from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5610, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([20,20,32,35]))
pari: [g,chi] = znchar(Mod(5459,5610))
Basic properties
Modulus: | \(5610\) | |
Conductor: | \(2805\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(40\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{2805}(2654,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5610.ek
\(\chi_{5610}(59,\cdot)\) \(\chi_{5610}(179,\cdot)\) \(\chi_{5610}(389,\cdot)\) \(\chi_{5610}(1379,\cdot)\) \(\chi_{5610}(1589,\cdot)\) \(\chi_{5610}(1709,\cdot)\) \(\chi_{5610}(1919,\cdot)\) \(\chi_{5610}(2099,\cdot)\) \(\chi_{5610}(2429,\cdot)\) \(\chi_{5610}(2909,\cdot)\) \(\chi_{5610}(3239,\cdot)\) \(\chi_{5610}(3419,\cdot)\) \(\chi_{5610}(3749,\cdot)\) \(\chi_{5610}(4139,\cdot)\) \(\chi_{5610}(4469,\cdot)\) \(\chi_{5610}(5459,\cdot)\)
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
\((1871,3367,1531,3301)\) → \((-1,-1,e\left(\frac{4}{5}\right),e\left(\frac{7}{8}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(13\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) | \(47\) |
\( \chi_{ 5610 }(5459, a) \) | \(-1\) | \(1\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{19}{40}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(i\) | \(e\left(\frac{9}{10}\right)\) |
sage: chi.jacobi_sum(n)