from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6600, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([10,0,10,19,10]))
pari: [g,chi] = znchar(Mod(263,6600))
Basic properties
| Modulus: | \(6600\) | |
| Conductor: | \(3300\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(20\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{3300}(263,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6600.mg
\(\chi_{6600}(263,\cdot)\) \(\chi_{6600}(527,\cdot)\) \(\chi_{6600}(1583,\cdot)\) \(\chi_{6600}(1847,\cdot)\) \(\chi_{6600}(2903,\cdot)\) \(\chi_{6600}(3167,\cdot)\) \(\chi_{6600}(4223,\cdot)\) \(\chi_{6600}(4487,\cdot)\)
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: | 20.20.4674008133741604614257812500000000000000000000.1 |
Values on generators
\((4951,3301,2201,2377,1201)\) → \((-1,1,-1,e\left(\frac{19}{20}\right),-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 6600 }(263, a) \) | \(1\) | \(1\) | \(-i\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(i\) |
sage: chi.jacobi_sum(n)