from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4020, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([11,0,0,13]))
pari: [g,chi] = znchar(Mod(271,4020))
Basic properties
| Modulus: | \(4020\) | |
| Conductor: | \(268\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(22\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{268}(3,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4020.bz
\(\chi_{4020}(271,\cdot)\) \(\chi_{4020}(511,\cdot)\) \(\chi_{4020}(991,\cdot)\) \(\chi_{4020}(1651,\cdot)\) \(\chi_{4020}(1951,\cdot)\) \(\chi_{4020}(2551,\cdot)\) \(\chi_{4020}(2671,\cdot)\) \(\chi_{4020}(3511,\cdot)\) \(\chi_{4020}(3871,\cdot)\) \(\chi_{4020}(3931,\cdot)\)
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: | 22.22.933750314983937236210361829997141120063111168.1 |
Values on generators
\((2011,2681,3217,1141)\) → \((-1,1,1,e\left(\frac{13}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 4020 }(271, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(1\) | \(e\left(\frac{3}{11}\right)\) | \(1\) | \(e\left(\frac{7}{22}\right)\) |
sage: chi.jacobi_sum(n)