from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2667, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([0,28,16]))
pari: [g,chi] = znchar(Mod(25,2667))
Basic properties
| Modulus: | \(2667\) | |
| 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 \(\chi_{889}(25,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2667.cv
\(\chi_{2667}(25,\cdot)\) \(\chi_{2667}(625,\cdot)\) \(\chi_{2667}(856,\cdot)\) \(\chi_{2667}(1054,\cdot)\) \(\chi_{2667}(1243,\cdot)\) \(\chi_{2667}(1444,\cdot)\) \(\chi_{2667}(1600,\cdot)\) \(\chi_{2667}(1738,\cdot)\) \(\chi_{2667}(1978,\cdot)\) \(\chi_{2667}(2209,\cdot)\) \(\chi_{2667}(2347,\cdot)\) \(\chi_{2667}(2662,\cdot)\)
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: | 21.21.808066270618405716993861719647864148675120272481133649.2 |
Values on generators
\((890,1144,2416)\) → \((1,e\left(\frac{2}{3}\right),e\left(\frac{8}{21}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 2667 }(25, a) \) | \(1\) | \(1\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{1}{3}\right)\) |
sage: chi.jacobi_sum(n)