from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(873, base_ring=CyclotomicField(32))
M = H._module
chi = DirichletCharacter(H, M([0,21]))
pari: [g,chi] = znchar(Mod(730,873))
Basic properties
| Modulus: | \(873\) | |
| Conductor: | \(97\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(32\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{97}(51,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 873.br
\(\chi_{873}(19,\cdot)\) \(\chi_{873}(28,\cdot)\) \(\chi_{873}(46,\cdot)\) \(\chi_{873}(55,\cdot)\) \(\chi_{873}(127,\cdot)\) \(\chi_{873}(271,\cdot)\) \(\chi_{873}(325,\cdot)\) \(\chi_{873}(343,\cdot)\) \(\chi_{873}(433,\cdot)\) \(\chi_{873}(451,\cdot)\) \(\chi_{873}(505,\cdot)\) \(\chi_{873}(649,\cdot)\) \(\chi_{873}(721,\cdot)\) \(\chi_{873}(730,\cdot)\) \(\chi_{873}(748,\cdot)\) \(\chi_{873}(757,\cdot)\)
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: | \(\Q(\zeta_{32})\) |
| Fixed field: | Number field defined by a degree 32 polynomial |
Values on generators
\((389,199)\) → \((1,e\left(\frac{21}{32}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 873 }(730, a) \) | \(-1\) | \(1\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{21}{32}\right)\) | \(e\left(\frac{11}{32}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{31}{32}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{13}{32}\right)\) | \(e\left(\frac{21}{32}\right)\) | \(i\) |
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)