from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1176, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,0,21,38]))
pari: [g,chi] = znchar(Mod(23,1176))
Basic properties
| Modulus: | \(1176\) | |
| Conductor: | \(588\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{588}(23,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1176.cb
\(\chi_{1176}(23,\cdot)\) \(\chi_{1176}(95,\cdot)\) \(\chi_{1176}(191,\cdot)\) \(\chi_{1176}(359,\cdot)\) \(\chi_{1176}(431,\cdot)\) \(\chi_{1176}(527,\cdot)\) \(\chi_{1176}(599,\cdot)\) \(\chi_{1176}(695,\cdot)\) \(\chi_{1176}(767,\cdot)\) \(\chi_{1176}(935,\cdot)\) \(\chi_{1176}(1031,\cdot)\) \(\chi_{1176}(1103,\cdot)\)
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
\((295,589,785,1081)\) → \((-1,1,-1,e\left(\frac{19}{21}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 1176 }(23, a) \) | \(1\) | \(1\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{20}{21}\right)\) |
sage: chi.jacobi_sum(n)