from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9464, base_ring=CyclotomicField(26))
M = H._module
chi = DirichletCharacter(H, M([0,13,0,3]))
pari: [g,chi] = znchar(Mod(701,9464))
Basic properties
Modulus: | \(9464\) | |
Conductor: | \(1352\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(26\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{1352}(701,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 9464.ey
\(\chi_{9464}(701,\cdot)\) \(\chi_{9464}(1429,\cdot)\) \(\chi_{9464}(2157,\cdot)\) \(\chi_{9464}(2885,\cdot)\) \(\chi_{9464}(3613,\cdot)\) \(\chi_{9464}(4341,\cdot)\) \(\chi_{9464}(5797,\cdot)\) \(\chi_{9464}(6525,\cdot)\) \(\chi_{9464}(7253,\cdot)\) \(\chi_{9464}(7981,\cdot)\) \(\chi_{9464}(8709,\cdot)\) \(\chi_{9464}(9437,\cdot)\)
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: | 26.26.2105688347980841837008408245509158917686052431514719567252107034624.1 |
Values on generators
\((2367,4733,2705,9297)\) → \((1,-1,1,e\left(\frac{3}{26}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) | \(27\) |
\( \chi_{ 9464 }(701, a) \) | \(1\) | \(1\) | \(e\left(\frac{21}{26}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(1\) | \(1\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{11}{26}\right)\) |
sage: chi.jacobi_sum(n)