from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7581, base_ring=CyclotomicField(18))
M = H._module
chi = DirichletCharacter(H, M([0,6,4]))
pari: [g,chi] = znchar(Mod(415,7581))
Basic properties
Modulus: | \(7581\) | |
Conductor: | \(133\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(9\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{133}(16,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 7581.bq
\(\chi_{7581}(415,\cdot)\) \(\chi_{7581}(4216,\cdot)\) \(\chi_{7581}(4792,\cdot)\) \(\chi_{7581}(5443,\cdot)\) \(\chi_{7581}(6010,\cdot)\) \(\chi_{7581}(7282,\cdot)\)
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: | \(\Q(\zeta_{9})\) |
Fixed field: | 9.9.1998099208210609.1 |
Values on generators
\((2528,6499,1807)\) → \((1,e\left(\frac{1}{3}\right),e\left(\frac{2}{9}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(20\) |
\( \chi_{ 7581 }(415, a) \) | \(1\) | \(1\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(1\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(1\) |
sage: chi.jacobi_sum(n)