from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2432, base_ring=CyclotomicField(32))
M = H._module
chi = DirichletCharacter(H, M([0,21,16]))
pari: [g,chi] = znchar(Mod(2165,2432))
Basic properties
Modulus: | \(2432\) | |
Conductor: | \(2432\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(32\) | 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 2432.bt
\(\chi_{2432}(37,\cdot)\) \(\chi_{2432}(189,\cdot)\) \(\chi_{2432}(341,\cdot)\) \(\chi_{2432}(493,\cdot)\) \(\chi_{2432}(645,\cdot)\) \(\chi_{2432}(797,\cdot)\) \(\chi_{2432}(949,\cdot)\) \(\chi_{2432}(1101,\cdot)\) \(\chi_{2432}(1253,\cdot)\) \(\chi_{2432}(1405,\cdot)\) \(\chi_{2432}(1557,\cdot)\) \(\chi_{2432}(1709,\cdot)\) \(\chi_{2432}(1861,\cdot)\) \(\chi_{2432}(2013,\cdot)\) \(\chi_{2432}(2165,\cdot)\) \(\chi_{2432}(2317,\cdot)\)
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: | 32.0.905288048831351058796666807211863041216387224344298280390835989733155786457088.1 |
Values on generators
\((1407,2053,1921)\) → \((1,e\left(\frac{21}{32}\right),-1)\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(21\) | \(23\) |
\( \chi_{ 2432 }(2165, a) \) | \(-1\) | \(1\) | \(e\left(\frac{15}{32}\right)\) | \(e\left(\frac{21}{32}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{25}{32}\right)\) | \(e\left(\frac{11}{32}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{1}{32}\right)\) | \(e\left(\frac{3}{16}\right)\) |
sage: chi.jacobi_sum(n)