from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4009, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([14,20]))
pari: [g,chi] = znchar(Mod(178,4009))
Basic properties
Modulus: | \(4009\) | |
Conductor: | \(4009\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(21\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4009.bv
\(\chi_{4009}(178,\cdot)\) \(\chi_{4009}(676,\cdot)\) \(\chi_{4009}(961,\cdot)\) \(\chi_{4009}(1109,\cdot)\) \(\chi_{4009}(1451,\cdot)\) \(\chi_{4009}(1550,\cdot)\) \(\chi_{4009}(2211,\cdot)\) \(\chi_{4009}(2500,\cdot)\) \(\chi_{4009}(3127,\cdot)\) \(\chi_{4009}(3621,\cdot)\) \(\chi_{4009}(3959,\cdot)\) \(\chi_{4009}(3978,\cdot)\)
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: | Number field defined by a degree 21 polynomial |
Values on generators
\((2111,1901)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{10}{21}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
\( \chi_{ 4009 }(178, a) \) | \(1\) | \(1\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(1\) | \(e\left(\frac{1}{7}\right)\) |
sage: chi.jacobi_sum(n)