from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7920, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([10,5,0,0,6]))
pari: [g,chi] = znchar(Mod(811,7920))
Basic properties
Modulus: | \(7920\) | |
Conductor: | \(176\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(20\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{176}(107,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 7920.jg
\(\chi_{7920}(811,\cdot)\) \(\chi_{7920}(1531,\cdot)\) \(\chi_{7920}(2251,\cdot)\) \(\chi_{7920}(3691,\cdot)\) \(\chi_{7920}(4771,\cdot)\) \(\chi_{7920}(5491,\cdot)\) \(\chi_{7920}(6211,\cdot)\) \(\chi_{7920}(7651,\cdot)\)
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: | 20.20.200317132330035063121671003054276608.1 |
Values on generators
\((991,5941,3521,6337,6481)\) → \((-1,i,1,1,e\left(\frac{3}{10}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
\( \chi_{ 7920 }(811, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(1\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(i\) |
sage: chi.jacobi_sum(n)