from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1175, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([9,10]))
pari: [g,chi] = znchar(Mod(187,1175))
Basic properties
| Modulus: | \(1175\) | |
| Conductor: | \(1175\) | 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 1175.l
\(\chi_{1175}(187,\cdot)\) \(\chi_{1175}(328,\cdot)\) \(\chi_{1175}(422,\cdot)\) \(\chi_{1175}(563,\cdot)\) \(\chi_{1175}(798,\cdot)\) \(\chi_{1175}(892,\cdot)\) \(\chi_{1175}(1033,\cdot)\) \(\chi_{1175}(1127,\cdot)\)
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: | 20.20.153083622676291412790305912494659423828125.1 |
Values on generators
\((377,851)\) → \((e\left(\frac{9}{20}\right),-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) |
| \( \chi_{ 1175 }(187, a) \) | \(1\) | \(1\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(i\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{1}{20}\right)\) |
sage: chi.jacobi_sum(n)