from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4033, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([9,23]))
pari: [g,chi] = znchar(Mod(2103,4033))
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.ee
\(\chi_{4033}(413,\cdot)\) \(\chi_{4033}(438,\cdot)\) \(\chi_{4033}(709,\cdot)\) \(\chi_{4033}(746,\cdot)\) \(\chi_{4033}(1289,\cdot)\) \(\chi_{4033}(1930,\cdot)\) \(\chi_{4033}(2103,\cdot)\) \(\chi_{4033}(2744,\cdot)\) \(\chi_{4033}(3287,\cdot)\) \(\chi_{4033}(3324,\cdot)\) \(\chi_{4033}(3595,\cdot)\) \(\chi_{4033}(3620,\cdot)\)
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
\((1963,2295)\) → \((i,e\left(\frac{23}{36}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
\( \chi_{ 4033 }(2103, a) \) | \(1\) | \(1\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{11}{36}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(1\) | \(e\left(\frac{4}{9}\right)\) | \(e\left(\frac{35}{36}\right)\) | \(e\left(\frac{19}{36}\right)\) |
sage: chi.jacobi_sum(n)