from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(573, base_ring=CyclotomicField(38))
M = H._module
chi = DirichletCharacter(H, M([19,22]))
pari: [g,chi] = znchar(Mod(536,573))
Basic properties
Modulus: | \(573\) | |
Conductor: | \(573\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(38\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 573.l
\(\chi_{573}(5,\cdot)\) \(\chi_{573}(32,\cdot)\) \(\chi_{573}(107,\cdot)\) \(\chi_{573}(125,\cdot)\) \(\chi_{573}(197,\cdot)\) \(\chi_{573}(221,\cdot)\) \(\chi_{573}(227,\cdot)\) \(\chi_{573}(260,\cdot)\) \(\chi_{573}(341,\cdot)\) \(\chi_{573}(344,\cdot)\) \(\chi_{573}(368,\cdot)\) \(\chi_{573}(371,\cdot)\) \(\chi_{573}(407,\cdot)\) \(\chi_{573}(434,\cdot)\) \(\chi_{573}(503,\cdot)\) \(\chi_{573}(518,\cdot)\) \(\chi_{573}(536,\cdot)\) \(\chi_{573}(542,\cdot)\)
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: | \(\Q(\zeta_{19})\) |
Fixed field: | 38.0.15223168714879313546692095257379448420591016496978036941227483116202289492875331751113046747.1 |
Values on generators
\((383,19)\) → \((-1,e\left(\frac{11}{19}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
\( \chi_{ 573 }(536, a) \) | \(-1\) | \(1\) | \(e\left(\frac{37}{38}\right)\) | \(e\left(\frac{18}{19}\right)\) | \(e\left(\frac{17}{38}\right)\) | \(1\) | \(e\left(\frac{35}{38}\right)\) | \(e\left(\frac{8}{19}\right)\) | \(e\left(\frac{27}{38}\right)\) | \(e\left(\frac{16}{19}\right)\) | \(e\left(\frac{37}{38}\right)\) | \(e\left(\frac{17}{19}\right)\) |
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)