from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(111, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([18,29]))
pari: [g,chi] = znchar(Mod(98,111))
Basic properties
Modulus: | \(111\) | |
Conductor: | \(111\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 111.q
\(\chi_{111}(2,\cdot)\) \(\chi_{111}(5,\cdot)\) \(\chi_{111}(17,\cdot)\) \(\chi_{111}(20,\cdot)\) \(\chi_{111}(32,\cdot)\) \(\chi_{111}(35,\cdot)\) \(\chi_{111}(50,\cdot)\) \(\chi_{111}(56,\cdot)\) \(\chi_{111}(59,\cdot)\) \(\chi_{111}(89,\cdot)\) \(\chi_{111}(92,\cdot)\) \(\chi_{111}(98,\cdot)\)
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: | \(\Q(\zeta_{36})\) |
Fixed field: | \(\Q(\zeta_{111})^+\) |
Values on generators
\((38,76)\) → \((-1,e\left(\frac{29}{36}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
\( \chi_{ 111 }(98, a) \) | \(1\) | \(1\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{2}{9}\right)\) |
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)