from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(972, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([0,11]))
pari: [g,chi] = znchar(Mod(17,972))
Basic properties
| Modulus: | \(972\) | |
| Conductor: | \(81\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(54\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{81}(23,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 972.o
\(\chi_{972}(17,\cdot)\) \(\chi_{972}(89,\cdot)\) \(\chi_{972}(125,\cdot)\) \(\chi_{972}(197,\cdot)\) \(\chi_{972}(233,\cdot)\) \(\chi_{972}(305,\cdot)\) \(\chi_{972}(341,\cdot)\) \(\chi_{972}(413,\cdot)\) \(\chi_{972}(449,\cdot)\) \(\chi_{972}(521,\cdot)\) \(\chi_{972}(557,\cdot)\) \(\chi_{972}(629,\cdot)\) \(\chi_{972}(665,\cdot)\) \(\chi_{972}(737,\cdot)\) \(\chi_{972}(773,\cdot)\) \(\chi_{972}(845,\cdot)\) \(\chi_{972}(881,\cdot)\) \(\chi_{972}(953,\cdot)\)
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
\((487,245)\) → \((1,e\left(\frac{11}{54}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 972 }(17, a) \) | \(-1\) | \(1\) | \(e\left(\frac{37}{54}\right)\) | \(e\left(\frac{7}{27}\right)\) | \(e\left(\frac{35}{54}\right)\) | \(e\left(\frac{17}{27}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{13}{54}\right)\) | \(e\left(\frac{10}{27}\right)\) | \(e\left(\frac{29}{54}\right)\) | \(e\left(\frac{2}{27}\right)\) |
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)