from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2793, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,3,28]))
pari: [g,chi] = znchar(Mod(125,2793))
Basic properties
| Modulus: | \(2793\) | |
| Conductor: | \(2793\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(42\) | 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 2793.dm
\(\chi_{2793}(83,\cdot)\) \(\chi_{2793}(125,\cdot)\) \(\chi_{2793}(482,\cdot)\) \(\chi_{2793}(524,\cdot)\) \(\chi_{2793}(923,\cdot)\) \(\chi_{2793}(1280,\cdot)\) \(\chi_{2793}(1679,\cdot)\) \(\chi_{2793}(1721,\cdot)\) \(\chi_{2793}(2078,\cdot)\) \(\chi_{2793}(2120,\cdot)\) \(\chi_{2793}(2477,\cdot)\) \(\chi_{2793}(2519,\cdot)\)
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 42 polynomial |
Values on generators
\((932,2110,2206)\) → \((-1,e\left(\frac{1}{14}\right),e\left(\frac{2}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(20\) |
| \( \chi_{ 2793 }(125, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{2}{7}\right)\) |
sage: chi.jacobi_sum(n)