from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(804, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([0,0,13]))
pari: [g,chi] = znchar(Mod(85,804))
Basic properties
Modulus: | \(804\) | |
Conductor: | \(67\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{67}(18,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 804.bc
\(\chi_{804}(13,\cdot)\) \(\chi_{804}(61,\cdot)\) \(\chi_{804}(85,\cdot)\) \(\chi_{804}(145,\cdot)\) \(\chi_{804}(229,\cdot)\) \(\chi_{804}(325,\cdot)\) \(\chi_{804}(337,\cdot)\) \(\chi_{804}(385,\cdot)\) \(\chi_{804}(409,\cdot)\) \(\chi_{804}(433,\cdot)\) \(\chi_{804}(481,\cdot)\) \(\chi_{804}(517,\cdot)\) \(\chi_{804}(577,\cdot)\) \(\chi_{804}(637,\cdot)\) \(\chi_{804}(649,\cdot)\) \(\chi_{804}(721,\cdot)\) \(\chi_{804}(733,\cdot)\) \(\chi_{804}(757,\cdot)\) \(\chi_{804}(769,\cdot)\) \(\chi_{804}(781,\cdot)\)
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
\((403,269,337)\) → \((1,1,e\left(\frac{13}{66}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
\( \chi_{ 804 }(85, a) \) | \(-1\) | \(1\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{35}{66}\right)\) | \(e\left(\frac{41}{66}\right)\) | \(e\left(\frac{49}{66}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{17}{33}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{17}{66}\right)\) |
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)