from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5520, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([0,0,0,0,2]))
pari: [g,chi] = znchar(Mod(1681,5520))
Basic properties
Modulus: | \(5520\) | |
Conductor: | \(23\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(11\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{23}(2,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5520.dc
\(\chi_{5520}(721,\cdot)\) \(\chi_{5520}(961,\cdot)\) \(\chi_{5520}(1681,\cdot)\) \(\chi_{5520}(1921,\cdot)\) \(\chi_{5520}(2401,\cdot)\) \(\chi_{5520}(2881,\cdot)\) \(\chi_{5520}(3121,\cdot)\) \(\chi_{5520}(3361,\cdot)\) \(\chi_{5520}(3601,\cdot)\) \(\chi_{5520}(5041,\cdot)\)
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: | \(\Q(\zeta_{23})^+\) |
Values on generators
\((4831,1381,1841,4417,1201)\) → \((1,1,1,1,e\left(\frac{1}{11}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
\( \chi_{ 5520 }(1681, a) \) | \(1\) | \(1\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{5}{11}\right)\) |
sage: chi.jacobi_sum(n)