from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4592, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([0,5,0,9]))
pari: [g,chi] = znchar(Mod(197,4592))
Basic properties
Modulus: | \(4592\) | |
Conductor: | \(656\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(20\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{656}(197,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4592.ek
\(\chi_{4592}(197,\cdot)\) \(\chi_{4592}(1597,\cdot)\) \(\chi_{4592}(1765,\cdot)\) \(\chi_{4592}(3165,\cdot)\) \(\chi_{4592}(3613,\cdot)\) \(\chi_{4592}(3669,\cdot)\) \(\chi_{4592}(4285,\cdot)\) \(\chi_{4592}(4341,\cdot)\)
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: | \(\Q(\zeta_{20})\) |
Fixed field: | 20.20.158322645890088916737377685620382389712556392448.2 |
Values on generators
\((575,3445,3937,785)\) → \((1,i,1,e\left(\frac{9}{20}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) |
\( \chi_{ 4592 }(197, a) \) | \(1\) | \(1\) | \(-1\) | \(e\left(\frac{3}{20}\right)\) | \(1\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{3}{10}\right)\) |
sage: chi.jacobi_sum(n)