from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3864, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,33,33,44,36]))
pari: [g,chi] = znchar(Mod(179,3864))
Basic properties
| Modulus: | \(3864\) | |
| Conductor: | \(3864\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3864.eo
\(\chi_{3864}(179,\cdot)\) \(\chi_{3864}(347,\cdot)\) \(\chi_{3864}(443,\cdot)\) \(\chi_{3864}(515,\cdot)\) \(\chi_{3864}(611,\cdot)\) \(\chi_{3864}(683,\cdot)\) \(\chi_{3864}(947,\cdot)\) \(\chi_{3864}(1283,\cdot)\) \(\chi_{3864}(1451,\cdot)\) \(\chi_{3864}(1619,\cdot)\) \(\chi_{3864}(1691,\cdot)\) \(\chi_{3864}(1787,\cdot)\) \(\chi_{3864}(2027,\cdot)\) \(\chi_{3864}(2699,\cdot)\) \(\chi_{3864}(2795,\cdot)\) \(\chi_{3864}(3131,\cdot)\) \(\chi_{3864}(3203,\cdot)\) \(\chi_{3864}(3371,\cdot)\) \(\chi_{3864}(3707,\cdot)\) \(\chi_{3864}(3803,\cdot)\)
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
\((967,1933,1289,2761,2857)\) → \((-1,-1,-1,e\left(\frac{2}{3}\right),e\left(\frac{6}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 3864 }(179, a) \) | \(1\) | \(1\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{5}{66}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{65}{66}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{29}{66}\right)\) | \(e\left(\frac{19}{66}\right)\) | \(e\left(\frac{1}{22}\right)\) |
sage: chi.jacobi_sum(n)