from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(201, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([0,20]))
pari: [g,chi] = znchar(Mod(160,201))
Basic properties
Modulus: | \(201\) | |
Conductor: | \(67\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(33\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{67}(26,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 201.m
\(\chi_{201}(4,\cdot)\) \(\chi_{201}(10,\cdot)\) \(\chi_{201}(16,\cdot)\) \(\chi_{201}(19,\cdot)\) \(\chi_{201}(49,\cdot)\) \(\chi_{201}(55,\cdot)\) \(\chi_{201}(73,\cdot)\) \(\chi_{201}(88,\cdot)\) \(\chi_{201}(100,\cdot)\) \(\chi_{201}(103,\cdot)\) \(\chi_{201}(106,\cdot)\) \(\chi_{201}(121,\cdot)\) \(\chi_{201}(127,\cdot)\) \(\chi_{201}(151,\cdot)\) \(\chi_{201}(157,\cdot)\) \(\chi_{201}(160,\cdot)\) \(\chi_{201}(169,\cdot)\) \(\chi_{201}(181,\cdot)\) \(\chi_{201}(190,\cdot)\) \(\chi_{201}(199,\cdot)\)
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 33 polynomial |
Values on generators
\((68,136)\) → \((1,e\left(\frac{10}{33}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
\( \chi_{ 201 }(160, a) \) | \(1\) | \(1\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{7}{33}\right)\) |
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)