from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(536, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,33,10]))
pari: [g,chi] = znchar(Mod(19,536))
Basic properties
| Modulus: | \(536\) | |
| Conductor: | \(536\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(66\) | 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 536.ba
\(\chi_{536}(19,\cdot)\) \(\chi_{536}(35,\cdot)\) \(\chi_{536}(83,\cdot)\) \(\chi_{536}(123,\cdot)\) \(\chi_{536}(155,\cdot)\) \(\chi_{536}(211,\cdot)\) \(\chi_{536}(227,\cdot)\) \(\chi_{536}(291,\cdot)\) \(\chi_{536}(307,\cdot)\) \(\chi_{536}(315,\cdot)\) \(\chi_{536}(323,\cdot)\) \(\chi_{536}(339,\cdot)\) \(\chi_{536}(371,\cdot)\) \(\chi_{536}(395,\cdot)\) \(\chi_{536}(419,\cdot)\) \(\chi_{536}(435,\cdot)\) \(\chi_{536}(451,\cdot)\) \(\chi_{536}(467,\cdot)\) \(\chi_{536}(475,\cdot)\) \(\chi_{536}(523,\cdot)\)
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
\((135,269,337)\) → \((-1,-1,e\left(\frac{5}{33}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 536 }(19, a) \) | \(-1\) | \(1\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{65}{66}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{25}{66}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{59}{66}\right)\) |
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)