from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2576, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([0,11,22,8]))
pari: [g,chi] = znchar(Mod(1637,2576))
Basic properties
| Modulus: | \(2576\) | |
| Conductor: | \(2576\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2576.cq
\(\chi_{2576}(13,\cdot)\) \(\chi_{2576}(349,\cdot)\) \(\chi_{2576}(629,\cdot)\) \(\chi_{2576}(685,\cdot)\) \(\chi_{2576}(853,\cdot)\) \(\chi_{2576}(909,\cdot)\) \(\chi_{2576}(1021,\cdot)\) \(\chi_{2576}(1133,\cdot)\) \(\chi_{2576}(1189,\cdot)\) \(\chi_{2576}(1245,\cdot)\) \(\chi_{2576}(1301,\cdot)\) \(\chi_{2576}(1637,\cdot)\) \(\chi_{2576}(1917,\cdot)\) \(\chi_{2576}(1973,\cdot)\) \(\chi_{2576}(2141,\cdot)\) \(\chi_{2576}(2197,\cdot)\) \(\chi_{2576}(2309,\cdot)\) \(\chi_{2576}(2421,\cdot)\) \(\chi_{2576}(2477,\cdot)\) \(\chi_{2576}(2533,\cdot)\)
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: | \(\Q(\zeta_{44})\) |
| Fixed field: | Number field defined by a degree 44 polynomial |
Values on generators
\((2255,645,1473,1569)\) → \((1,i,-1,e\left(\frac{2}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(25\) | \(27\) |
| \( \chi_{ 2576 }(1637, a) \) | \(-1\) | \(1\) | \(e\left(\frac{7}{44}\right)\) | \(e\left(\frac{41}{44}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{39}{44}\right)\) | \(e\left(\frac{35}{44}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{43}{44}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{21}{44}\right)\) |
sage: chi.jacobi_sum(n)