chi_{900}(227,\cdot)\) \(\chi_{900}(263,\cdot)\) \(\chi_{900}(347,\cdot)\) \(\chi_{900}(383,\cdot)\) \(\chi_{900}(527,\cdot)\) \(\chi_{900}(563,\cdot)\) \(\chi_{900}(587,\cdot)\) \(\chi_{900}(623,\cdot)\) \(\chi_{900}(767,\cdot)\) \(\chi_{900}(803,\cdot)\) \(\chi_{900}(887,\cdot)\)
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: | \(\Q(\zeta_{60})\) |
| Fixed field: | Number field defined by a degree 60 polynomial |
Values on generators
\((451,101,577)\) → \((-1,e\left(\frac{1}{6}\right),e\left(\frac{11}{20}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 900 }(623, a) \) | \(-1\) | \(1\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{1}{30}\right)\) |
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)