from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3015, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,33,61]))
pari: [g,chi] = znchar(Mod(44,3015))
Basic properties
| Modulus: | \(3015\) | |
| 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 \(\chi_{1005}(44,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3015.dk
\(\chi_{3015}(44,\cdot)\) \(\chi_{3015}(314,\cdot)\) \(\chi_{3015}(404,\cdot)\) \(\chi_{3015}(584,\cdot)\) \(\chi_{3015}(854,\cdot)\) \(\chi_{3015}(899,\cdot)\) \(\chi_{3015}(989,\cdot)\) \(\chi_{3015}(1079,\cdot)\) \(\chi_{3015}(1304,\cdot)\) \(\chi_{3015}(1439,\cdot)\) \(\chi_{3015}(1619,\cdot)\) \(\chi_{3015}(1709,\cdot)\) \(\chi_{3015}(1754,\cdot)\) \(\chi_{3015}(1799,\cdot)\) \(\chi_{3015}(1889,\cdot)\) \(\chi_{3015}(2339,\cdot)\) \(\chi_{3015}(2564,\cdot)\) \(\chi_{3015}(2609,\cdot)\) \(\chi_{3015}(2654,\cdot)\) \(\chi_{3015}(2834,\cdot)\)
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
\((1676,1207,136)\) → \((-1,-1,e\left(\frac{61}{66}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 3015 }(44, a) \) | \(1\) | \(1\) | \(e\left(\frac{61}{66}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{8}{33}\right)\) |
sage: chi.jacobi_sum(n)