from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8001, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([0,14,22]))
pari: [g,chi] = znchar(Mod(100,8001))
Basic properties
Modulus: | \(8001\) | |
Conductor: | \(889\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(21\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{889}(100,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8001.hw
\(\chi_{8001}(100,\cdot)\) \(\chi_{8001}(1549,\cdot)\) \(\chi_{8001}(1999,\cdot)\) \(\chi_{8001}(2881,\cdot)\) \(\chi_{8001}(3124,\cdot)\) \(\chi_{8001}(3502,\cdot)\) \(\chi_{8001}(6157,\cdot)\) \(\chi_{8001}(6400,\cdot)\) \(\chi_{8001}(6472,\cdot)\) \(\chi_{8001}(7102,\cdot)\) \(\chi_{8001}(7912,\cdot)\) \(\chi_{8001}(7921,\cdot)\)
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: | 21.21.808066270618405716993861719647864148675120272481133649.1 |
Values on generators
\((3557,1144,7750)\) → \((1,e\left(\frac{1}{3}\right),e\left(\frac{11}{21}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
\( \chi_{ 8001 }(100, a) \) | \(1\) | \(1\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{2}{3}\right)\) |
sage: chi.jacobi_sum(n)