from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4706, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([15,4]))
pari: [g,chi] = znchar(Mod(421,4706))
Basic properties
Modulus: | \(4706\) | |
Conductor: | \(2353\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(20\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{2353}(421,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4706.cu
\(\chi_{4706}(125,\cdot)\) \(\chi_{4706}(135,\cdot)\) \(\chi_{4706}(421,\cdot)\) \(\chi_{4706}(1945,\cdot)\) \(\chi_{4706}(2033,\cdot)\) \(\chi_{4706}(2231,\cdot)\) \(\chi_{4706}(3021,\cdot)\) \(\chi_{4706}(3843,\cdot)\)
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: | \(\Q(\zeta_{20})\) |
Fixed field: | Number field defined by a degree 20 polynomial |
Values on generators
\((2173,183)\) → \((-i,e\left(\frac{1}{5}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(15\) | \(17\) | \(19\) | \(21\) | \(23\) |
\( \chi_{ 4706 }(421, a) \) | \(-1\) | \(1\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(i\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(-1\) | \(-i\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) |
sage: chi.jacobi_sum(n)