from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(366, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,41]))
pari: [g,chi] = znchar(Mod(209,366))
Basic properties
| Modulus: | \(366\) | |
| Conductor: | \(183\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{183}(26,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 366.x
\(\chi_{366}(17,\cdot)\) \(\chi_{366}(35,\cdot)\) \(\chi_{366}(59,\cdot)\) \(\chi_{366}(71,\cdot)\) \(\chi_{366}(173,\cdot)\) \(\chi_{366}(185,\cdot)\) \(\chi_{366}(209,\cdot)\) \(\chi_{366}(227,\cdot)\) \(\chi_{366}(251,\cdot)\) \(\chi_{366}(275,\cdot)\) \(\chi_{366}(287,\cdot)\) \(\chi_{366}(299,\cdot)\) \(\chi_{366}(311,\cdot)\) \(\chi_{366}(323,\cdot)\) \(\chi_{366}(335,\cdot)\) \(\chi_{366}(359,\cdot)\)
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
\((245,307)\) → \((-1,e\left(\frac{41}{60}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 366 }(209, a) \) | \(1\) | \(1\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{29}{60}\right)\) | \(-i\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{19}{60}\right)\) |
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)