from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4592, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([0,0,0,17]))
pari: [g,chi] = znchar(Mod(225,4592))
Basic properties
| Modulus: | \(4592\) | |
| Conductor: | \(41\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(20\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{41}(20,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4592.ez
\(\chi_{4592}(225,\cdot)\) \(\chi_{4592}(449,\cdot)\) \(\chi_{4592}(897,\cdot)\) \(\chi_{4592}(1345,\cdot)\) \(\chi_{4592}(2017,\cdot)\) \(\chi_{4592}(2465,\cdot)\) \(\chi_{4592}(2913,\cdot)\) \(\chi_{4592}(3137,\cdot)\)
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: | Number field defined by a degree 20 polynomial |
Values on generators
\((575,3445,3937,785)\) → \((1,1,1,e\left(\frac{17}{20}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) |
| \( \chi_{ 4592 }(225, a) \) | \(1\) | \(1\) | \(-i\) | \(e\left(\frac{7}{10}\right)\) | \(-1\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{2}{5}\right)\) |
sage: chi.jacobi_sum(n)