from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4592, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([0,30,20,31]))
pari: [g,chi] = znchar(Mod(13,4592))
Basic properties
Modulus: | \(4592\) | |
Conductor: | \(4592\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(40\) | 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 4592.gy
\(\chi_{4592}(13,\cdot)\) \(\chi_{4592}(181,\cdot)\) \(\chi_{4592}(293,\cdot)\) \(\chi_{4592}(685,\cdot)\) \(\chi_{4592}(909,\cdot)\) \(\chi_{4592}(965,\cdot)\) \(\chi_{4592}(1413,\cdot)\) \(\chi_{4592}(1469,\cdot)\) \(\chi_{4592}(1693,\cdot)\) \(\chi_{4592}(2085,\cdot)\) \(\chi_{4592}(2197,\cdot)\) \(\chi_{4592}(2365,\cdot)\) \(\chi_{4592}(3373,\cdot)\) \(\chi_{4592}(3429,\cdot)\) \(\chi_{4592}(3541,\cdot)\) \(\chi_{4592}(3597,\cdot)\)
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: | \(\Q(\zeta_{40})\) |
Fixed field: | Number field defined by a degree 40 polynomial |
Values on generators
\((575,3445,3937,785)\) → \((1,-i,-1,e\left(\frac{31}{40}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) |
\( \chi_{ 4592 }(13, a) \) | \(1\) | \(1\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(-i\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{3}{5}\right)\) |
sage: chi.jacobi_sum(n)