from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5586, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([0,20,28]))
pari: [g,chi] = znchar(Mod(163,5586))
Basic properties
Modulus: | \(5586\) | |
Conductor: | \(931\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(21\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{931}(163,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5586.cx
\(\chi_{5586}(163,\cdot)\) \(\chi_{5586}(235,\cdot)\) \(\chi_{5586}(1033,\cdot)\) \(\chi_{5586}(1759,\cdot)\) \(\chi_{5586}(2557,\cdot)\) \(\chi_{5586}(2629,\cdot)\) \(\chi_{5586}(3355,\cdot)\) \(\chi_{5586}(3427,\cdot)\) \(\chi_{5586}(4153,\cdot)\) \(\chi_{5586}(4225,\cdot)\) \(\chi_{5586}(4951,\cdot)\) \(\chi_{5586}(5023,\cdot)\)
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: | \(\Q(\zeta_{21})\) |
Fixed field: | 21.21.103818783062189717738091671292152377422379176428329.2 |
Values on generators
\((3725,4903,4999)\) → \((1,e\left(\frac{10}{21}\right),e\left(\frac{2}{3}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
\( \chi_{ 5586 }(163, a) \) | \(1\) | \(1\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{17}{21}\right)\) |
sage: chi.jacobi_sum(n)