from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2600, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([10,10,3,15]))
pari: [g,chi] = znchar(Mod(83,2600))
Basic properties
| Modulus: | \(2600\) | |
| Conductor: | \(2600\) | 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: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2600.dy
\(\chi_{2600}(83,\cdot)\) \(\chi_{2600}(603,\cdot)\) \(\chi_{2600}(827,\cdot)\) \(\chi_{2600}(1123,\cdot)\) \(\chi_{2600}(1347,\cdot)\) \(\chi_{2600}(1867,\cdot)\) \(\chi_{2600}(2163,\cdot)\) \(\chi_{2600}(2387,\cdot)\)
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: | 20.0.159955915669033615625000000000000000000000000000000.2 |
Values on generators
\((1951,1301,1977,1601)\) → \((-1,-1,e\left(\frac{3}{20}\right),-i)\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 2600 }(83, a) \) | \(-1\) | \(1\) | \(e\left(\frac{1}{20}\right)\) | \(-1\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{4}{5}\right)\) |
sage: chi.jacobi_sum(n)