from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8036, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([0,32,0]))
pari: [g,chi] = znchar(Mod(821,8036))
Basic properties
Modulus: | \(8036\) | |
Conductor: | \(49\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(21\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{49}(37,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8036.bx
\(\chi_{8036}(821,\cdot)\) \(\chi_{8036}(1313,\cdot)\) \(\chi_{8036}(1969,\cdot)\) \(\chi_{8036}(2461,\cdot)\) \(\chi_{8036}(3609,\cdot)\) \(\chi_{8036}(4265,\cdot)\) \(\chi_{8036}(4757,\cdot)\) \(\chi_{8036}(5413,\cdot)\) \(\chi_{8036}(5905,\cdot)\) \(\chi_{8036}(6561,\cdot)\) \(\chi_{8036}(7053,\cdot)\) \(\chi_{8036}(7709,\cdot)\)
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
\((4019,493,785)\) → \((1,e\left(\frac{16}{21}\right),1)\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) |
\( \chi_{ 8036 }(821, a) \) | \(1\) | \(1\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{4}{21}\right)\) |
sage: chi.jacobi_sum(n)