from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1380, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([0,0,11,13]))
pari: [g,chi] = znchar(Mod(1309,1380))
Basic properties
Modulus: | \(1380\) | |
Conductor: | \(115\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(22\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{115}(44,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1380.be
\(\chi_{1380}(109,\cdot)\) \(\chi_{1380}(589,\cdot)\) \(\chi_{1380}(649,\cdot)\) \(\chi_{1380}(709,\cdot)\) \(\chi_{1380}(769,\cdot)\) \(\chi_{1380}(889,\cdot)\) \(\chi_{1380}(1009,\cdot)\) \(\chi_{1380}(1069,\cdot)\) \(\chi_{1380}(1249,\cdot)\) \(\chi_{1380}(1309,\cdot)\)
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: | 22.0.1927323443393334271838358868310546875.1 |
Values on generators
\((691,461,277,1201)\) → \((1,1,-1,e\left(\frac{13}{22}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
\( \chi_{ 1380 }(1309, a) \) | \(-1\) | \(1\) | \(e\left(\frac{8}{11}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{5}{11}\right)\) |
sage: chi.jacobi_sum(n)