from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6031, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([33,50]))
pari: [g,chi] = znchar(Mod(132,6031))
Basic properties
Modulus: | \(6031\) | |
Conductor: | \(6031\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(54\) | 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 6031.ep
\(\chi_{6031}(132,\cdot)\) \(\chi_{6031}(188,\cdot)\) \(\chi_{6031}(802,\cdot)\) \(\chi_{6031}(876,\cdot)\) \(\chi_{6031}(879,\cdot)\) \(\chi_{6031}(1325,\cdot)\) \(\chi_{6031}(1695,\cdot)\) \(\chi_{6031}(1928,\cdot)\) \(\chi_{6031}(2581,\cdot)\) \(\chi_{6031}(2926,\cdot)\) \(\chi_{6031}(3092,\cdot)\) \(\chi_{6031}(3445,\cdot)\) \(\chi_{6031}(3989,\cdot)\) \(\chi_{6031}(4221,\cdot)\) \(\chi_{6031}(4407,\cdot)\) \(\chi_{6031}(4690,\cdot)\) \(\chi_{6031}(5168,\cdot)\) \(\chi_{6031}(5904,\cdot)\)
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: | \(\Q(\zeta_{27})\) |
Fixed field: | Number field defined by a degree 54 polynomial |
Values on generators
\((816,5218)\) → \((e\left(\frac{11}{18}\right),e\left(\frac{25}{27}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
\( \chi_{ 6031 }(132, a) \) | \(1\) | \(1\) | \(e\left(\frac{29}{54}\right)\) | \(e\left(\frac{11}{27}\right)\) | \(e\left(\frac{2}{27}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{4}{27}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{22}{27}\right)\) | \(e\left(\frac{13}{27}\right)\) | \(e\left(\frac{23}{27}\right)\) |
sage: chi.jacobi_sum(n)