from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2160, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([18,9,20,9]))
pari: [g,chi] = znchar(Mod(187,2160))
Basic properties
| Modulus: | \(2160\) | |
| Conductor: | \(2160\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2160.eg
\(\chi_{2160}(187,\cdot)\) \(\chi_{2160}(403,\cdot)\) \(\chi_{2160}(427,\cdot)\) \(\chi_{2160}(643,\cdot)\) \(\chi_{2160}(907,\cdot)\) \(\chi_{2160}(1123,\cdot)\) \(\chi_{2160}(1147,\cdot)\) \(\chi_{2160}(1363,\cdot)\) \(\chi_{2160}(1627,\cdot)\) \(\chi_{2160}(1843,\cdot)\) \(\chi_{2160}(1867,\cdot)\) \(\chi_{2160}(2083,\cdot)\)
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
\((271,1621,2081,1297)\) → \((-1,i,e\left(\frac{5}{9}\right),i)\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 2160 }(187, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{17}{18}\right)\) |
sage: chi.jacobi_sum(n)