from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6030, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([44,33,24]))
pari: [g,chi] = znchar(Mod(349,6030))
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 \(\chi_{3015}(349,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6030.dl
\(\chi_{6030}(349,\cdot)\) \(\chi_{6030}(679,\cdot)\) \(\chi_{6030}(799,\cdot)\) \(\chi_{6030}(1069,\cdot)\) \(\chi_{6030}(1399,\cdot)\) \(\chi_{6030}(1429,\cdot)\) \(\chi_{6030}(1489,\cdot)\) \(\chi_{6030}(1699,\cdot)\) \(\chi_{6030}(1849,\cdot)\) \(\chi_{6030}(2689,\cdot)\) \(\chi_{6030}(2839,\cdot)\) \(\chi_{6030}(3409,\cdot)\) \(\chi_{6030}(3499,\cdot)\) \(\chi_{6030}(3859,\cdot)\) \(\chi_{6030}(4369,\cdot)\) \(\chi_{6030}(4819,\cdot)\) \(\chi_{6030}(4849,\cdot)\) \(\chi_{6030}(5089,\cdot)\) \(\chi_{6030}(5449,\cdot)\) \(\chi_{6030}(5719,\cdot)\)
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
\((4691,1207,3151)\) → \((e\left(\frac{2}{3}\right),-1,e\left(\frac{4}{11}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
\( \chi_{ 6030 }(349, a) \) | \(1\) | \(1\) | \(e\left(\frac{35}{66}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{49}{66}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(-1\) | \(e\left(\frac{20}{33}\right)\) |
sage: chi.jacobi_sum(n)