from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1640, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([10,0,10,11]))
pari: [g,chi] = znchar(Mod(1399,1640))
Basic properties
| Modulus: | \(1640\) | |
| Conductor: | \(820\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(20\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{820}(579,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1640.dh
\(\chi_{1640}(39,\cdot)\) \(\chi_{1640}(159,\cdot)\) \(\chi_{1640}(279,\cdot)\) \(\chi_{1640}(759,\cdot)\) \(\chi_{1640}(799,\cdot)\) \(\chi_{1640}(1279,\cdot)\) \(\chi_{1640}(1399,\cdot)\) \(\chi_{1640}(1519,\cdot)\)
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: | 20.0.44998002377408544344171356675696640000000000.1 |
Values on generators
\((1231,821,657,1441)\) → \((-1,1,-1,e\left(\frac{11}{20}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 1640 }(1399, a) \) | \(-1\) | \(1\) | \(i\) | \(e\left(\frac{9}{20}\right)\) | \(-1\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(-i\) |
sage: chi.jacobi_sum(n)