from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4400, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([0,5,9,6]))
pari: [g,chi] = znchar(Mod(437,4400))
Basic properties
Modulus: | \(4400\) | |
Conductor: | \(4400\) | 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 4400.jz
\(\chi_{4400}(437,\cdot)\) \(\chi_{4400}(1877,\cdot)\) \(\chi_{4400}(2813,\cdot)\) \(\chi_{4400}(2917,\cdot)\) \(\chi_{4400}(3053,\cdot)\) \(\chi_{4400}(3373,\cdot)\) \(\chi_{4400}(3533,\cdot)\) \(\chi_{4400}(4197,\cdot)\)
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: | 20.20.582999585691243011742105600000000000000000000000000000000000.3 |
Values on generators
\((2751,3301,177,1201)\) → \((1,i,e\left(\frac{9}{20}\right),e\left(\frac{3}{10}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
\( \chi_{ 4400 }(437, a) \) | \(1\) | \(1\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(-i\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{9}{10}\right)\) | \(-i\) |
sage: chi.jacobi_sum(n)