from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6030, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([0,33,10]))
pari: [g,chi] = znchar(Mod(19,6030))
Basic properties
| Modulus: | \(6030\) | |
| Conductor: | \(335\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{335}(19,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6030.di
\(\chi_{6030}(19,\cdot)\) \(\chi_{6030}(199,\cdot)\) \(\chi_{6030}(289,\cdot)\) \(\chi_{6030}(559,\cdot)\) \(\chi_{6030}(1009,\cdot)\) \(\chi_{6030}(1279,\cdot)\) \(\chi_{6030}(1729,\cdot)\) \(\chi_{6030}(1819,\cdot)\) \(\chi_{6030}(1909,\cdot)\) \(\chi_{6030}(1999,\cdot)\) \(\chi_{6030}(2179,\cdot)\) \(\chi_{6030}(2539,\cdot)\) \(\chi_{6030}(2629,\cdot)\) \(\chi_{6030}(2719,\cdot)\) \(\chi_{6030}(3799,\cdot)\) \(\chi_{6030}(3979,\cdot)\) \(\chi_{6030}(4069,\cdot)\) \(\chi_{6030}(4879,\cdot)\) \(\chi_{6030}(5329,\cdot)\) \(\chi_{6030}(5779,\cdot)\)
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
\((4691,1207,3151)\) → \((1,-1,e\left(\frac{5}{33}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 6030 }(19, a) \) | \(1\) | \(1\) | \(e\left(\frac{65}{66}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{25}{66}\right)\) | \(e\left(\frac{13}{66}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{49}{66}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{1}{33}\right)\) |
sage: chi.jacobi_sum(n)