from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4033, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([23,27]))
pari: [g,chi] = znchar(Mod(1929,4033))
Basic properties
Modulus: | \(4033\) | |
Conductor: | \(4033\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(36\) | 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 4033.gb
\(\chi_{4033}(294,\cdot)\) \(\chi_{4033}(1014,\cdot)\) \(\chi_{4033}(1166,\cdot)\) \(\chi_{4033}(1275,\cdot)\) \(\chi_{4033}(1493,\cdot)\) \(\chi_{4033}(1929,\cdot)\) \(\chi_{4033}(2104,\cdot)\) \(\chi_{4033}(2540,\cdot)\) \(\chi_{4033}(2758,\cdot)\) \(\chi_{4033}(2867,\cdot)\) \(\chi_{4033}(3019,\cdot)\) \(\chi_{4033}(3739,\cdot)\)
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: | \(\Q(\zeta_{36})\) |
Fixed field: | Number field defined by a degree 36 polynomial |
Values on generators
\((1963,2295)\) → \((e\left(\frac{23}{36}\right),-i)\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
\( \chi_{ 4033 }(1929, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(1\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{5}{12}\right)\) |
sage: chi.jacobi_sum(n)