from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1134, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([53,27]))
pari: [g,chi] = znchar(Mod(41,1134))
Basic properties
Modulus: | \(1134\) | |
Conductor: | \(567\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(54\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{567}(41,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1134.bq
\(\chi_{1134}(41,\cdot)\) \(\chi_{1134}(83,\cdot)\) \(\chi_{1134}(167,\cdot)\) \(\chi_{1134}(209,\cdot)\) \(\chi_{1134}(293,\cdot)\) \(\chi_{1134}(335,\cdot)\) \(\chi_{1134}(419,\cdot)\) \(\chi_{1134}(461,\cdot)\) \(\chi_{1134}(545,\cdot)\) \(\chi_{1134}(587,\cdot)\) \(\chi_{1134}(671,\cdot)\) \(\chi_{1134}(713,\cdot)\) \(\chi_{1134}(797,\cdot)\) \(\chi_{1134}(839,\cdot)\) \(\chi_{1134}(923,\cdot)\) \(\chi_{1134}(965,\cdot)\) \(\chi_{1134}(1049,\cdot)\) \(\chi_{1134}(1091,\cdot)\)
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
\((407,325)\) → \((e\left(\frac{53}{54}\right),-1)\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
\( \chi_{ 1134 }(41, a) \) | \(1\) | \(1\) | \(e\left(\frac{2}{27}\right)\) | \(e\left(\frac{41}{54}\right)\) | \(e\left(\frac{19}{54}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{43}{54}\right)\) | \(e\left(\frac{4}{27}\right)\) | \(e\left(\frac{17}{54}\right)\) | \(e\left(\frac{7}{54}\right)\) | \(e\left(\frac{2}{9}\right)\) |
sage: chi.jacobi_sum(n)