from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(770, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([15,10,54]))
pari: [g,chi] = znchar(Mod(17,770))
Basic properties
Modulus: | \(770\) | |
Conductor: | \(385\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{385}(17,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 770.bs
\(\chi_{770}(17,\cdot)\) \(\chi_{770}(73,\cdot)\) \(\chi_{770}(117,\cdot)\) \(\chi_{770}(173,\cdot)\) \(\chi_{770}(227,\cdot)\) \(\chi_{770}(283,\cdot)\) \(\chi_{770}(327,\cdot)\) \(\chi_{770}(437,\cdot)\) \(\chi_{770}(453,\cdot)\) \(\chi_{770}(523,\cdot)\) \(\chi_{770}(563,\cdot)\) \(\chi_{770}(607,\cdot)\) \(\chi_{770}(633,\cdot)\) \(\chi_{770}(677,\cdot)\) \(\chi_{770}(717,\cdot)\) \(\chi_{770}(733,\cdot)\)
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
\((617,661,211)\) → \((i,e\left(\frac{1}{6}\right),e\left(\frac{9}{10}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(9\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) | \(37\) |
\( \chi_{ 770 }(17, a) \) | \(-1\) | \(1\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{23}{60}\right)\) |
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)