from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6003, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([22,14,33]))
pari: [g,chi] = znchar(Mod(17,6003))
Basic properties
Modulus: | \(6003\) | |
Conductor: | \(2001\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{2001}(17,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6003.ce
\(\chi_{6003}(17,\cdot)\) \(\chi_{6003}(539,\cdot)\) \(\chi_{6003}(1259,\cdot)\) \(\chi_{6003}(1322,\cdot)\) \(\chi_{6003}(1583,\cdot)\) \(\chi_{6003}(1781,\cdot)\) \(\chi_{6003}(2366,\cdot)\) \(\chi_{6003}(2564,\cdot)\) \(\chi_{6003}(2627,\cdot)\) \(\chi_{6003}(2825,\cdot)\) \(\chi_{6003}(3149,\cdot)\) \(\chi_{6003}(3608,\cdot)\) \(\chi_{6003}(3671,\cdot)\) \(\chi_{6003}(3869,\cdot)\) \(\chi_{6003}(4193,\cdot)\) \(\chi_{6003}(4391,\cdot)\) \(\chi_{6003}(4454,\cdot)\) \(\chi_{6003}(4913,\cdot)\) \(\chi_{6003}(5435,\cdot)\) \(\chi_{6003}(5696,\cdot)\)
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
\((668,3133,4555)\) → \((-1,e\left(\frac{7}{22}\right),-i)\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
\( \chi_{ 6003 }(17, a) \) | \(-1\) | \(1\) | \(e\left(\frac{39}{44}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{29}{44}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{5}{44}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{41}{44}\right)\) | \(e\left(\frac{6}{11}\right)\) |
sage: chi.jacobi_sum(n)