from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2160, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([18,9,10,18]))
pari: [g,chi] = znchar(Mod(59,2160))
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.du
\(\chi_{2160}(59,\cdot)\) \(\chi_{2160}(299,\cdot)\) \(\chi_{2160}(419,\cdot)\) \(\chi_{2160}(659,\cdot)\) \(\chi_{2160}(779,\cdot)\) \(\chi_{2160}(1019,\cdot)\) \(\chi_{2160}(1139,\cdot)\) \(\chi_{2160}(1379,\cdot)\) \(\chi_{2160}(1499,\cdot)\) \(\chi_{2160}(1739,\cdot)\) \(\chi_{2160}(1859,\cdot)\) \(\chi_{2160}(2099,\cdot)\)
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: | 36.36.21102921020234122106053112702928025095164093219943162661428472578048000000000000000000.1 |
Values on generators
\((271,1621,2081,1297)\) → \((-1,i,e\left(\frac{5}{18}\right),-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 2160 }(59, a) \) | \(1\) | \(1\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{1}{36}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{2}{9}\right)\) |
sage: chi.jacobi_sum(n)