from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3234, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([0,1,0]))
pari: [g,chi] = znchar(Mod(199,3234))
Basic properties
| Modulus: | \(3234\) | |
| Conductor: | \(49\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{49}(3,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3234.bs
\(\chi_{3234}(199,\cdot)\) \(\chi_{3234}(397,\cdot)\) \(\chi_{3234}(661,\cdot)\) \(\chi_{3234}(859,\cdot)\) \(\chi_{3234}(1123,\cdot)\) \(\chi_{3234}(1321,\cdot)\) \(\chi_{3234}(1585,\cdot)\) \(\chi_{3234}(2047,\cdot)\) \(\chi_{3234}(2245,\cdot)\) \(\chi_{3234}(2509,\cdot)\) \(\chi_{3234}(2707,\cdot)\) \(\chi_{3234}(3169,\cdot)\)
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: | \(\Q(\zeta_{21})\) |
| Fixed field: | Number field defined by a degree 42 polynomial |
Values on generators
\((1079,199,2059)\) → \((1,e\left(\frac{1}{42}\right),1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 3234 }(199, a) \) | \(-1\) | \(1\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{25}{42}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{5}{14}\right)\) |
sage: chi.jacobi_sum(n)