from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7728, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([0,11,0,11,7]))
pari: [g,chi] = znchar(Mod(937,7728))
Basic properties
| Modulus: | \(7728\) | |
| Conductor: | \(1288\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(22\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1288}(293,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 7728.ej
\(\chi_{7728}(937,\cdot)\) \(\chi_{7728}(3625,\cdot)\) \(\chi_{7728}(3961,\cdot)\) \(\chi_{7728}(4297,\cdot)\) \(\chi_{7728}(4633,\cdot)\) \(\chi_{7728}(5305,\cdot)\) \(\chi_{7728}(5977,\cdot)\) \(\chi_{7728}(6313,\cdot)\) \(\chi_{7728}(7321,\cdot)\) \(\chi_{7728}(7657,\cdot)\)
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: | \(\Q(\zeta_{11})\) |
| Fixed field: | Number field defined by a degree 22 polynomial |
Values on generators
\((4831,5797,5153,6625,6721)\) → \((1,-1,1,-1,e\left(\frac{7}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 7728 }(937, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{7}{22}\right)\) |
sage: chi.jacobi_sum(n)