from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4029, base_ring=CyclotomicField(26))
M = H._module
chi = DirichletCharacter(H, M([13,13,25]))
pari: [g,chi] = znchar(Mod(1937,4029))
Basic properties
| Modulus: | \(4029\) | |
| Conductor: | \(4029\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(26\) | 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 4029.bo
\(\chi_{4029}(254,\cdot)\) \(\chi_{4029}(407,\cdot)\) \(\chi_{4029}(611,\cdot)\) \(\chi_{4029}(1019,\cdot)\) \(\chi_{4029}(1121,\cdot)\) \(\chi_{4029}(1325,\cdot)\) \(\chi_{4029}(1376,\cdot)\) \(\chi_{4029}(1886,\cdot)\) \(\chi_{4029}(1937,\cdot)\) \(\chi_{4029}(3059,\cdot)\) \(\chi_{4029}(3569,\cdot)\) \(\chi_{4029}(3977,\cdot)\)
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
\((2687,3556,3163)\) → \((-1,-1,e\left(\frac{25}{26}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 4029 }(1937, a) \) | \(1\) | \(1\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{6}{13}\right)\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{25}{26}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{21}{26}\right)\) | \(e\left(\frac{5}{13}\right)\) |
sage: chi.jacobi_sum(n)