from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1053, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([26,0]))
pari: [g,chi] = znchar(Mod(40,1053))
Basic properties
| Modulus: | \(1053\) | |
| Conductor: | \(81\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(27\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{81}(40,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1053.bq
\(\chi_{1053}(40,\cdot)\) \(\chi_{1053}(79,\cdot)\) \(\chi_{1053}(157,\cdot)\) \(\chi_{1053}(196,\cdot)\) \(\chi_{1053}(274,\cdot)\) \(\chi_{1053}(313,\cdot)\) \(\chi_{1053}(391,\cdot)\) \(\chi_{1053}(430,\cdot)\) \(\chi_{1053}(508,\cdot)\) \(\chi_{1053}(547,\cdot)\) \(\chi_{1053}(625,\cdot)\) \(\chi_{1053}(664,\cdot)\) \(\chi_{1053}(742,\cdot)\) \(\chi_{1053}(781,\cdot)\) \(\chi_{1053}(859,\cdot)\) \(\chi_{1053}(898,\cdot)\) \(\chi_{1053}(976,\cdot)\) \(\chi_{1053}(1015,\cdot)\)
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 27 polynomial |
Values on generators
\((326,730)\) → \((e\left(\frac{13}{27}\right),1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(14\) | \(16\) | \(17\) |
| \( \chi_{ 1053 }(40, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{27}\right)\) | \(e\left(\frac{26}{27}\right)\) | \(e\left(\frac{2}{27}\right)\) | \(e\left(\frac{19}{27}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{7}{27}\right)\) | \(e\left(\frac{5}{27}\right)\) | \(e\left(\frac{25}{27}\right)\) | \(e\left(\frac{8}{9}\right)\) |
sage: chi.jacobi_sum(n)