from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(588, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([0,0,10]))
pari: [g,chi] = znchar(Mod(445,588))
Basic properties
Modulus: | \(588\) | |
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 \(\chi_{49}(4,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 588.y
\(\chi_{588}(25,\cdot)\) \(\chi_{588}(37,\cdot)\) \(\chi_{588}(109,\cdot)\) \(\chi_{588}(121,\cdot)\) \(\chi_{588}(193,\cdot)\) \(\chi_{588}(205,\cdot)\) \(\chi_{588}(277,\cdot)\) \(\chi_{588}(289,\cdot)\) \(\chi_{588}(445,\cdot)\) \(\chi_{588}(457,\cdot)\) \(\chi_{588}(529,\cdot)\) \(\chi_{588}(541,\cdot)\)
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
\((295,197,493)\) → \((1,1,e\left(\frac{5}{21}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
\( \chi_{ 588 }(445, a) \) | \(1\) | \(1\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{13}{21}\right)\) |
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)