from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1134, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([22,0]))
pari: [g,chi] = znchar(Mod(43,1134))
Basic properties
| Modulus: | \(1134\) | |
| 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 \(\chi_{81}(43,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1134.bh
\(\chi_{1134}(43,\cdot)\) \(\chi_{1134}(85,\cdot)\) \(\chi_{1134}(169,\cdot)\) \(\chi_{1134}(211,\cdot)\) \(\chi_{1134}(295,\cdot)\) \(\chi_{1134}(337,\cdot)\) \(\chi_{1134}(421,\cdot)\) \(\chi_{1134}(463,\cdot)\) \(\chi_{1134}(547,\cdot)\) \(\chi_{1134}(589,\cdot)\) \(\chi_{1134}(673,\cdot)\) \(\chi_{1134}(715,\cdot)\) \(\chi_{1134}(799,\cdot)\) \(\chi_{1134}(841,\cdot)\) \(\chi_{1134}(925,\cdot)\) \(\chi_{1134}(967,\cdot)\) \(\chi_{1134}(1051,\cdot)\) \(\chi_{1134}(1093,\cdot)\)
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
\((407,325)\) → \((e\left(\frac{11}{27}\right),1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 1134 }(43, a) \) | \(1\) | \(1\) | \(e\left(\frac{10}{27}\right)\) | \(e\left(\frac{8}{27}\right)\) | \(e\left(\frac{7}{27}\right)\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{13}{27}\right)\) | \(e\left(\frac{20}{27}\right)\) | \(e\left(\frac{2}{27}\right)\) | \(e\left(\frac{4}{27}\right)\) | \(e\left(\frac{1}{9}\right)\) |
sage: chi.jacobi_sum(n)