from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3015, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([0,0,1]))
pari: [g,chi] = znchar(Mod(136,3015))
Basic properties
Modulus: | \(3015\) | |
Conductor: | \(67\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{67}(2,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3015.dd
\(\chi_{3015}(46,\cdot)\) \(\chi_{3015}(136,\cdot)\) \(\chi_{3015}(316,\cdot)\) \(\chi_{3015}(586,\cdot)\) \(\chi_{3015}(631,\cdot)\) \(\chi_{3015}(721,\cdot)\) \(\chi_{3015}(811,\cdot)\) \(\chi_{3015}(1036,\cdot)\) \(\chi_{3015}(1171,\cdot)\) \(\chi_{3015}(1351,\cdot)\) \(\chi_{3015}(1441,\cdot)\) \(\chi_{3015}(1486,\cdot)\) \(\chi_{3015}(1531,\cdot)\) \(\chi_{3015}(1621,\cdot)\) \(\chi_{3015}(2071,\cdot)\) \(\chi_{3015}(2296,\cdot)\) \(\chi_{3015}(2341,\cdot)\) \(\chi_{3015}(2386,\cdot)\) \(\chi_{3015}(2566,\cdot)\) \(\chi_{3015}(2791,\cdot)\)
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
\((1676,1207,136)\) → \((1,1,e\left(\frac{1}{66}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
\( \chi_{ 3015 }(136, a) \) | \(-1\) | \(1\) | \(e\left(\frac{1}{66}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{23}{66}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{19}{66}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{5}{33}\right)\) |
sage: chi.jacobi_sum(n)