from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6760, base_ring=CyclotomicField(26))
M = H._module
chi = DirichletCharacter(H, M([0,0,13,2]))
pari: [g,chi] = znchar(Mod(209,6760))
Basic properties
| Modulus: | \(6760\) | |
| Conductor: | \(845\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(26\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{845}(209,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6760.dp
\(\chi_{6760}(209,\cdot)\) \(\chi_{6760}(729,\cdot)\) \(\chi_{6760}(1249,\cdot)\) \(\chi_{6760}(1769,\cdot)\) \(\chi_{6760}(2289,\cdot)\) \(\chi_{6760}(2809,\cdot)\) \(\chi_{6760}(3329,\cdot)\) \(\chi_{6760}(3849,\cdot)\) \(\chi_{6760}(4369,\cdot)\) \(\chi_{6760}(4889,\cdot)\) \(\chi_{6760}(5929,\cdot)\) \(\chi_{6760}(6449,\cdot)\)
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: | \(\Q(\zeta_{13})\) |
| Fixed field: | Number field defined by a degree 26 polynomial |
Values on generators
\((5071,3381,4057,5241)\) → \((1,1,-1,e\left(\frac{1}{13}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 6760 }(209, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{19}{26}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(e\left(\frac{19}{26}\right)\) | \(1\) | \(e\left(\frac{10}{13}\right)\) | \(-1\) | \(e\left(\frac{3}{26}\right)\) | \(e\left(\frac{1}{13}\right)\) |
sage: chi.jacobi_sum(n)