from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2160, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([0,27,22,9]))
pari: [g,chi] = znchar(Mod(77,2160))
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.ec
\(\chi_{2160}(77,\cdot)\) \(\chi_{2160}(293,\cdot)\) \(\chi_{2160}(317,\cdot)\) \(\chi_{2160}(533,\cdot)\) \(\chi_{2160}(797,\cdot)\) \(\chi_{2160}(1013,\cdot)\) \(\chi_{2160}(1037,\cdot)\) \(\chi_{2160}(1253,\cdot)\) \(\chi_{2160}(1517,\cdot)\) \(\chi_{2160}(1733,\cdot)\) \(\chi_{2160}(1757,\cdot)\) \(\chi_{2160}(1973,\cdot)\)
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: | 36.36.41216642617644769738384985747906299013992369570201489573102485504000000000000000000000000000.1 |
Values on generators
\((271,1621,2081,1297)\) → \((1,-i,e\left(\frac{11}{18}\right),i)\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 2160 }(77, a) \) | \(1\) | \(1\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{8}{9}\right)\) |
sage: chi.jacobi_sum(n)