from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5328, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([0,27,0,28]))
pari: [g,chi] = znchar(Mod(2269,5328))
Basic properties
Modulus: | \(5328\) | |
Conductor: | \(592\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{592}(493,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5328.lq
\(\chi_{5328}(181,\cdot)\) \(\chi_{5328}(1045,\cdot)\) \(\chi_{5328}(1117,\cdot)\) \(\chi_{5328}(1477,\cdot)\) \(\chi_{5328}(2125,\cdot)\) \(\chi_{5328}(2269,\cdot)\) \(\chi_{5328}(2845,\cdot)\) \(\chi_{5328}(3709,\cdot)\) \(\chi_{5328}(3781,\cdot)\) \(\chi_{5328}(4141,\cdot)\) \(\chi_{5328}(4789,\cdot)\) \(\chi_{5328}(4933,\cdot)\)
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: | \(\Q(\zeta_{36})\) |
Fixed field: | Number field defined by a degree 36 polynomial |
Values on generators
\((1999,1333,2369,1297)\) → \((1,-i,1,e\left(\frac{7}{9}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
\( \chi_{ 5328 }(2269, a) \) | \(1\) | \(1\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(1\) |
sage: chi.jacobi_sum(n)