from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(930, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,45,58]))
pari: [g,chi] = znchar(Mod(83,930))
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 \(\chi_{465}(83,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 930.bu
\(\chi_{930}(17,\cdot)\) \(\chi_{930}(53,\cdot)\) \(\chi_{930}(83,\cdot)\) \(\chi_{930}(137,\cdot)\) \(\chi_{930}(167,\cdot)\) \(\chi_{930}(197,\cdot)\) \(\chi_{930}(203,\cdot)\) \(\chi_{930}(323,\cdot)\) \(\chi_{930}(353,\cdot)\) \(\chi_{930}(383,\cdot)\) \(\chi_{930}(437,\cdot)\) \(\chi_{930}(623,\cdot)\) \(\chi_{930}(737,\cdot)\) \(\chi_{930}(797,\cdot)\) \(\chi_{930}(827,\cdot)\) \(\chi_{930}(923,\cdot)\)
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
\((311,187,871)\) → \((-1,-i,e\left(\frac{29}{30}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(37\) | \(41\) | \(43\) |
\( \chi_{ 930 }(83, a) \) | \(-1\) | \(1\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{53}{60}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{37}{60}\right)\) |
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)