from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1012, base_ring=CyclotomicField(10))
M = H._module
chi = DirichletCharacter(H, M([5,6,5]))
pari: [g,chi] = znchar(Mod(735,1012))
Basic properties
Modulus: | \(1012\) | |
Conductor: | \(1012\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(10\) | 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 1012.o
\(\chi_{1012}(91,\cdot)\) \(\chi_{1012}(367,\cdot)\) \(\chi_{1012}(643,\cdot)\) \(\chi_{1012}(735,\cdot)\)
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: | \(\Q(\zeta_{5})\) |
Fixed field: | 10.10.1412799778009275392.1 |
Values on generators
\((507,277,925)\) → \((-1,e\left(\frac{3}{5}\right),-1)\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) | \(25\) |
\( \chi_{ 1012 }(735, a) \) | \(1\) | \(1\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(-1\) | \(e\left(\frac{4}{5}\right)\) |
sage: chi.jacobi_sum(n)