from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3015, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([44,0,4]))
pari: [g,chi] = znchar(Mod(16,3015))
Basic properties
| Modulus: | \(3015\) | |
| Conductor: | \(603\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(33\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{603}(16,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3015.cq
\(\chi_{3015}(16,\cdot)\) \(\chi_{3015}(106,\cdot)\) \(\chi_{3015}(151,\cdot)\) \(\chi_{3015}(256,\cdot)\) \(\chi_{3015}(301,\cdot)\) \(\chi_{3015}(391,\cdot)\) \(\chi_{3015}(421,\cdot)\) \(\chi_{3015}(571,\cdot)\) \(\chi_{3015}(601,\cdot)\) \(\chi_{3015}(706,\cdot)\) \(\chi_{3015}(931,\cdot)\) \(\chi_{3015}(961,\cdot)\) \(\chi_{3015}(1456,\cdot)\) \(\chi_{3015}(1681,\cdot)\) \(\chi_{3015}(1696,\cdot)\) \(\chi_{3015}(2131,\cdot)\) \(\chi_{3015}(2191,\cdot)\) \(\chi_{3015}(2221,\cdot)\) \(\chi_{3015}(2371,\cdot)\) \(\chi_{3015}(2416,\cdot)\)
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 33 polynomial |
Values on generators
\((1676,1207,136)\) → \((e\left(\frac{2}{3}\right),1,e\left(\frac{2}{33}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 3015 }(16, a) \) | \(1\) | \(1\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{8}{33}\right)\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{20}{33}\right)\) |
sage: chi.jacobi_sum(n)