from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7600, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([10,5,2,10]))
pari: [g,chi] = znchar(Mod(379,7600))
Basic properties
Modulus: | \(7600\) | |
Conductor: | \(7600\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(20\) | 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 7600.ep
\(\chi_{7600}(379,\cdot)\) \(\chi_{7600}(1139,\cdot)\) \(\chi_{7600}(2659,\cdot)\) \(\chi_{7600}(3419,\cdot)\) \(\chi_{7600}(4179,\cdot)\) \(\chi_{7600}(4939,\cdot)\) \(\chi_{7600}(6459,\cdot)\) \(\chi_{7600}(7219,\cdot)\)
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: | Number field defined by a degree 20 polynomial |
Values on generators
\((4751,5701,5777,401)\) → \((-1,i,e\left(\frac{1}{10}\right),-1)\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(21\) | \(23\) | \(27\) | \(29\) |
\( \chi_{ 7600 }(379, a) \) | \(1\) | \(1\) | \(e\left(\frac{9}{20}\right)\) | \(-1\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{9}{20}\right)\) |
sage: chi.jacobi_sum(n)