from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2160, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([0,9,16,18]))
pari: [g,chi] = znchar(Mod(229,2160))
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.eo
\(\chi_{2160}(229,\cdot)\) \(\chi_{2160}(349,\cdot)\) \(\chi_{2160}(589,\cdot)\) \(\chi_{2160}(709,\cdot)\) \(\chi_{2160}(949,\cdot)\) \(\chi_{2160}(1069,\cdot)\) \(\chi_{2160}(1309,\cdot)\) \(\chi_{2160}(1429,\cdot)\) \(\chi_{2160}(1669,\cdot)\) \(\chi_{2160}(1789,\cdot)\) \(\chi_{2160}(2029,\cdot)\) \(\chi_{2160}(2149,\cdot)\)
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
\((271,1621,2081,1297)\) → \((1,i,e\left(\frac{4}{9}\right),-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 2160 }(229, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{29}{36}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{1}{18}\right)\) |
sage: chi.jacobi_sum(n)