from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4005, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([44,33,42]))
pari: [g,chi] = znchar(Mod(3004,4005))
Basic properties
| Modulus: | \(4005\) | |
| Conductor: | \(4005\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(66\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4005.da
\(\chi_{4005}(4,\cdot)\) \(\chi_{4005}(364,\cdot)\) \(\chi_{4005}(484,\cdot)\) \(\chi_{4005}(1024,\cdot)\) \(\chi_{4005}(1084,\cdot)\) \(\chi_{4005}(1159,\cdot)\) \(\chi_{4005}(1339,\cdot)\) \(\chi_{4005}(1399,\cdot)\) \(\chi_{4005}(1669,\cdot)\) \(\chi_{4005}(1699,\cdot)\) \(\chi_{4005}(2419,\cdot)\) \(\chi_{4005}(2524,\cdot)\) \(\chi_{4005}(2659,\cdot)\) \(\chi_{4005}(2734,\cdot)\) \(\chi_{4005}(3004,\cdot)\) \(\chi_{4005}(3154,\cdot)\) \(\chi_{4005}(3694,\cdot)\) \(\chi_{4005}(3829,\cdot)\) \(\chi_{4005}(3859,\cdot)\) \(\chi_{4005}(3994,\cdot)\)
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 66 polynomial |
Values on generators
\((3116,802,181)\) → \((e\left(\frac{2}{3}\right),-1,e\left(\frac{7}{11}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 4005 }(3004, a) \) | \(1\) | \(1\) | \(e\left(\frac{23}{66}\right)\) | \(e\left(\frac{23}{33}\right)\) | \(e\left(\frac{47}{66}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{31}{66}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{3}{11}\right)\) |
sage: chi.jacobi_sum(n)