from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9295, base_ring=CyclotomicField(26))
M = H._module
chi = DirichletCharacter(H, M([13,0,9]))
pari: [g,chi] = znchar(Mod(584,9295))
Basic properties
| Modulus: | \(9295\) | |
| 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 \(\chi_{845}(584,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 9295.cm
\(\chi_{9295}(584,\cdot)\) \(\chi_{9295}(1299,\cdot)\) \(\chi_{9295}(2014,\cdot)\) \(\chi_{9295}(2729,\cdot)\) \(\chi_{9295}(3444,\cdot)\) \(\chi_{9295}(4159,\cdot)\) \(\chi_{9295}(4874,\cdot)\) \(\chi_{9295}(5589,\cdot)\) \(\chi_{9295}(6304,\cdot)\) \(\chi_{9295}(7019,\cdot)\) \(\chi_{9295}(7734,\cdot)\) \(\chi_{9295}(9164,\cdot)\)
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
\((7437,4226,6931)\) → \((-1,1,e\left(\frac{9}{26}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(12\) | \(14\) | \(16\) |
| \( \chi_{ 9295 }(584, a) \) | \(1\) | \(1\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{3}{26}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{5}{13}\right)\) |
sage: chi.jacobi_sum(n)