from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8002, base_ring=CyclotomicField(50))
M = H._module
chi = DirichletCharacter(H, M([14]))
pari: [g,chi] = znchar(Mod(391,8002))
Basic properties
Modulus: | \(8002\) | |
Conductor: | \(4001\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(25\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{4001}(391,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8002.i
\(\chi_{8002}(201,\cdot)\) \(\chi_{8002}(391,\cdot)\) \(\chi_{8002}(395,\cdot)\) \(\chi_{8002}(617,\cdot)\) \(\chi_{8002}(843,\cdot)\) \(\chi_{8002}(1187,\cdot)\) \(\chi_{8002}(1257,\cdot)\) \(\chi_{8002}(1531,\cdot)\) \(\chi_{8002}(2407,\cdot)\) \(\chi_{8002}(3365,\cdot)\) \(\chi_{8002}(3655,\cdot)\) \(\chi_{8002}(3687,\cdot)\) \(\chi_{8002}(3987,\cdot)\) \(\chi_{8002}(4197,\cdot)\) \(\chi_{8002}(4595,\cdot)\) \(\chi_{8002}(4749,\cdot)\) \(\chi_{8002}(6473,\cdot)\) \(\chi_{8002}(6529,\cdot)\) \(\chi_{8002}(6573,\cdot)\) \(\chi_{8002}(7377,\cdot)\)
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: | \(\Q(\zeta_{25})\) |
Fixed field: | Number field defined by a degree 25 polynomial |
Values on generators
\(3\) → \(e\left(\frac{7}{25}\right)\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
\( \chi_{ 8002 }(391, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{25}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{21}{25}\right)\) | \(e\left(\frac{14}{25}\right)\) | \(1\) | \(e\left(\frac{23}{25}\right)\) | \(e\left(\frac{17}{25}\right)\) | \(e\left(\frac{17}{25}\right)\) | \(e\left(\frac{18}{25}\right)\) | \(e\left(\frac{3}{25}\right)\) |
sage: chi.jacobi_sum(n)