from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6030, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([44,33,4]))
pari: [g,chi] = znchar(Mod(619,6030))
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 \(\chi_{3015}(619,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6030.dh
\(\chi_{6030}(619,\cdot)\) \(\chi_{6030}(709,\cdot)\) \(\chi_{6030}(859,\cdot)\) \(\chi_{6030}(1309,\cdot)\) \(\chi_{6030}(2059,\cdot)\) \(\chi_{6030}(2299,\cdot)\) \(\chi_{6030}(3019,\cdot)\) \(\chi_{6030}(3769,\cdot)\) \(\chi_{6030}(3919,\cdot)\) \(\chi_{6030}(4009,\cdot)\) \(\chi_{6030}(4039,\cdot)\) \(\chi_{6030}(4189,\cdot)\) \(\chi_{6030}(4219,\cdot)\) \(\chi_{6030}(4549,\cdot)\) \(\chi_{6030}(4579,\cdot)\) \(\chi_{6030}(5299,\cdot)\) \(\chi_{6030}(5749,\cdot)\) \(\chi_{6030}(5809,\cdot)\) \(\chi_{6030}(5839,\cdot)\) \(\chi_{6030}(5989,\cdot)\)
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
\((4691,1207,3151)\) → \((e\left(\frac{2}{3}\right),-1,e\left(\frac{2}{33}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
\( \chi_{ 6030 }(619, a) \) | \(1\) | \(1\) | \(e\left(\frac{37}{66}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{65}{66}\right)\) | \(e\left(\frac{25}{66}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{35}{66}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{6}{11}\right)\) |
sage: chi.jacobi_sum(n)