from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4005, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([44,0,48]))
pari: [g,chi] = znchar(Mod(16,4005))
Basic properties
Modulus: | \(4005\) | |
Conductor: | \(801\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(33\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{801}(16,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4005.cm
\(\chi_{4005}(16,\cdot)\) \(\chi_{4005}(121,\cdot)\) \(\chi_{4005}(256,\cdot)\) \(\chi_{4005}(331,\cdot)\) \(\chi_{4005}(601,\cdot)\) \(\chi_{4005}(751,\cdot)\) \(\chi_{4005}(1291,\cdot)\) \(\chi_{4005}(1426,\cdot)\) \(\chi_{4005}(1456,\cdot)\) \(\chi_{4005}(1591,\cdot)\) \(\chi_{4005}(1606,\cdot)\) \(\chi_{4005}(1966,\cdot)\) \(\chi_{4005}(2086,\cdot)\) \(\chi_{4005}(2626,\cdot)\) \(\chi_{4005}(2686,\cdot)\) \(\chi_{4005}(2761,\cdot)\) \(\chi_{4005}(2941,\cdot)\) \(\chi_{4005}(3001,\cdot)\) \(\chi_{4005}(3271,\cdot)\) \(\chi_{4005}(3301,\cdot)\)
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: | \(\Q(\zeta_{33})\) |
Fixed field: | Number field defined by a degree 33 polynomial |
Values on generators
\((3116,802,181)\) → \((e\left(\frac{2}{3}\right),1,e\left(\frac{8}{11}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
\( \chi_{ 4005 }(16, a) \) | \(1\) | \(1\) | \(e\left(\frac{10}{33}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{19}{33}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{25}{33}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{5}{11}\right)\) |
sage: chi.jacobi_sum(n)