from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5184, base_ring=CyclotomicField(16))
M = H._module
chi = DirichletCharacter(H, M([0,1,0]))
pari: [g,chi] = znchar(Mod(325,5184))
Basic properties
Modulus: | \(5184\) | |
Conductor: | \(64\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(16\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{64}(5,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5184.be
\(\chi_{5184}(325,\cdot)\) \(\chi_{5184}(973,\cdot)\) \(\chi_{5184}(1621,\cdot)\) \(\chi_{5184}(2269,\cdot)\) \(\chi_{5184}(2917,\cdot)\) \(\chi_{5184}(3565,\cdot)\) \(\chi_{5184}(4213,\cdot)\) \(\chi_{5184}(4861,\cdot)\)
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: | \(\Q(\zeta_{16})\) |
Fixed field: | \(\Q(\zeta_{64})^+\) |
Values on generators
\((2431,325,1217)\) → \((1,e\left(\frac{1}{16}\right),1)\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
\( \chi_{ 5184 }(325, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{16}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(-i\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(-1\) |
sage: chi.jacobi_sum(n)