from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6675, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([10,7,10]))
pari: [g,chi] = znchar(Mod(3203,6675))
Basic properties
Modulus: | \(6675\) | |
Conductor: | \(6675\) | 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 6675.bs
\(\chi_{6675}(533,\cdot)\) \(\chi_{6675}(1067,\cdot)\) \(\chi_{6675}(2402,\cdot)\) \(\chi_{6675}(3203,\cdot)\) \(\chi_{6675}(3737,\cdot)\) \(\chi_{6675}(4538,\cdot)\) \(\chi_{6675}(5072,\cdot)\) \(\chi_{6675}(5873,\cdot)\)
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
\((4451,802,2851)\) → \((-1,e\left(\frac{7}{20}\right),-1)\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
\( \chi_{ 6675 }(3203, a) \) | \(1\) | \(1\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(i\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{1}{20}\right)\) | \(e\left(\frac{4}{5}\right)\) |
sage: chi.jacobi_sum(n)