from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3864, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,0,0,11,18]))
pari: [g,chi] = znchar(Mod(31,3864))
Basic properties
| Modulus: | \(3864\) | |
| Conductor: | \(644\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{644}(31,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3864.dy
\(\chi_{3864}(31,\cdot)\) \(\chi_{3864}(271,\cdot)\) \(\chi_{3864}(439,\cdot)\) \(\chi_{3864}(535,\cdot)\) \(\chi_{3864}(607,\cdot)\) \(\chi_{3864}(703,\cdot)\) \(\chi_{3864}(775,\cdot)\) \(\chi_{3864}(1039,\cdot)\) \(\chi_{3864}(1375,\cdot)\) \(\chi_{3864}(1543,\cdot)\) \(\chi_{3864}(1711,\cdot)\) \(\chi_{3864}(1783,\cdot)\) \(\chi_{3864}(1879,\cdot)\) \(\chi_{3864}(2119,\cdot)\) \(\chi_{3864}(2791,\cdot)\) \(\chi_{3864}(2887,\cdot)\) \(\chi_{3864}(3223,\cdot)\) \(\chi_{3864}(3295,\cdot)\) \(\chi_{3864}(3463,\cdot)\) \(\chi_{3864}(3799,\cdot)\)
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
\((967,1933,1289,2761,2857)\) → \((-1,1,1,e\left(\frac{1}{6}\right),e\left(\frac{3}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 3864 }(31, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{66}\right)\) | \(e\left(\frac{41}{66}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{5}{66}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{17}{22}\right)\) |
sage: chi.jacobi_sum(n)