from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(644, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([0,55,21]))
pari: [g,chi] = znchar(Mod(201,644))
Basic properties
| Modulus: | \(644\) | |
| Conductor: | \(161\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{161}(40,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 644.bc
\(\chi_{644}(5,\cdot)\) \(\chi_{644}(17,\cdot)\) \(\chi_{644}(33,\cdot)\) \(\chi_{644}(61,\cdot)\) \(\chi_{644}(89,\cdot)\) \(\chi_{644}(129,\cdot)\) \(\chi_{644}(145,\cdot)\) \(\chi_{644}(157,\cdot)\) \(\chi_{644}(201,\cdot)\) \(\chi_{644}(241,\cdot)\) \(\chi_{644}(297,\cdot)\) \(\chi_{644}(313,\cdot)\) \(\chi_{644}(341,\cdot)\) \(\chi_{644}(425,\cdot)\) \(\chi_{644}(465,\cdot)\) \(\chi_{644}(481,\cdot)\) \(\chi_{644}(493,\cdot)\) \(\chi_{644}(521,\cdot)\) \(\chi_{644}(549,\cdot)\) \(\chi_{644}(605,\cdot)\)
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: | Number field defined by a degree 66 polynomial |
Values on generators
\((323,185,281)\) → \((1,e\left(\frac{5}{6}\right),e\left(\frac{7}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(25\) | \(27\) |
| \( \chi_{ 644 }(201, a) \) | \(1\) | \(1\) | \(e\left(\frac{61}{66}\right)\) | \(e\left(\frac{16}{33}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{13}{66}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{17}{22}\right)\) |
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)