from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8021, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([44,54]))
pari: [g,chi] = znchar(Mod(113,8021))
Basic properties
Modulus: | \(8021\) | |
Conductor: | \(8021\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(33\) | 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 8021.br
\(\chi_{8021}(113,\cdot)\) \(\chi_{8021}(1576,\cdot)\) \(\chi_{8021}(2193,\cdot)\) \(\chi_{8021}(2499,\cdot)\) \(\chi_{8021}(3116,\cdot)\) \(\chi_{8021}(3877,\cdot)\) \(\chi_{8021}(4046,\cdot)\) \(\chi_{8021}(4494,\cdot)\) \(\chi_{8021}(4663,\cdot)\) \(\chi_{8021}(4670,\cdot)\) \(\chi_{8021}(4748,\cdot)\) \(\chi_{8021}(5287,\cdot)\) \(\chi_{8021}(5365,\cdot)\) \(\chi_{8021}(6659,\cdot)\) \(\chi_{8021}(7179,\cdot)\) \(\chi_{8021}(7205,\cdot)\) \(\chi_{8021}(7276,\cdot)\) \(\chi_{8021}(7517,\cdot)\) \(\chi_{8021}(7796,\cdot)\) \(\chi_{8021}(7822,\cdot)\)
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
\((6788,2471)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{9}{11}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
\( \chi_{ 8021 }(113, a) \) | \(1\) | \(1\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{31}{33}\right)\) |
sage: chi.jacobi_sum(n)