from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8021, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([0,35]))
pari: [g,chi] = znchar(Mod(157,8021))
Basic properties
Modulus: | \(8021\) | |
Conductor: | \(617\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{617}(157,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8021.bv
\(\chi_{8021}(157,\cdot)\) \(\chi_{8021}(352,\cdot)\) \(\chi_{8021}(1249,\cdot)\) \(\chi_{8021}(1522,\cdot)\) \(\chi_{8021}(1561,\cdot)\) \(\chi_{8021}(1782,\cdot)\) \(\chi_{8021}(1951,\cdot)\) \(\chi_{8021}(2003,\cdot)\) \(\chi_{8021}(2692,\cdot)\) \(\chi_{8021}(2861,\cdot)\) \(\chi_{8021}(3550,\cdot)\) \(\chi_{8021}(3602,\cdot)\) \(\chi_{8021}(3771,\cdot)\) \(\chi_{8021}(3992,\cdot)\) \(\chi_{8021}(4031,\cdot)\) \(\chi_{8021}(4304,\cdot)\) \(\chi_{8021}(5201,\cdot)\) \(\chi_{8021}(5396,\cdot)\) \(\chi_{8021}(6631,\cdot)\) \(\chi_{8021}(6943,\cdot)\)
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: | \(\Q(\zeta_{44})\) |
Fixed field: | Number field defined by a degree 44 polynomial |
Values on generators
\((6788,2471)\) → \((1,e\left(\frac{35}{44}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
\( \chi_{ 8021 }(157, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{35}{44}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{15}{44}\right)\) | \(i\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{35}{44}\right)\) | \(e\left(\frac{15}{22}\right)\) |
sage: chi.jacobi_sum(n)