from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6030, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([22,33,46]))
pari: [g,chi] = znchar(Mod(49,6030))
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 \(\chi_{3015}(49,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6030.dx
\(\chi_{6030}(49,\cdot)\) \(\chi_{6030}(169,\cdot)\) \(\chi_{6030}(529,\cdot)\) \(\chi_{6030}(1759,\cdot)\) \(\chi_{6030}(1789,\cdot)\) \(\chi_{6030}(1969,\cdot)\) \(\chi_{6030}(2029,\cdot)\) \(\chi_{6030}(2209,\cdot)\) \(\chi_{6030}(2569,\cdot)\) \(\chi_{6030}(2869,\cdot)\) \(\chi_{6030}(3289,\cdot)\) \(\chi_{6030}(3319,\cdot)\) \(\chi_{6030}(3739,\cdot)\) \(\chi_{6030}(3829,\cdot)\) \(\chi_{6030}(4309,\cdot)\) \(\chi_{6030}(4639,\cdot)\) \(\chi_{6030}(4729,\cdot)\) \(\chi_{6030}(5029,\cdot)\) \(\chi_{6030}(5929,\cdot)\) \(\chi_{6030}(6019,\cdot)\)
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
\((4691,1207,3151)\) → \((e\left(\frac{1}{3}\right),-1,e\left(\frac{23}{33}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 6030 }(49, a) \) | \(1\) | \(1\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{7}{66}\right)\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(1\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{20}{33}\right)\) |
sage: chi.jacobi_sum(n)