from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(207, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([22,21]))
pari: [g,chi] = znchar(Mod(40,207))
Basic properties
Modulus: | \(207\) | |
Conductor: | \(207\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(66\) | 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 207.p
\(\chi_{207}(7,\cdot)\) \(\chi_{207}(34,\cdot)\) \(\chi_{207}(40,\cdot)\) \(\chi_{207}(43,\cdot)\) \(\chi_{207}(61,\cdot)\) \(\chi_{207}(67,\cdot)\) \(\chi_{207}(76,\cdot)\) \(\chi_{207}(79,\cdot)\) \(\chi_{207}(88,\cdot)\) \(\chi_{207}(97,\cdot)\) \(\chi_{207}(103,\cdot)\) \(\chi_{207}(106,\cdot)\) \(\chi_{207}(112,\cdot)\) \(\chi_{207}(130,\cdot)\) \(\chi_{207}(148,\cdot)\) \(\chi_{207}(157,\cdot)\) \(\chi_{207}(166,\cdot)\) \(\chi_{207}(175,\cdot)\) \(\chi_{207}(178,\cdot)\) \(\chi_{207}(205,\cdot)\)
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: | \(\Q(\zeta_{33})\) |
Fixed field: | Number field defined by a degree 66 polynomial |
Values on generators
\((47,28)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{7}{22}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
\( \chi_{ 207 }(40, a) \) | \(-1\) | \(1\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{65}{66}\right)\) | \(e\left(\frac{25}{66}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{13}{66}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{23}{66}\right)\) | \(e\left(\frac{29}{33}\right)\) |
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)