from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(441, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,17]))
pari: [g,chi] = znchar(Mod(26,441))
Basic properties
Modulus: | \(441\) | |
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 \(\chi_{147}(26,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 441.bg
\(\chi_{441}(17,\cdot)\) \(\chi_{441}(26,\cdot)\) \(\chi_{441}(89,\cdot)\) \(\chi_{441}(143,\cdot)\) \(\chi_{441}(152,\cdot)\) \(\chi_{441}(206,\cdot)\) \(\chi_{441}(269,\cdot)\) \(\chi_{441}(278,\cdot)\) \(\chi_{441}(332,\cdot)\) \(\chi_{441}(341,\cdot)\) \(\chi_{441}(395,\cdot)\) \(\chi_{441}(404,\cdot)\)
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
\((344,199)\) → \((-1,e\left(\frac{17}{42}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
\( \chi_{ 441 }(26, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{42}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{2}{21}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{1}{6}\right)\) |
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)