from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3800, base_ring=CyclotomicField(20))
M = H._module
chi = DirichletCharacter(H, M([0,10,9,10]))
pari: [g,chi] = znchar(Mod(37,3800))
Basic properties
Modulus: | \(3800\) | |
Conductor: | \(3800\) | 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 3800.di
\(\chi_{3800}(37,\cdot)\) \(\chi_{3800}(797,\cdot)\) \(\chi_{3800}(1253,\cdot)\) \(\chi_{3800}(2013,\cdot)\) \(\chi_{3800}(2317,\cdot)\) \(\chi_{3800}(2773,\cdot)\) \(\chi_{3800}(3077,\cdot)\) \(\chi_{3800}(3533,\cdot)\)
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: | 20.20.19159582055628125000000000000000000000000000000.1 |
Values on generators
\((951,1901,1977,401)\) → \((1,-1,e\left(\frac{9}{20}\right),-1)\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(21\) | \(23\) | \(27\) | \(29\) |
\( \chi_{ 3800 }(37, a) \) | \(1\) | \(1\) | \(e\left(\frac{3}{20}\right)\) | \(i\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{9}{10}\right)\) |
sage: chi.jacobi_sum(n)