from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7175, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([19,10,6]))
pari: [g,chi] = znchar(Mod(5088,7175))
Basic properties
| Modulus: | \(7175\) | |
| Conductor: | \(7175\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(20\) | 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 7175.ex
\(\chi_{7175}(517,\cdot)\) \(\chi_{7175}(3303,\cdot)\) \(\chi_{7175}(3352,\cdot)\) \(\chi_{7175}(5088,\cdot)\) \(\chi_{7175}(5648,\cdot)\) \(\chi_{7175}(5662,\cdot)\) \(\chi_{7175}(5683,\cdot)\) \(\chi_{7175}(6747,\cdot)\)
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
\((6602,3076,5951)\) → \((e\left(\frac{19}{20}\right),-1,e\left(\frac{3}{10}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) | \(16\) |
| \( \chi_{ 7175 }(5088, a) \) | \(1\) | \(1\) | \(-i\) | \(e\left(\frac{13}{20}\right)\) | \(-1\) | \(e\left(\frac{2}{5}\right)\) | \(i\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(1\) |
sage: chi.jacobi_sum(n)