from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3015, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([11,33,21]))
pari: [g,chi] = znchar(Mod(119,3015))
Basic properties
| Modulus: | \(3015\) | |
| Conductor: | \(3015\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3015.dj
\(\chi_{3015}(119,\cdot)\) \(\chi_{3015}(209,\cdot)\) \(\chi_{3015}(254,\cdot)\) \(\chi_{3015}(779,\cdot)\) \(\chi_{3015}(914,\cdot)\) \(\chi_{3015}(929,\cdot)\) \(\chi_{3015}(1184,\cdot)\) \(\chi_{3015}(1544,\cdot)\) \(\chi_{3015}(1769,\cdot)\) \(\chi_{3015}(1784,\cdot)\) \(\chi_{3015}(1814,\cdot)\) \(\chi_{3015}(1919,\cdot)\) \(\chi_{3015}(2129,\cdot)\) \(\chi_{3015}(2189,\cdot)\) \(\chi_{3015}(2219,\cdot)\) \(\chi_{3015}(2264,\cdot)\) \(\chi_{3015}(2549,\cdot)\) \(\chi_{3015}(2774,\cdot)\) \(\chi_{3015}(2819,\cdot)\) \(\chi_{3015}(2939,\cdot)\)
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
\((1676,1207,136)\) → \((e\left(\frac{1}{6}\right),-1,e\left(\frac{7}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 3015 }(119, a) \) | \(1\) | \(1\) | \(e\left(\frac{65}{66}\right)\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{31}{66}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{2}{11}\right)\) |
sage: chi.jacobi_sum(n)