from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3484, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([0,44,29]))
pari: [g,chi] = znchar(Mod(113,3484))
Basic properties
| Modulus: | \(3484\) | |
| Conductor: | \(871\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{871}(113,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3484.dj
\(\chi_{3484}(113,\cdot)\) \(\chi_{3484}(165,\cdot)\) \(\chi_{3484}(185,\cdot)\) \(\chi_{3484}(497,\cdot)\) \(\chi_{3484}(549,\cdot)\) \(\chi_{3484}(653,\cdot)\) \(\chi_{3484}(861,\cdot)\) \(\chi_{3484}(945,\cdot)\) \(\chi_{3484}(1589,\cdot)\) \(\chi_{3484}(1693,\cdot)\) \(\chi_{3484}(1829,\cdot)\) \(\chi_{3484}(2089,\cdot)\) \(\chi_{3484}(2213,\cdot)\) \(\chi_{3484}(2245,\cdot)\) \(\chi_{3484}(2453,\cdot)\) \(\chi_{3484}(2473,\cdot)\) \(\chi_{3484}(2557,\cdot)\) \(\chi_{3484}(2609,\cdot)\) \(\chi_{3484}(3181,\cdot)\) \(\chi_{3484}(3461,\cdot)\)
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
\((1743,1341,3017)\) → \((1,e\left(\frac{2}{3}\right),e\left(\frac{29}{66}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(15\) | \(17\) | \(19\) | \(21\) | \(23\) |
| \( \chi_{ 3484 }(113, a) \) | \(-1\) | \(1\) | \(e\left(\frac{53}{66}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{29}{66}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{32}{33}\right)\) |
sage: chi.jacobi_sum(n)