from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4018, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([4,21]))
pari: [g,chi] = znchar(Mod(81,4018))
Basic properties
| Modulus: | \(4018\) | |
| Conductor: | \(2009\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{2009}(81,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4018.bl
\(\chi_{4018}(81,\cdot)\) \(\chi_{4018}(163,\cdot)\) \(\chi_{4018}(737,\cdot)\) \(\chi_{4018}(1229,\cdot)\) \(\chi_{4018}(1311,\cdot)\) \(\chi_{4018}(1803,\cdot)\) \(\chi_{4018}(1885,\cdot)\) \(\chi_{4018}(2377,\cdot)\) \(\chi_{4018}(2459,\cdot)\) \(\chi_{4018}(2951,\cdot)\) \(\chi_{4018}(3033,\cdot)\) \(\chi_{4018}(3525,\cdot)\)
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 42 polynomial |
Values on generators
\((493,785)\) → \((e\left(\frac{2}{21}\right),-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) |
| \( \chi_{ 4018 }(81, a) \) | \(1\) | \(1\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{11}{21}\right)\) |
sage: chi.jacobi_sum(n)