from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6030, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,33,25]))
pari: [g,chi] = znchar(Mod(899,6030))
Basic properties
| Modulus: | \(6030\) | |
| Conductor: | \(1005\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1005}(899,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6030.du
\(\chi_{6030}(899,\cdot)\) \(\chi_{6030}(989,\cdot)\) \(\chi_{6030}(1079,\cdot)\) \(\chi_{6030}(1439,\cdot)\) \(\chi_{6030}(1619,\cdot)\) \(\chi_{6030}(1709,\cdot)\) \(\chi_{6030}(1799,\cdot)\) \(\chi_{6030}(1889,\cdot)\) \(\chi_{6030}(2339,\cdot)\) \(\chi_{6030}(2609,\cdot)\) \(\chi_{6030}(3059,\cdot)\) \(\chi_{6030}(3329,\cdot)\) \(\chi_{6030}(3419,\cdot)\) \(\chi_{6030}(3599,\cdot)\) \(\chi_{6030}(3869,\cdot)\) \(\chi_{6030}(4319,\cdot)\) \(\chi_{6030}(4769,\cdot)\) \(\chi_{6030}(5579,\cdot)\) \(\chi_{6030}(5669,\cdot)\) \(\chi_{6030}(5849,\cdot)\)
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
\((4691,1207,3151)\) → \((-1,-1,e\left(\frac{25}{66}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 6030 }(899, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{53}{66}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{19}{33}\right)\) |
sage: chi.jacobi_sum(n)