from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6031, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([33,4]))
pari: [g,chi] = znchar(Mod(21,6031))
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.fd
\(\chi_{6031}(21,\cdot)\) \(\chi_{6031}(25,\cdot)\) \(\chi_{6031}(391,\cdot)\) \(\chi_{6031}(965,\cdot)\) \(\chi_{6031}(1039,\cdot)\) \(\chi_{6031}(1114,\cdot)\) \(\chi_{6031}(1177,\cdot)\) \(\chi_{6031}(1436,\cdot)\) \(\chi_{6031}(2685,\cdot)\) \(\chi_{6031}(2763,\cdot)\) \(\chi_{6031}(3223,\cdot)\) \(\chi_{6031}(3395,\cdot)\) \(\chi_{6031}(4058,\cdot)\) \(\chi_{6031}(4396,\cdot)\) \(\chi_{6031}(4912,\cdot)\) \(\chi_{6031}(5331,\cdot)\) \(\chi_{6031}(5443,\cdot)\) \(\chi_{6031}(5874,\cdot)\)
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
\((816,5218)\) → \((e\left(\frac{11}{18}\right),e\left(\frac{2}{27}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
\( \chi_{ 6031 }(21, a) \) | \(1\) | \(1\) | \(e\left(\frac{37}{54}\right)\) | \(e\left(\frac{10}{27}\right)\) | \(e\left(\frac{10}{27}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{26}{27}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{20}{27}\right)\) | \(e\left(\frac{23}{27}\right)\) | \(e\left(\frac{22}{27}\right)\) |
sage: chi.jacobi_sum(n)