from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(322, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([44,54]))
pari: [g,chi] = znchar(Mod(305,322))
Basic properties
Modulus: | \(322\) | |
Conductor: | \(161\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(33\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{161}(144,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 322.m
\(\chi_{322}(9,\cdot)\) \(\chi_{322}(25,\cdot)\) \(\chi_{322}(39,\cdot)\) \(\chi_{322}(81,\cdot)\) \(\chi_{322}(95,\cdot)\) \(\chi_{322}(121,\cdot)\) \(\chi_{322}(123,\cdot)\) \(\chi_{322}(151,\cdot)\) \(\chi_{322}(163,\cdot)\) \(\chi_{322}(165,\cdot)\) \(\chi_{322}(177,\cdot)\) \(\chi_{322}(179,\cdot)\) \(\chi_{322}(193,\cdot)\) \(\chi_{322}(219,\cdot)\) \(\chi_{322}(233,\cdot)\) \(\chi_{322}(261,\cdot)\) \(\chi_{322}(289,\cdot)\) \(\chi_{322}(303,\cdot)\) \(\chi_{322}(305,\cdot)\) \(\chi_{322}(317,\cdot)\)
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: | 33.33.277966181338944111003326058293667039541136678070715028736001.1 |
Values on generators
\((185,281)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{9}{11}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(25\) | \(27\) |
\( \chi_{ 322 }(305, a) \) | \(1\) | \(1\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{3}{11}\right)\) |
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)