from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6030, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([11,33,21]))
pari: [g,chi] = znchar(Mod(119,6030))
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 \(\chi_{3015}(119,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6030.dt
\(\chi_{6030}(119,\cdot)\) \(\chi_{6030}(209,\cdot)\) \(\chi_{6030}(779,\cdot)\) \(\chi_{6030}(929,\cdot)\) \(\chi_{6030}(1769,\cdot)\) \(\chi_{6030}(1919,\cdot)\) \(\chi_{6030}(2129,\cdot)\) \(\chi_{6030}(2189,\cdot)\) \(\chi_{6030}(2219,\cdot)\) \(\chi_{6030}(2549,\cdot)\) \(\chi_{6030}(2819,\cdot)\) \(\chi_{6030}(2939,\cdot)\) \(\chi_{6030}(3269,\cdot)\) \(\chi_{6030}(3929,\cdot)\) \(\chi_{6030}(4199,\cdot)\) \(\chi_{6030}(4559,\cdot)\) \(\chi_{6030}(4799,\cdot)\) \(\chi_{6030}(4829,\cdot)\) \(\chi_{6030}(5279,\cdot)\) \(\chi_{6030}(5789,\cdot)\)
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
\((4691,1207,3151)\) → \((e\left(\frac{1}{6}\right),-1,e\left(\frac{7}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 6030 }(119, a) \) | \(1\) | \(1\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{19}{66}\right)\) | \(-1\) | \(e\left(\frac{23}{33}\right)\) |
sage: chi.jacobi_sum(n)