from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6030, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([55,33,35]))
pari: [g,chi] = znchar(Mod(599,6030))
Basic properties
| Modulus: | \(6030\) | |
| Conductor: | \(3015\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{3015}(599,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6030.dv
\(\chi_{6030}(599,\cdot)\) \(\chi_{6030}(1319,\cdot)\) \(\chi_{6030}(1559,\cdot)\) \(\chi_{6030}(2309,\cdot)\) \(\chi_{6030}(2759,\cdot)\) \(\chi_{6030}(2909,\cdot)\) \(\chi_{6030}(2999,\cdot)\) \(\chi_{6030}(3659,\cdot)\) \(\chi_{6030}(3809,\cdot)\) \(\chi_{6030}(3839,\cdot)\) \(\chi_{6030}(3899,\cdot)\) \(\chi_{6030}(4349,\cdot)\) \(\chi_{6030}(5069,\cdot)\) \(\chi_{6030}(5099,\cdot)\) \(\chi_{6030}(5429,\cdot)\) \(\chi_{6030}(5459,\cdot)\) \(\chi_{6030}(5609,\cdot)\) \(\chi_{6030}(5639,\cdot)\) \(\chi_{6030}(5729,\cdot)\) \(\chi_{6030}(5879,\cdot)\)
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
\((4691,1207,3151)\) → \((e\left(\frac{5}{6}\right),-1,e\left(\frac{35}{66}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 6030 }(599, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{3}{11}\right)\) |
sage: chi.jacobi_sum(n)