from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(201, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,59]))
pari: [g,chi] = znchar(Mod(11,201))
Basic properties
| Modulus: | \(201\) | |
| Conductor: | \(201\) | 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: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 201.p
\(\chi_{201}(2,\cdot)\) \(\chi_{201}(11,\cdot)\) \(\chi_{201}(20,\cdot)\) \(\chi_{201}(32,\cdot)\) \(\chi_{201}(41,\cdot)\) \(\chi_{201}(44,\cdot)\) \(\chi_{201}(50,\cdot)\) \(\chi_{201}(74,\cdot)\) \(\chi_{201}(80,\cdot)\) \(\chi_{201}(95,\cdot)\) \(\chi_{201}(98,\cdot)\) \(\chi_{201}(101,\cdot)\) \(\chi_{201}(113,\cdot)\) \(\chi_{201}(128,\cdot)\) \(\chi_{201}(146,\cdot)\) \(\chi_{201}(152,\cdot)\) \(\chi_{201}(182,\cdot)\) \(\chi_{201}(185,\cdot)\) \(\chi_{201}(191,\cdot)\) \(\chi_{201}(197,\cdot)\)
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
\((68,136)\) → \((-1,e\left(\frac{59}{66}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 201 }(11, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{37}{66}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{65}{66}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{19}{33}\right)\) |
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)