from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4056, base_ring=CyclotomicField(26))
M = H._module
chi = DirichletCharacter(H, M([13,0,0,25]))
pari: [g,chi] = znchar(Mod(103,4056))
Basic properties
| Modulus: | \(4056\) | |
| Conductor: | \(676\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(26\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{676}(103,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4056.ce
\(\chi_{4056}(103,\cdot)\) \(\chi_{4056}(415,\cdot)\) \(\chi_{4056}(727,\cdot)\) \(\chi_{4056}(1039,\cdot)\) \(\chi_{4056}(1663,\cdot)\) \(\chi_{4056}(1975,\cdot)\) \(\chi_{4056}(2287,\cdot)\) \(\chi_{4056}(2599,\cdot)\) \(\chi_{4056}(2911,\cdot)\) \(\chi_{4056}(3223,\cdot)\) \(\chi_{4056}(3535,\cdot)\) \(\chi_{4056}(3847,\cdot)\)
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: | 26.0.257042034665630107056690459656879750694098197206386665924329472.1 |
Values on generators
\((1015,2029,2705,3889)\) → \((-1,1,1,e\left(\frac{25}{26}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 4056 }(103, a) \) | \(-1\) | \(1\) | \(e\left(\frac{17}{26}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(1\) | \(-1\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{1}{26}\right)\) |
sage: chi.jacobi_sum(n)