from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3025, base_ring=CyclotomicField(22))
M = H._module
chi = DirichletCharacter(H, M([11,1]))
pari: [g,chi] = znchar(Mod(274,3025))
Basic properties
Modulus: | \(3025\) | |
Conductor: | \(605\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(22\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{605}(274,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3025.br
\(\chi_{3025}(274,\cdot)\) \(\chi_{3025}(549,\cdot)\) \(\chi_{3025}(824,\cdot)\) \(\chi_{3025}(1099,\cdot)\) \(\chi_{3025}(1374,\cdot)\) \(\chi_{3025}(1649,\cdot)\) \(\chi_{3025}(1924,\cdot)\) \(\chi_{3025}(2199,\cdot)\) \(\chi_{3025}(2474,\cdot)\) \(\chi_{3025}(2749,\cdot)\)
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: | 22.0.243091704711882553644913533390559631408320849609375.1 |
Values on generators
\((727,2301)\) → \((-1,e\left(\frac{1}{22}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(12\) | \(13\) | \(14\) |
\( \chi_{ 3025 }(274, a) \) | \(-1\) | \(1\) | \(e\left(\frac{6}{11}\right)\) | \(-1\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(1\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{4}{11}\right)\) |
sage: chi.jacobi_sum(n)