from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(804, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([33,0,35]))
pari: [g,chi] = znchar(Mod(331,804))
Basic properties
| Modulus: | \(804\) | |
| Conductor: | \(268\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{268}(63,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 804.bf
\(\chi_{804}(7,\cdot)\) \(\chi_{804}(31,\cdot)\) \(\chi_{804}(79,\cdot)\) \(\chi_{804}(115,\cdot)\) \(\chi_{804}(175,\cdot)\) \(\chi_{804}(235,\cdot)\) \(\chi_{804}(247,\cdot)\) \(\chi_{804}(319,\cdot)\) \(\chi_{804}(331,\cdot)\) \(\chi_{804}(355,\cdot)\) \(\chi_{804}(367,\cdot)\) \(\chi_{804}(379,\cdot)\) \(\chi_{804}(415,\cdot)\) \(\chi_{804}(463,\cdot)\) \(\chi_{804}(487,\cdot)\) \(\chi_{804}(547,\cdot)\) \(\chi_{804}(631,\cdot)\) \(\chi_{804}(727,\cdot)\) \(\chi_{804}(739,\cdot)\) \(\chi_{804}(787,\cdot)\)
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
\((403,269,337)\) → \((-1,1,e\left(\frac{35}{66}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
| \( \chi_{ 804 }(331, a) \) | \(1\) | \(1\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{5}{66}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{53}{66}\right)\) | \(e\left(\frac{23}{66}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{14}{33}\right)\) |
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)