from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3015, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([55,0,46]))
pari: [g,chi] = znchar(Mod(1121,3015))
Basic properties
| Modulus: | \(3015\) | |
| Conductor: | \(603\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{603}(518,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3015.dc
\(\chi_{3015}(56,\cdot)\) \(\chi_{3015}(86,\cdot)\) \(\chi_{3015}(236,\cdot)\) \(\chi_{3015}(266,\cdot)\) \(\chi_{3015}(371,\cdot)\) \(\chi_{3015}(596,\cdot)\) \(\chi_{3015}(626,\cdot)\) \(\chi_{3015}(1121,\cdot)\) \(\chi_{3015}(1346,\cdot)\) \(\chi_{3015}(1361,\cdot)\) \(\chi_{3015}(1796,\cdot)\) \(\chi_{3015}(1856,\cdot)\) \(\chi_{3015}(1886,\cdot)\) \(\chi_{3015}(2036,\cdot)\) \(\chi_{3015}(2081,\cdot)\) \(\chi_{3015}(2696,\cdot)\) \(\chi_{3015}(2786,\cdot)\) \(\chi_{3015}(2831,\cdot)\) \(\chi_{3015}(2936,\cdot)\) \(\chi_{3015}(2981,\cdot)\)
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
\((1676,1207,136)\) → \((e\left(\frac{5}{6}\right),1,e\left(\frac{23}{33}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 3015 }(1121, a) \) | \(-1\) | \(1\) | \(e\left(\frac{35}{66}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{7}{66}\right)\) | \(e\left(\frac{32}{33}\right)\) |
sage: chi.jacobi_sum(n)