from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4235, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,11,63]))
pari: [g,chi] = znchar(Mod(934,4235))
Basic properties
Modulus: | \(4235\) | |
Conductor: | \(4235\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(66\) | 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 4235.co
\(\chi_{4235}(54,\cdot)\) \(\chi_{4235}(164,\cdot)\) \(\chi_{4235}(439,\cdot)\) \(\chi_{4235}(549,\cdot)\) \(\chi_{4235}(824,\cdot)\) \(\chi_{4235}(934,\cdot)\) \(\chi_{4235}(1319,\cdot)\) \(\chi_{4235}(1594,\cdot)\) \(\chi_{4235}(1704,\cdot)\) \(\chi_{4235}(1979,\cdot)\) \(\chi_{4235}(2089,\cdot)\) \(\chi_{4235}(2364,\cdot)\) \(\chi_{4235}(2474,\cdot)\) \(\chi_{4235}(2749,\cdot)\) \(\chi_{4235}(2859,\cdot)\) \(\chi_{4235}(3134,\cdot)\) \(\chi_{4235}(3244,\cdot)\) \(\chi_{4235}(3519,\cdot)\) \(\chi_{4235}(3904,\cdot)\) \(\chi_{4235}(4014,\cdot)\)
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 66 polynomial |
Values on generators
\((2542,1816,2906)\) → \((-1,e\left(\frac{1}{6}\right),e\left(\frac{21}{22}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(12\) | \(13\) | \(16\) | \(17\) |
\( \chi_{ 4235 }(934, a) \) | \(1\) | \(1\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{19}{33}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{29}{66}\right)\) |
sage: chi.jacobi_sum(n)