from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9072, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([27,0,52,27]))
pari: [g,chi] = znchar(Mod(223,9072))
Basic properties
| Modulus: | \(9072\) | |
| Conductor: | \(2268\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(54\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{2268}(223,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 9072.hx
\(\chi_{9072}(223,\cdot)\) \(\chi_{9072}(895,\cdot)\) \(\chi_{9072}(1231,\cdot)\) \(\chi_{9072}(1903,\cdot)\) \(\chi_{9072}(2239,\cdot)\) \(\chi_{9072}(2911,\cdot)\) \(\chi_{9072}(3247,\cdot)\) \(\chi_{9072}(3919,\cdot)\) \(\chi_{9072}(4255,\cdot)\) \(\chi_{9072}(4927,\cdot)\) \(\chi_{9072}(5263,\cdot)\) \(\chi_{9072}(5935,\cdot)\) \(\chi_{9072}(6271,\cdot)\) \(\chi_{9072}(6943,\cdot)\) \(\chi_{9072}(7279,\cdot)\) \(\chi_{9072}(7951,\cdot)\) \(\chi_{9072}(8287,\cdot)\) \(\chi_{9072}(8959,\cdot)\)
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: | \(\Q(\zeta_{27})\) |
| Fixed field: | Number field defined by a degree 54 polynomial |
Values on generators
\((1135,6805,3809,2593)\) → \((-1,1,e\left(\frac{26}{27}\right),-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 9072 }(223, a) \) | \(1\) | \(1\) | \(e\left(\frac{35}{54}\right)\) | \(e\left(\frac{1}{54}\right)\) | \(e\left(\frac{11}{54}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{5}{54}\right)\) | \(e\left(\frac{8}{27}\right)\) | \(e\left(\frac{17}{27}\right)\) | \(e\left(\frac{7}{27}\right)\) | \(e\left(\frac{4}{9}\right)\) |
sage: chi.jacobi_sum(n)