from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(87362, base_ring=CyclotomicField(1710))
M = H._module
chi = DirichletCharacter(H, M([1368,695]))
pari: [g,chi] = znchar(Mod(3,87362))
Basic properties
Modulus: | \(87362\) | |
Conductor: | \(3971\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(1710\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{3971}(3,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 87362.da
\(\chi_{87362}(3,\cdot)\) \(\chi_{87362}(269,\cdot)\) \(\chi_{87362}(565,\cdot)\) \(\chi_{87362}(735,\cdot)\) \(\chi_{87362}(971,\cdot)\) \(\chi_{87362}(1219,\cdot)\) \(\chi_{87362}(1237,\cdot)\) \(\chi_{87362}(1533,\cdot)\)
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: | $\Q(\zeta_{855})$ |
Fixed field: | Number field defined by a degree 1710 polynomial (not computed) |
Values on generators
\((21661,22023)\) → \((e\left(\frac{4}{5}\right),e\left(\frac{139}{342}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(13\) | \(15\) | \(17\) | \(21\) | \(23\) | \(25\) |
\( \chi_{ 87362 }(3, a) \) | \(-1\) | \(1\) | \(e\left(\frac{1529}{1710}\right)\) | \(e\left(\frac{421}{855}\right)\) | \(e\left(\frac{161}{285}\right)\) | \(e\left(\frac{674}{855}\right)\) | \(e\left(\frac{883}{1710}\right)\) | \(e\left(\frac{661}{1710}\right)\) | \(e\left(\frac{46}{855}\right)\) | \(e\left(\frac{157}{342}\right)\) | \(e\left(\frac{58}{171}\right)\) | \(e\left(\frac{842}{855}\right)\) |
sage: chi.jacobi_sum(n)