from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6760, base_ring=CyclotomicField(26))
M = H._module
chi = DirichletCharacter(H, M([0,0,0,25]))
pari: [g,chi] = znchar(Mod(441,6760))
Basic properties
| Modulus: | \(6760\) | |
| Conductor: | \(169\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(26\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{169}(103,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6760.di
\(\chi_{6760}(441,\cdot)\) \(\chi_{6760}(961,\cdot)\) \(\chi_{6760}(1481,\cdot)\) \(\chi_{6760}(2001,\cdot)\) \(\chi_{6760}(2521,\cdot)\) \(\chi_{6760}(3561,\cdot)\) \(\chi_{6760}(4081,\cdot)\) \(\chi_{6760}(4601,\cdot)\) \(\chi_{6760}(5121,\cdot)\) \(\chi_{6760}(5641,\cdot)\) \(\chi_{6760}(6161,\cdot)\) \(\chi_{6760}(6681,\cdot)\)
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: | 26.26.3830224792147131369362629348887201408953937846517364173.1 |
Values on generators
\((5071,3381,4057,5241)\) → \((1,1,1,e\left(\frac{25}{26}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 6760 }(441, a) \) | \(1\) | \(1\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{23}{26}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(-1\) | \(e\left(\frac{3}{26}\right)\) | \(1\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{6}{13}\right)\) |
sage: chi.jacobi_sum(n)