from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5775, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,3,10,42]))
pari: [g,chi] = znchar(Mod(227,5775))
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.mw
\(\chi_{5775}(227,\cdot)\) \(\chi_{5775}(437,\cdot)\) \(\chi_{5775}(992,\cdot)\) \(\chi_{5775}(1172,\cdot)\) \(\chi_{5775}(1403,\cdot)\) \(\chi_{5775}(2273,\cdot)\) \(\chi_{5775}(2642,\cdot)\) \(\chi_{5775}(3533,\cdot)\) \(\chi_{5775}(3638,\cdot)\) \(\chi_{5775}(3923,\cdot)\) \(\chi_{5775}(4352,\cdot)\) \(\chi_{5775}(4562,\cdot)\) \(\chi_{5775}(5183,\cdot)\) \(\chi_{5775}(5288,\cdot)\) \(\chi_{5775}(5297,\cdot)\) \(\chi_{5775}(5528,\cdot)\)
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
\((3851,4852,4126,3676)\) → \((-1,e\left(\frac{1}{20}\right),e\left(\frac{1}{6}\right),e\left(\frac{7}{10}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(13\) | \(16\) | \(17\) | \(19\) | \(23\) | \(26\) | \(29\) |
\( \chi_{ 5775 }(227, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(-i\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(-1\) |
sage: chi.jacobi_sum(n)