from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5390, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([0,16,0]))
pari: [g,chi] = znchar(Mod(221,5390))
Basic properties
| Modulus: | \(5390\) | |
| Conductor: | \(49\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(21\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{49}(25,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5390.bt
\(\chi_{5390}(221,\cdot)\) \(\chi_{5390}(331,\cdot)\) \(\chi_{5390}(991,\cdot)\) \(\chi_{5390}(1101,\cdot)\) \(\chi_{5390}(1761,\cdot)\) \(\chi_{5390}(1871,\cdot)\) \(\chi_{5390}(2531,\cdot)\) \(\chi_{5390}(2641,\cdot)\) \(\chi_{5390}(4071,\cdot)\) \(\chi_{5390}(4181,\cdot)\) \(\chi_{5390}(4841,\cdot)\) \(\chi_{5390}(4951,\cdot)\)
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: | \(\Q(\zeta_{21})\) |
| Fixed field: | Number field defined by a degree 21 polynomial |
Values on generators
\((2157,4511,981)\) → \((1,e\left(\frac{8}{21}\right),1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(9\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 5390 }(221, a) \) | \(1\) | \(1\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{4}{21}\right)\) |
sage: chi.jacobi_sum(n)