from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7938, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,13]))
pari: [g,chi] = znchar(Mod(647,7938))
Basic properties
Modulus: | \(7938\) | |
Conductor: | \(147\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{147}(59,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 7938.ca
\(\chi_{7938}(647,\cdot)\) \(\chi_{7938}(971,\cdot)\) \(\chi_{7938}(1781,\cdot)\) \(\chi_{7938}(2105,\cdot)\) \(\chi_{7938}(2915,\cdot)\) \(\chi_{7938}(3239,\cdot)\) \(\chi_{7938}(4373,\cdot)\) \(\chi_{7938}(5183,\cdot)\) \(\chi_{7938}(6317,\cdot)\) \(\chi_{7938}(6641,\cdot)\) \(\chi_{7938}(7451,\cdot)\) \(\chi_{7938}(7775,\cdot)\)
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: | \(\Q(\zeta_{147})^+\) |
Values on generators
\((6077,3727)\) → \((-1,e\left(\frac{13}{42}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
\( \chi_{ 7938 }(647, a) \) | \(1\) | \(1\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{19}{21}\right)\) |
sage: chi.jacobi_sum(n)