from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5775, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,3,20,24]))
pari: [g,chi] = znchar(Mod(5252,5775))
Basic properties
Modulus: | \(5775\) | |
Conductor: | \(5775\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5775.lj
\(\chi_{5775}(158,\cdot)\) \(\chi_{5775}(872,\cdot)\) \(\chi_{5775}(1103,\cdot)\) \(\chi_{5775}(1523,\cdot)\) \(\chi_{5775}(1808,\cdot)\) \(\chi_{5775}(2363,\cdot)\) \(\chi_{5775}(2522,\cdot)\) \(\chi_{5775}(2753,\cdot)\) \(\chi_{5775}(3173,\cdot)\) \(\chi_{5775}(3287,\cdot)\) \(\chi_{5775}(3392,\cdot)\) \(\chi_{5775}(3602,\cdot)\) \(\chi_{5775}(4013,\cdot)\) \(\chi_{5775}(4937,\cdot)\) \(\chi_{5775}(5042,\cdot)\) \(\chi_{5775}(5252,\cdot)\)
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
\((3851,4852,4126,3676)\) → \((-1,e\left(\frac{1}{20}\right),e\left(\frac{1}{3}\right),e\left(\frac{2}{5}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(13\) | \(16\) | \(17\) | \(19\) | \(23\) | \(26\) | \(29\) |
\( \chi_{ 5775 }(5252, a) \) | \(1\) | \(1\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{43}{60}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{2}{5}\right)\) |
sage: chi.jacobi_sum(n)