from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8018, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([14,36]))
pari: [g,chi] = znchar(Mod(691,8018))
Basic properties
| Modulus: | \(8018\) | |
| Conductor: | \(4009\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(21\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{4009}(691,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8018.bu
\(\chi_{8018}(691,\cdot)\) \(\chi_{8018}(1113,\cdot)\) \(\chi_{8018}(3313,\cdot)\) \(\chi_{8018}(3731,\cdot)\) \(\chi_{8018}(3735,\cdot)\) \(\chi_{8018}(3921,\cdot)\) \(\chi_{8018}(3997,\cdot)\) \(\chi_{8018}(4153,\cdot)\) \(\chi_{8018}(4343,\cdot)\) \(\chi_{8018}(4419,\cdot)\) \(\chi_{8018}(6923,\cdot)\) \(\chi_{8018}(7345,\cdot)\)
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
\((2111,1901)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{6}{7}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(21\) | \(23\) |
| \( \chi_{ 8018 }(691, a) \) | \(1\) | \(1\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) |
sage: chi.jacobi_sum(n)