from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3216, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([0,0,0,1]))
pari: [g,chi] = znchar(Mod(2353,3216))
Basic properties
| Modulus: | \(3216\) | |
| Conductor: | \(67\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(22\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{67}(8,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3216.cl
\(\chi_{3216}(673,\cdot)\) \(\chi_{3216}(913,\cdot)\) \(\chi_{3216}(1057,\cdot)\) \(\chi_{3216}(1249,\cdot)\) \(\chi_{3216}(1345,\cdot)\) \(\chi_{3216}(1393,\cdot)\) \(\chi_{3216}(1921,\cdot)\) \(\chi_{3216}(2305,\cdot)\) \(\chi_{3216}(2353,\cdot)\) \(\chi_{3216}(3073,\cdot)\)
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: | \(\Q(\zeta_{11})\) |
| Fixed field: | Number field defined by a degree 22 polynomial |
Values on generators
\((2815,805,1073,337)\) → \((1,1,1,e\left(\frac{1}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 3216 }(2353, a) \) | \(-1\) | \(1\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(1\) | \(e\left(\frac{3}{22}\right)\) |
sage: chi.jacobi_sum(n)