from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(209, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([9,20]))
pari: [g,chi] = znchar(Mod(30,209))
Basic properties
Modulus: | \(209\) | |
Conductor: | \(209\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(30\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 209.s
\(\chi_{209}(7,\cdot)\) \(\chi_{209}(30,\cdot)\) \(\chi_{209}(68,\cdot)\) \(\chi_{209}(83,\cdot)\) \(\chi_{209}(106,\cdot)\) \(\chi_{209}(140,\cdot)\) \(\chi_{209}(178,\cdot)\) \(\chi_{209}(182,\cdot)\)
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: | \(\Q(\zeta_{15})\) |
Fixed field: | 30.0.492804333687064066258882287420865640909549691972387971.1 |
Values on generators
\((134,78)\) → \((e\left(\frac{3}{10}\right),e\left(\frac{2}{3}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(12\) |
\( \chi_{ 209 }(30, a) \) | \(-1\) | \(1\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(1\) |
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)