from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2176, base_ring=CyclotomicField(32))
M = H._module
chi = DirichletCharacter(H, M([0,11,16]))
pari: [g,chi] = znchar(Mod(2141,2176))
Basic properties
Modulus: | \(2176\) | |
Conductor: | \(2176\) | 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: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2176.ej
\(\chi_{2176}(101,\cdot)\) \(\chi_{2176}(237,\cdot)\) \(\chi_{2176}(373,\cdot)\) \(\chi_{2176}(509,\cdot)\) \(\chi_{2176}(645,\cdot)\) \(\chi_{2176}(781,\cdot)\) \(\chi_{2176}(917,\cdot)\) \(\chi_{2176}(1053,\cdot)\) \(\chi_{2176}(1189,\cdot)\) \(\chi_{2176}(1325,\cdot)\) \(\chi_{2176}(1461,\cdot)\) \(\chi_{2176}(1597,\cdot)\) \(\chi_{2176}(1733,\cdot)\) \(\chi_{2176}(1869,\cdot)\) \(\chi_{2176}(2005,\cdot)\) \(\chi_{2176}(2141,\cdot)\)
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: | 32.32.152725625984366375633872344931707463035520335286488090842213114237373265215488.1 |
Values on generators
\((511,1157,513)\) → \((1,e\left(\frac{11}{32}\right),-1)\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(19\) | \(21\) | \(23\) |
\( \chi_{ 2176 }(2141, a) \) | \(1\) | \(1\) | \(e\left(\frac{17}{32}\right)\) | \(e\left(\frac{27}{32}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{23}{32}\right)\) | \(e\left(\frac{5}{32}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{29}{32}\right)\) | \(e\left(\frac{15}{32}\right)\) | \(e\left(\frac{5}{16}\right)\) |
sage: chi.jacobi_sum(n)