from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(972, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([27,52]))
pari: [g,chi] = znchar(Mod(19,972))
Basic properties
| Modulus: | \(972\) | |
| Conductor: | \(324\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(54\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{324}(223,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 972.n
\(\chi_{972}(19,\cdot)\) \(\chi_{972}(91,\cdot)\) \(\chi_{972}(127,\cdot)\) \(\chi_{972}(199,\cdot)\) \(\chi_{972}(235,\cdot)\) \(\chi_{972}(307,\cdot)\) \(\chi_{972}(343,\cdot)\) \(\chi_{972}(415,\cdot)\) \(\chi_{972}(451,\cdot)\) \(\chi_{972}(523,\cdot)\) \(\chi_{972}(559,\cdot)\) \(\chi_{972}(631,\cdot)\) \(\chi_{972}(667,\cdot)\) \(\chi_{972}(739,\cdot)\) \(\chi_{972}(775,\cdot)\) \(\chi_{972}(847,\cdot)\) \(\chi_{972}(883,\cdot)\) \(\chi_{972}(955,\cdot)\)
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
\((487,245)\) → \((-1,e\left(\frac{26}{27}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 972 }(19, a) \) | \(-1\) | \(1\) | \(e\left(\frac{4}{27}\right)\) | \(e\left(\frac{49}{54}\right)\) | \(e\left(\frac{1}{54}\right)\) | \(e\left(\frac{19}{27}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{5}{54}\right)\) | \(e\left(\frac{8}{27}\right)\) | \(e\left(\frac{17}{27}\right)\) | \(e\left(\frac{41}{54}\right)\) |
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)