from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5225, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([45,12,10]))
pari: [g,chi] = znchar(Mod(2193,5225))
Basic properties
Modulus: | \(5225\) | |
Conductor: | \(1045\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{1045}(103,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5225.gv
\(\chi_{5225}(768,\cdot)\) \(\chi_{5225}(1357,\cdot)\) \(\chi_{5225}(1532,\cdot)\) \(\chi_{5225}(2007,\cdot)\) \(\chi_{5225}(2193,\cdot)\) \(\chi_{5225}(2368,\cdot)\) \(\chi_{5225}(2843,\cdot)\) \(\chi_{5225}(2957,\cdot)\) \(\chi_{5225}(3732,\cdot)\) \(\chi_{5225}(3793,\cdot)\) \(\chi_{5225}(4207,\cdot)\) \(\chi_{5225}(4382,\cdot)\) \(\chi_{5225}(4568,\cdot)\) \(\chi_{5225}(5043,\cdot)\) \(\chi_{5225}(5157,\cdot)\) \(\chi_{5225}(5218,\cdot)\)
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: | \(\Q(\zeta_{60})\) |
Fixed field: | Number field defined by a degree 60 polynomial |
Values on generators
\((2927,2851,4676)\) → \((-i,e\left(\frac{1}{5}\right),e\left(\frac{1}{6}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(12\) | \(13\) | \(14\) |
\( \chi_{ 5225 }(2193, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(i\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{4}{15}\right)\) |
sage: chi.jacobi_sum(n)