from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(98, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([16]))
pari: [g,chi] = znchar(Mod(25,98))
Basic properties
Modulus: | \(98\) | |
Conductor: | \(49\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(21\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{49}(25,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 98.g
\(\chi_{98}(9,\cdot)\) \(\chi_{98}(11,\cdot)\) \(\chi_{98}(23,\cdot)\) \(\chi_{98}(25,\cdot)\) \(\chi_{98}(37,\cdot)\) \(\chi_{98}(39,\cdot)\) \(\chi_{98}(51,\cdot)\) \(\chi_{98}(53,\cdot)\) \(\chi_{98}(65,\cdot)\) \(\chi_{98}(81,\cdot)\) \(\chi_{98}(93,\cdot)\) \(\chi_{98}(95,\cdot)\)
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
\(3\) → \(e\left(\frac{8}{21}\right)\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) |
\( \chi_{ 98 }(25, a) \) | \(1\) | \(1\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{2}{21}\right)\) |
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)