from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1407, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([11,0,10]))
pari: [g,chi] = znchar(Mod(92,1407))
Basic properties
Modulus: | \(1407\) | |
Conductor: | \(201\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(22\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{201}(92,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1407.bu
\(\chi_{1407}(92,\cdot)\) \(\chi_{1407}(260,\cdot)\) \(\chi_{1407}(344,\cdot)\) \(\chi_{1407}(491,\cdot)\) \(\chi_{1407}(533,\cdot)\) \(\chi_{1407}(617,\cdot)\) \(\chi_{1407}(911,\cdot)\) \(\chi_{1407}(953,\cdot)\) \(\chi_{1407}(1163,\cdot)\) \(\chi_{1407}(1268,\cdot)\)
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: | 22.0.588613008557716517601287069955881168578347.1 |
Values on generators
\((470,1207,337)\) → \((-1,1,e\left(\frac{5}{11}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
\( \chi_{ 1407 }(92, a) \) | \(-1\) | \(1\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{6}{11}\right)\) |
sage: chi.jacobi_sum(n)