from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(35937, base_ring=CyclotomicField(1210))
M = H._module
chi = DirichletCharacter(H, M([0,559]))
pari: [g,chi] = znchar(Mod(28,35937))
Basic properties
| Modulus: | \(35937\) | |
| Conductor: | \(1331\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(1210\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1331}(28,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 35937.cg
\(\chi_{35937}(28,\cdot)\) \(\chi_{35937}(217,\cdot)\) \(\chi_{35937}(244,\cdot)\) \(\chi_{35937}(271,\cdot)\) \(\chi_{35937}(325,\cdot)\) \(\chi_{35937}(514,\cdot)\) \(\chi_{35937}(541,\cdot)\) \(\chi_{35937}(568,\cdot)\) \(\chi_{35937}(622,\cdot)\) \(\chi_{35937}(811,\cdot)\) \(\chi_{35937}(865,\cdot)\) \(\chi_{35937}(919,\cdot)\) \(\chi_{35937}(1108,\cdot)\) \(\chi_{35937}(1135,\cdot)\) \(\chi_{35937}(1162,\cdot)\) \(\chi_{35937}(1216,\cdot)\) \(\chi_{35937}(1405,\cdot)\) \(\chi_{35937}(1432,\cdot)\) \(\chi_{35937}(1459,\cdot)\) \(\chi_{35937}(1513,\cdot)\)
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: | $\Q(\zeta_{605})$ |
| Fixed field: | Number field defined by a degree 1210 polynomial (not computed) |
Values on generators
\((22628,13312)\) → \((1,e\left(\frac{559}{1210}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(13\) | \(14\) | \(16\) | \(17\) |
| \( \chi_{ 35937 }(28, a) \) | \(-1\) | \(1\) | \(e\left(\frac{559}{1210}\right)\) | \(e\left(\frac{559}{605}\right)\) | \(e\left(\frac{3}{605}\right)\) | \(e\left(\frac{393}{1210}\right)\) | \(e\left(\frac{467}{1210}\right)\) | \(e\left(\frac{113}{242}\right)\) | \(e\left(\frac{1019}{1210}\right)\) | \(e\left(\frac{476}{605}\right)\) | \(e\left(\frac{513}{605}\right)\) | \(e\left(\frac{551}{1210}\right)\) |
sage: chi.jacobi_sum(n)