from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1200, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([0,15,10,1]))
pari: [g,chi] = znchar(Mod(77,1200))
Basic properties
| Modulus: | \(1200\) | |
| Conductor: | \(1200\) | 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 1200.cm
\(\chi_{1200}(53,\cdot)\) \(\chi_{1200}(77,\cdot)\) \(\chi_{1200}(317,\cdot)\) \(\chi_{1200}(533,\cdot)\) \(\chi_{1200}(773,\cdot)\) \(\chi_{1200}(797,\cdot)\) \(\chi_{1200}(1013,\cdot)\) \(\chi_{1200}(1037,\cdot)\)
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: | 20.20.6191736422400000000000000000000000000000000000.1 |
Values on generators
\((751,901,401,577)\) → \((1,-i,-1,e\left(\frac{1}{20}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 1200 }(77, a) \) | \(1\) | \(1\) | \(-i\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{1}{5}\right)\) |
sage: chi.jacobi_sum(n)