from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7938, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([28,4]))
pari: [g,chi] = znchar(Mod(865,7938))
Basic properties
Modulus: | \(7938\) | |
Conductor: | \(441\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(21\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{441}(277,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 7938.bn
\(\chi_{7938}(865,\cdot)\) \(\chi_{7938}(919,\cdot)\) \(\chi_{7938}(1999,\cdot)\) \(\chi_{7938}(2053,\cdot)\) \(\chi_{7938}(3133,\cdot)\) \(\chi_{7938}(3187,\cdot)\) \(\chi_{7938}(4267,\cdot)\) \(\chi_{7938}(4321,\cdot)\) \(\chi_{7938}(5401,\cdot)\) \(\chi_{7938}(5455,\cdot)\) \(\chi_{7938}(6589,\cdot)\) \(\chi_{7938}(7669,\cdot)\)
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
\((6077,3727)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{2}{21}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
\( \chi_{ 7938 }(865, a) \) | \(1\) | \(1\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(1\) | \(e\left(\frac{1}{21}\right)\) |
sage: chi.jacobi_sum(n)