from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3800, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,30,9,20]))
pari: [g,chi] = znchar(Mod(83,3800))
Basic properties
Modulus: | \(3800\) | |
Conductor: | \(3800\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(60\) | 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.ej
\(\chi_{3800}(83,\cdot)\) \(\chi_{3800}(163,\cdot)\) \(\chi_{3800}(387,\cdot)\) \(\chi_{3800}(467,\cdot)\) \(\chi_{3800}(923,\cdot)\) \(\chi_{3800}(1147,\cdot)\) \(\chi_{3800}(1227,\cdot)\) \(\chi_{3800}(1603,\cdot)\) \(\chi_{3800}(1683,\cdot)\) \(\chi_{3800}(1987,\cdot)\) \(\chi_{3800}(2363,\cdot)\) \(\chi_{3800}(2667,\cdot)\) \(\chi_{3800}(2747,\cdot)\) \(\chi_{3800}(3123,\cdot)\) \(\chi_{3800}(3203,\cdot)\) \(\chi_{3800}(3427,\cdot)\)
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: | \(\Q(\zeta_{60})\) |
Fixed field: | Number field defined by a degree 60 polynomial |
Values on generators
\((951,1901,1977,401)\) → \((-1,-1,e\left(\frac{3}{20}\right),e\left(\frac{1}{3}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(21\) | \(23\) | \(27\) | \(29\) |
\( \chi_{ 3800 }(83, a) \) | \(1\) | \(1\) | \(e\left(\frac{23}{60}\right)\) | \(i\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{49}{60}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{7}{15}\right)\) |
sage: chi.jacobi_sum(n)