from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3015, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,0,20]))
pari: [g,chi] = znchar(Mod(26,3015))
Basic properties
| Modulus: | \(3015\) | |
| Conductor: | \(201\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{201}(26,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3015.di
\(\chi_{3015}(26,\cdot)\) \(\chi_{3015}(71,\cdot)\) \(\chi_{3015}(116,\cdot)\) \(\chi_{3015}(341,\cdot)\) \(\chi_{3015}(791,\cdot)\) \(\chi_{3015}(881,\cdot)\) \(\chi_{3015}(926,\cdot)\) \(\chi_{3015}(971,\cdot)\) \(\chi_{3015}(1061,\cdot)\) \(\chi_{3015}(1241,\cdot)\) \(\chi_{3015}(1376,\cdot)\) \(\chi_{3015}(1601,\cdot)\) \(\chi_{3015}(1691,\cdot)\) \(\chi_{3015}(1781,\cdot)\) \(\chi_{3015}(1826,\cdot)\) \(\chi_{3015}(2096,\cdot)\) \(\chi_{3015}(2276,\cdot)\) \(\chi_{3015}(2366,\cdot)\) \(\chi_{3015}(2636,\cdot)\) \(\chi_{3015}(2861,\cdot)\)
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
\((1676,1207,136)\) → \((-1,1,e\left(\frac{10}{33}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 3015 }(26, a) \) | \(-1\) | \(1\) | \(e\left(\frac{53}{66}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{25}{66}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{1}{33}\right)\) |
sage: chi.jacobi_sum(n)