from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(328, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([0,0,19]))
pari: [g,chi] = znchar(Mod(321,328))
Basic properties
Modulus: | \(328\) | |
Conductor: | \(41\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(40\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{41}(34,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 328.be
\(\chi_{328}(17,\cdot)\) \(\chi_{328}(65,\cdot)\) \(\chi_{328}(89,\cdot)\) \(\chi_{328}(97,\cdot)\) \(\chi_{328}(129,\cdot)\) \(\chi_{328}(145,\cdot)\) \(\chi_{328}(153,\cdot)\) \(\chi_{328}(177,\cdot)\) \(\chi_{328}(193,\cdot)\) \(\chi_{328}(217,\cdot)\) \(\chi_{328}(233,\cdot)\) \(\chi_{328}(257,\cdot)\) \(\chi_{328}(265,\cdot)\) \(\chi_{328}(281,\cdot)\) \(\chi_{328}(313,\cdot)\) \(\chi_{328}(321,\cdot)\)
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: | \(\Q(\zeta_{40})\) |
Fixed field: | Number field defined by a degree 40 polynomial |
Values on generators
\((247,165,129)\) → \((1,1,e\left(\frac{19}{40}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
\( \chi_{ 328 }(321, a) \) | \(-1\) | \(1\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{21}{40}\right)\) | \(i\) | \(e\left(\frac{17}{40}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{23}{40}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(e\left(\frac{11}{40}\right)\) | \(e\left(\frac{13}{20}\right)\) |
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)