from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2793, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,5,14]))
pari: [g,chi] = znchar(Mod(1223,2793))
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.dw
\(\chi_{2793}(26,\cdot)\) \(\chi_{2793}(353,\cdot)\) \(\chi_{2793}(425,\cdot)\) \(\chi_{2793}(752,\cdot)\) \(\chi_{2793}(824,\cdot)\) \(\chi_{2793}(1151,\cdot)\) \(\chi_{2793}(1223,\cdot)\) \(\chi_{2793}(1622,\cdot)\) \(\chi_{2793}(1949,\cdot)\) \(\chi_{2793}(2021,\cdot)\) \(\chi_{2793}(2348,\cdot)\) \(\chi_{2793}(2747,\cdot)\)
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
\((932,2110,2206)\) → \((-1,e\left(\frac{5}{42}\right),e\left(\frac{1}{3}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(20\) |
| \( \chi_{ 2793 }(1223, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{1}{7}\right)\) |
sage: chi.jacobi_sum(n)