from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(804, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([0,33,34]))
pari: [g,chi] = znchar(Mod(65,804))
Basic properties
| Modulus: | \(804\) | |
| Conductor: | \(201\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{201}(65,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 804.be
\(\chi_{804}(17,\cdot)\) \(\chi_{804}(65,\cdot)\) \(\chi_{804}(77,\cdot)\) \(\chi_{804}(173,\cdot)\) \(\chi_{804}(257,\cdot)\) \(\chi_{804}(317,\cdot)\) \(\chi_{804}(341,\cdot)\) \(\chi_{804}(389,\cdot)\) \(\chi_{804}(425,\cdot)\) \(\chi_{804}(437,\cdot)\) \(\chi_{804}(449,\cdot)\) \(\chi_{804}(473,\cdot)\) \(\chi_{804}(485,\cdot)\) \(\chi_{804}(557,\cdot)\) \(\chi_{804}(569,\cdot)\) \(\chi_{804}(629,\cdot)\) \(\chi_{804}(689,\cdot)\) \(\chi_{804}(725,\cdot)\) \(\chi_{804}(773,\cdot)\) \(\chi_{804}(797,\cdot)\)
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
\((403,269,337)\) → \((1,-1,e\left(\frac{17}{33}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 804 }(65, a) \) | \(-1\) | \(1\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{59}{66}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{31}{66}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{61}{66}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{7}{33}\right)\) |
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)