from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6600, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([10,0,0,7,4]))
pari: [g,chi] = znchar(Mod(103,6600))
Basic properties
| Modulus: | \(6600\) | |
| Conductor: | \(1100\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(20\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1100}(103,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6600.kj
\(\chi_{6600}(103,\cdot)\) \(\chi_{6600}(823,\cdot)\) \(\chi_{6600}(2887,\cdot)\) \(\chi_{6600}(3727,\cdot)\) \(\chi_{6600}(4183,\cdot)\) \(\chi_{6600}(5263,\cdot)\) \(\chi_{6600}(5647,\cdot)\) \(\chi_{6600}(5767,\cdot)\)
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: | 20.20.140227447093420901489257812500000000000000000000.3 |
Values on generators
\((4951,3301,2201,2377,1201)\) → \((-1,1,1,e\left(\frac{7}{20}\right),e\left(\frac{1}{5}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 6600 }(103, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(-1\) | \(e\left(\frac{11}{20}\right)\) | \(1\) | \(-i\) |
sage: chi.jacobi_sum(n)