from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2624, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([0,0,13]))
pari: [g,chi] = znchar(Mod(65,2624))
Basic properties
Modulus: | \(2624\) | |
Conductor: | \(41\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(40\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{41}(24,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2624.dq
\(\chi_{2624}(65,\cdot)\) \(\chi_{2624}(129,\cdot)\) \(\chi_{2624}(193,\cdot)\) \(\chi_{2624}(257,\cdot)\) \(\chi_{2624}(321,\cdot)\) \(\chi_{2624}(641,\cdot)\) \(\chi_{2624}(833,\cdot)\) \(\chi_{2624}(1217,\cdot)\) \(\chi_{2624}(1409,\cdot)\) \(\chi_{2624}(1729,\cdot)\) \(\chi_{2624}(1793,\cdot)\) \(\chi_{2624}(1857,\cdot)\) \(\chi_{2624}(1921,\cdot)\) \(\chi_{2624}(1985,\cdot)\) \(\chi_{2624}(2113,\cdot)\) \(\chi_{2624}(2561,\cdot)\)
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: | \(\Q(\zeta_{40})\) |
Fixed field: | Number field defined by a degree 40 polynomial |
Values on generators
\((575,1477,129)\) → \((1,1,e\left(\frac{13}{40}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
\( \chi_{ 2624 }(65, a) \) | \(-1\) | \(1\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{27}{40}\right)\) | \(-i\) | \(e\left(\frac{39}{40}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{1}{40}\right)\) | \(e\left(\frac{29}{40}\right)\) | \(e\left(\frac{37}{40}\right)\) | \(e\left(\frac{11}{20}\right)\) |
sage: chi.jacobi_sum(n)