from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5472, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([18,27,18,28]))
pari: [g,chi] = znchar(Mod(215,5472))
Basic properties
Modulus: | \(5472\) | |
Conductor: | \(912\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{912}(899,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5472.ij
\(\chi_{5472}(215,\cdot)\) \(\chi_{5472}(359,\cdot)\) \(\chi_{5472}(503,\cdot)\) \(\chi_{5472}(935,\cdot)\) \(\chi_{5472}(1943,\cdot)\) \(\chi_{5472}(2087,\cdot)\) \(\chi_{5472}(2951,\cdot)\) \(\chi_{5472}(3095,\cdot)\) \(\chi_{5472}(3239,\cdot)\) \(\chi_{5472}(3671,\cdot)\) \(\chi_{5472}(4679,\cdot)\) \(\chi_{5472}(4823,\cdot)\)
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: | \(\Q(\zeta_{36})\) |
Fixed field: | Number field defined by a degree 36 polynomial |
Values on generators
\((4447,2053,1217,3745)\) → \((-1,-i,-1,e\left(\frac{7}{9}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
\( \chi_{ 5472 }(215, a) \) | \(1\) | \(1\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{13}{36}\right)\) |
sage: chi.jacobi_sum(n)