from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4020, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([11,11,11,13]))
pari: [g,chi] = znchar(Mod(539,4020))
Basic properties
Modulus: | \(4020\) | |
Conductor: | \(4020\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(22\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4020.ci
\(\chi_{4020}(119,\cdot)\) \(\chi_{4020}(179,\cdot)\) \(\chi_{4020}(539,\cdot)\) \(\chi_{4020}(779,\cdot)\) \(\chi_{4020}(1259,\cdot)\) \(\chi_{4020}(1919,\cdot)\) \(\chi_{4020}(2219,\cdot)\) \(\chi_{4020}(2819,\cdot)\) \(\chi_{4020}(2939,\cdot)\) \(\chi_{4020}(3779,\cdot)\)
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
\((2011,2681,3217,1141)\) → \((-1,-1,-1,e\left(\frac{13}{22}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
\( \chi_{ 4020 }(539, a) \) | \(-1\) | \(1\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(-1\) | \(e\left(\frac{3}{11}\right)\) | \(-1\) | \(e\left(\frac{9}{11}\right)\) |
sage: chi.jacobi_sum(n)