from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5775, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([0,1,10,0]))
pari: [g,chi] = znchar(Mod(727,5775))
Basic properties
Modulus: | \(5775\) | |
Conductor: | \(175\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(20\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{175}(27,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5775.gd
\(\chi_{5775}(727,\cdot)\) \(\chi_{5775}(958,\cdot)\) \(\chi_{5775}(2113,\cdot)\) \(\chi_{5775}(3037,\cdot)\) \(\chi_{5775}(4192,\cdot)\) \(\chi_{5775}(4423,\cdot)\) \(\chi_{5775}(5347,\cdot)\) \(\chi_{5775}(5578,\cdot)\)
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: | \(\Q(\zeta_{20})\) |
Fixed field: | 20.20.822111175511963665485382080078125.1 |
Values on generators
\((3851,4852,4126,3676)\) → \((1,e\left(\frac{1}{20}\right),-1,1)\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(13\) | \(16\) | \(17\) | \(19\) | \(23\) | \(26\) | \(29\) |
\( \chi_{ 5775 }(727, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(-1\) | \(e\left(\frac{1}{10}\right)\) |
sage: chi.jacobi_sum(n)