from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(930, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,45,56]))
pari: [g,chi] = znchar(Mod(503,930))
Basic properties
| Modulus: | \(930\) | |
| Conductor: | \(465\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{465}(38,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 930.bs
\(\chi_{930}(107,\cdot)\) \(\chi_{930}(113,\cdot)\) \(\chi_{930}(143,\cdot)\) \(\chi_{930}(173,\cdot)\) \(\chi_{930}(227,\cdot)\) \(\chi_{930}(257,\cdot)\) \(\chi_{930}(293,\cdot)\) \(\chi_{930}(317,\cdot)\) \(\chi_{930}(413,\cdot)\) \(\chi_{930}(443,\cdot)\) \(\chi_{930}(503,\cdot)\) \(\chi_{930}(617,\cdot)\) \(\chi_{930}(803,\cdot)\) \(\chi_{930}(857,\cdot)\) \(\chi_{930}(887,\cdot)\) \(\chi_{930}(917,\cdot)\)
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
\((311,187,871)\) → \((-1,-i,e\left(\frac{14}{15}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 930 }(503, a) \) | \(1\) | \(1\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{47}{60}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{59}{60}\right)\) |
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)