from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9800, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,30,57,40]))
pari: [g,chi] = znchar(Mod(5163,9800))
Basic properties
Modulus: | \(9800\) | |
Conductor: | \(1400\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{1400}(963,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 9800.fd
\(\chi_{9800}(67,\cdot)\) \(\chi_{9800}(667,\cdot)\) \(\chi_{9800}(2027,\cdot)\) \(\chi_{9800}(2627,\cdot)\) \(\chi_{9800}(3203,\cdot)\) \(\chi_{9800}(3803,\cdot)\) \(\chi_{9800}(3987,\cdot)\) \(\chi_{9800}(4587,\cdot)\) \(\chi_{9800}(5163,\cdot)\) \(\chi_{9800}(5763,\cdot)\) \(\chi_{9800}(5947,\cdot)\) \(\chi_{9800}(6547,\cdot)\) \(\chi_{9800}(7123,\cdot)\) \(\chi_{9800}(7723,\cdot)\) \(\chi_{9800}(9083,\cdot)\) \(\chi_{9800}(9683,\cdot)\)
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
\((7351,4901,1177,5001)\) → \((-1,-1,e\left(\frac{19}{20}\right),e\left(\frac{2}{3}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) |
\( \chi_{ 9800 }(5163, a) \) | \(1\) | \(1\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{13}{30}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{23}{30}\right)\) |
sage: chi.jacobi_sum(n)