from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(588, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([0,21,32]))
pari: [g,chi] = znchar(Mod(233,588))
Basic properties
Modulus: | \(588\) | |
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 \(\chi_{147}(86,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 588.z
\(\chi_{588}(53,\cdot)\) \(\chi_{588}(65,\cdot)\) \(\chi_{588}(137,\cdot)\) \(\chi_{588}(149,\cdot)\) \(\chi_{588}(221,\cdot)\) \(\chi_{588}(233,\cdot)\) \(\chi_{588}(305,\cdot)\) \(\chi_{588}(317,\cdot)\) \(\chi_{588}(389,\cdot)\) \(\chi_{588}(401,\cdot)\) \(\chi_{588}(473,\cdot)\) \(\chi_{588}(485,\cdot)\)
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: | 42.0.176602720807616761537805583365440112858555316650025456145851095351761290003.1 |
Values on generators
\((295,197,493)\) → \((1,-1,e\left(\frac{16}{21}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
\( \chi_{ 588 }(233, a) \) | \(-1\) | \(1\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{8}{21}\right)\) |
sage: chi.jacobi_sum(n)
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
Jacobi sum
sage: chi.jacobi_sum(n)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)