from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4864, base_ring=CyclotomicField(32))
M = H._module
chi = DirichletCharacter(H, M([0,17,0]))
pari: [g,chi] = znchar(Mod(153,4864))
Basic properties
| Modulus: | \(4864\) | |
| Conductor: | \(128\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(32\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{128}(69,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4864.bs
\(\chi_{4864}(153,\cdot)\) \(\chi_{4864}(457,\cdot)\) \(\chi_{4864}(761,\cdot)\) \(\chi_{4864}(1065,\cdot)\) \(\chi_{4864}(1369,\cdot)\) \(\chi_{4864}(1673,\cdot)\) \(\chi_{4864}(1977,\cdot)\) \(\chi_{4864}(2281,\cdot)\) \(\chi_{4864}(2585,\cdot)\) \(\chi_{4864}(2889,\cdot)\) \(\chi_{4864}(3193,\cdot)\) \(\chi_{4864}(3497,\cdot)\) \(\chi_{4864}(3801,\cdot)\) \(\chi_{4864}(4105,\cdot)\) \(\chi_{4864}(4409,\cdot)\) \(\chi_{4864}(4713,\cdot)\)
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: | \(\Q(\zeta_{128})^+\) |
Values on generators
\((3839,2053,4353)\) → \((1,e\left(\frac{17}{32}\right),1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(21\) | \(23\) |
| \( \chi_{ 4864 }(153, a) \) | \(1\) | \(1\) | \(e\left(\frac{19}{32}\right)\) | \(e\left(\frac{17}{32}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{5}{32}\right)\) | \(e\left(\frac{31}{32}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{29}{32}\right)\) | \(e\left(\frac{7}{16}\right)\) |
sage: chi.jacobi_sum(n)