from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1610, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([0,44,60]))
pari: [g,chi] = znchar(Mod(81,1610))
Basic properties
| Modulus: | \(1610\) | |
| Conductor: | \(161\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(33\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{161}(81,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1610.bg
\(\chi_{1610}(81,\cdot)\) \(\chi_{1610}(121,\cdot)\) \(\chi_{1610}(151,\cdot)\) \(\chi_{1610}(261,\cdot)\) \(\chi_{1610}(331,\cdot)\) \(\chi_{1610}(361,\cdot)\) \(\chi_{1610}(501,\cdot)\) \(\chi_{1610}(541,\cdot)\) \(\chi_{1610}(611,\cdot)\) \(\chi_{1610}(821,\cdot)\) \(\chi_{1610}(961,\cdot)\) \(\chi_{1610}(991,\cdot)\) \(\chi_{1610}(1061,\cdot)\) \(\chi_{1610}(1131,\cdot)\) \(\chi_{1610}(1271,\cdot)\) \(\chi_{1610}(1411,\cdot)\) \(\chi_{1610}(1451,\cdot)\) \(\chi_{1610}(1481,\cdot)\) \(\chi_{1610}(1521,\cdot)\) \(\chi_{1610}(1591,\cdot)\)
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: | \(\Q(\zeta_{33})\) |
| Fixed field: | 33.33.277966181338944111003326058293667039541136678070715028736001.1 |
Values on generators
\((967,1151,281)\) → \((1,e\left(\frac{2}{3}\right),e\left(\frac{10}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(27\) | \(29\) | \(31\) | \(33\) |
| \( \chi_{ 1610 }(81, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{2}{33}\right)\) |
sage: chi.jacobi_sum(n)