from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(700, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,9,50]))
pari: [g,chi] = znchar(Mod(33,700))
Basic properties
Modulus: | \(700\) | |
Conductor: | \(175\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{175}(33,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 700.bv
\(\chi_{700}(17,\cdot)\) \(\chi_{700}(33,\cdot)\) \(\chi_{700}(73,\cdot)\) \(\chi_{700}(117,\cdot)\) \(\chi_{700}(173,\cdot)\) \(\chi_{700}(213,\cdot)\) \(\chi_{700}(297,\cdot)\) \(\chi_{700}(313,\cdot)\) \(\chi_{700}(353,\cdot)\) \(\chi_{700}(397,\cdot)\) \(\chi_{700}(437,\cdot)\) \(\chi_{700}(453,\cdot)\) \(\chi_{700}(537,\cdot)\) \(\chi_{700}(577,\cdot)\) \(\chi_{700}(633,\cdot)\) \(\chi_{700}(677,\cdot)\)
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: | \(\Q(\zeta_{60})\) |
Fixed field: | Number field defined by a degree 60 polynomial |
Values on generators
\((351,477,101)\) → \((1,e\left(\frac{3}{20}\right),e\left(\frac{5}{6}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) |
\( \chi_{ 700 }(33, a) \) | \(1\) | \(1\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{13}{20}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{1}{30}\right)\) |
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)