from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4033, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([27,47]))
pari: [g,chi] = znchar(Mod(3699,4033))
Basic properties
Modulus: | \(4033\) | |
Conductor: | \(4033\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(54\) | 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.hd
\(\chi_{4033}(36,\cdot)\) \(\chi_{4033}(628,\cdot)\) \(\chi_{4033}(850,\cdot)\) \(\chi_{4033}(1368,\cdot)\) \(\chi_{4033}(1664,\cdot)\) \(\chi_{4033}(1775,\cdot)\) \(\chi_{4033}(2145,\cdot)\) \(\chi_{4033}(2700,\cdot)\) \(\chi_{4033}(2737,\cdot)\) \(\chi_{4033}(2922,\cdot)\) \(\chi_{4033}(3181,\cdot)\) \(\chi_{4033}(3255,\cdot)\) \(\chi_{4033}(3440,\cdot)\) \(\chi_{4033}(3588,\cdot)\) \(\chi_{4033}(3625,\cdot)\) \(\chi_{4033}(3699,\cdot)\) \(\chi_{4033}(3810,\cdot)\) \(\chi_{4033}(3921,\cdot)\)
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: | \(\Q(\zeta_{27})\) |
Fixed field: | Number field defined by a degree 54 polynomial |
Values on generators
\((1963,2295)\) → \((-1,e\left(\frac{47}{54}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
\( \chi_{ 4033 }(3699, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{7}{27}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{35}{54}\right)\) | \(e\left(\frac{10}{27}\right)\) | \(e\left(\frac{22}{27}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{14}{27}\right)\) | \(e\left(\frac{41}{54}\right)\) | \(e\left(\frac{13}{54}\right)\) |
sage: chi.jacobi_sum(n)