from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2576, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([22,11,22,18]))
pari: [g,chi] = znchar(Mod(1483,2576))
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.cn
\(\chi_{2576}(83,\cdot)\) \(\chi_{2576}(195,\cdot)\) \(\chi_{2576}(251,\cdot)\) \(\chi_{2576}(419,\cdot)\) \(\chi_{2576}(475,\cdot)\) \(\chi_{2576}(755,\cdot)\) \(\chi_{2576}(1091,\cdot)\) \(\chi_{2576}(1147,\cdot)\) \(\chi_{2576}(1203,\cdot)\) \(\chi_{2576}(1259,\cdot)\) \(\chi_{2576}(1371,\cdot)\) \(\chi_{2576}(1483,\cdot)\) \(\chi_{2576}(1539,\cdot)\) \(\chi_{2576}(1707,\cdot)\) \(\chi_{2576}(1763,\cdot)\) \(\chi_{2576}(2043,\cdot)\) \(\chi_{2576}(2379,\cdot)\) \(\chi_{2576}(2435,\cdot)\) \(\chi_{2576}(2491,\cdot)\) \(\chi_{2576}(2547,\cdot)\)
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
\((2255,645,1473,1569)\) → \((-1,i,-1,e\left(\frac{9}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(25\) | \(27\) |
| \( \chi_{ 2576 }(1483, a) \) | \(-1\) | \(1\) | \(e\left(\frac{13}{44}\right)\) | \(e\left(\frac{7}{44}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{19}{44}\right)\) | \(e\left(\frac{43}{44}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{39}{44}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{39}{44}\right)\) |
sage: chi.jacobi_sum(n)