from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6013, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([0,18]))
pari: [g,chi] = znchar(Mod(169,6013))
Basic properties
Modulus: | \(6013\) | |
Conductor: | \(859\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(11\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{859}(169,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6013.u
\(\chi_{6013}(169,\cdot)\) \(\chi_{6013}(1128,\cdot)\) \(\chi_{6013}(1338,\cdot)\) \(\chi_{6013}(1779,\cdot)\) \(\chi_{6013}(2003,\cdot)\) \(\chi_{6013}(3641,\cdot)\) \(\chi_{6013}(4229,\cdot)\) \(\chi_{6013}(4383,\cdot)\) \(\chi_{6013}(4509,\cdot)\) \(\chi_{6013}(5167,\cdot)\)
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: | \(\Q(\zeta_{11})\) |
Fixed field: | 11.11.218741727855135890482344953401.1 |
Values on generators
\((5155,3438)\) → \((1,e\left(\frac{9}{11}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(11\) | \(12\) |
\( \chi_{ 6013 }(169, a) \) | \(1\) | \(1\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(1\) | \(e\left(\frac{7}{11}\right)\) | \(1\) |
sage: chi.jacobi_sum(n)