from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6030, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([11,33,47]))
pari: [g,chi] = znchar(Mod(299,6030))
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 \(\chi_{3015}(299,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6030.cy
\(\chi_{6030}(299,\cdot)\) \(\chi_{6030}(329,\cdot)\) \(\chi_{6030}(749,\cdot)\) \(\chi_{6030}(1049,\cdot)\) \(\chi_{6030}(1409,\cdot)\) \(\chi_{6030}(1589,\cdot)\) \(\chi_{6030}(1649,\cdot)\) \(\chi_{6030}(1829,\cdot)\) \(\chi_{6030}(1859,\cdot)\) \(\chi_{6030}(3089,\cdot)\) \(\chi_{6030}(3449,\cdot)\) \(\chi_{6030}(3569,\cdot)\) \(\chi_{6030}(3629,\cdot)\) \(\chi_{6030}(3719,\cdot)\) \(\chi_{6030}(4619,\cdot)\) \(\chi_{6030}(4919,\cdot)\) \(\chi_{6030}(5009,\cdot)\) \(\chi_{6030}(5339,\cdot)\) \(\chi_{6030}(5819,\cdot)\) \(\chi_{6030}(5909,\cdot)\)
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
\((4691,1207,3151)\) → \((e\left(\frac{1}{6}\right),-1,e\left(\frac{47}{66}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
\( \chi_{ 6030 }(299, a) \) | \(1\) | \(1\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{19}{33}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(-1\) | \(e\left(\frac{53}{66}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{19}{33}\right)\) |
sage: chi.jacobi_sum(n)