from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2800, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([15,0,21,25]))
pari: [g,chi] = znchar(Mod(159,2800))
Basic properties
Modulus: | \(2800\) | |
Conductor: | \(700\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(30\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{700}(159,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2800.es
\(\chi_{2800}(159,\cdot)\) \(\chi_{2800}(479,\cdot)\) \(\chi_{2800}(719,\cdot)\) \(\chi_{2800}(1039,\cdot)\) \(\chi_{2800}(1279,\cdot)\) \(\chi_{2800}(1839,\cdot)\) \(\chi_{2800}(2159,\cdot)\) \(\chi_{2800}(2719,\cdot)\)
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: | \(\Q(\zeta_{15})\) |
Fixed field: | 30.30.639471349555952501681804656982421875000000000000000000000000000000.1 |
Values on generators
\((351,2101,2577,801)\) → \((-1,1,e\left(\frac{7}{10}\right),e\left(\frac{5}{6}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) |
\( \chi_{ 2800 }(159, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{14}{15}\right)\) |
sage: chi.jacobi_sum(n)