from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(573, base_ring=CyclotomicField(38))
M = H._module
chi = DirichletCharacter(H, M([0,10]))
pari: [g,chi] = znchar(Mod(196,573))
Basic properties
Modulus: | \(573\) | |
Conductor: | \(191\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(19\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{191}(5,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 573.i
\(\chi_{573}(25,\cdot)\) \(\chi_{573}(52,\cdot)\) \(\chi_{573}(121,\cdot)\) \(\chi_{573}(136,\cdot)\) \(\chi_{573}(154,\cdot)\) \(\chi_{573}(160,\cdot)\) \(\chi_{573}(196,\cdot)\) \(\chi_{573}(223,\cdot)\) \(\chi_{573}(298,\cdot)\) \(\chi_{573}(316,\cdot)\) \(\chi_{573}(388,\cdot)\) \(\chi_{573}(412,\cdot)\) \(\chi_{573}(418,\cdot)\) \(\chi_{573}(451,\cdot)\) \(\chi_{573}(532,\cdot)\) \(\chi_{573}(535,\cdot)\) \(\chi_{573}(559,\cdot)\) \(\chi_{573}(562,\cdot)\)
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: | Number field defined by a degree 19 polynomial |
Values on generators
\((383,19)\) → \((1,e\left(\frac{5}{19}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
\( \chi_{ 573 }(196, a) \) | \(1\) | \(1\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{3}{19}\right)\) | \(e\left(\frac{3}{19}\right)\) | \(1\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{14}{19}\right)\) | \(e\left(\frac{7}{19}\right)\) | \(e\left(\frac{9}{19}\right)\) | \(e\left(\frac{11}{19}\right)\) | \(e\left(\frac{6}{19}\right)\) |
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)