from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(405, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([41,27]))
pari: [g,chi] = znchar(Mod(59,405))
Basic properties
| Modulus: | \(405\) | |
| Conductor: | \(405\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(54\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 405.v
\(\chi_{405}(14,\cdot)\) \(\chi_{405}(29,\cdot)\) \(\chi_{405}(59,\cdot)\) \(\chi_{405}(74,\cdot)\) \(\chi_{405}(104,\cdot)\) \(\chi_{405}(119,\cdot)\) \(\chi_{405}(149,\cdot)\) \(\chi_{405}(164,\cdot)\) \(\chi_{405}(194,\cdot)\) \(\chi_{405}(209,\cdot)\) \(\chi_{405}(239,\cdot)\) \(\chi_{405}(254,\cdot)\) \(\chi_{405}(284,\cdot)\) \(\chi_{405}(299,\cdot)\) \(\chi_{405}(329,\cdot)\) \(\chi_{405}(344,\cdot)\) \(\chi_{405}(374,\cdot)\) \(\chi_{405}(389,\cdot)\)
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: | \(\Q(\zeta_{27})\) |
| Fixed field: | Number field defined by a degree 54 polynomial |
Values on generators
\((326,82)\) → \((e\left(\frac{41}{54}\right),-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 405 }(59, a) \) | \(-1\) | \(1\) | \(e\left(\frac{7}{27}\right)\) | \(e\left(\frac{14}{27}\right)\) | \(e\left(\frac{35}{54}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{47}{54}\right)\) | \(e\left(\frac{31}{54}\right)\) | \(e\left(\frac{49}{54}\right)\) | \(e\left(\frac{1}{27}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{4}{9}\right)\) |
sage: chi.jacobi_sum(n)
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
Jacobi sum
sage: chi.jacobi_sum(n)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)