from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4033, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([48,41]))
pari: [g,chi] = znchar(Mod(2745,4033))
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.hv
\(\chi_{4033}(197,\cdot)\) \(\chi_{4033}(305,\cdot)\) \(\chi_{4033}(530,\cdot)\) \(\chi_{4033}(737,\cdot)\) \(\chi_{4033}(884,\cdot)\) \(\chi_{4033}(1230,\cdot)\) \(\chi_{4033}(1344,\cdot)\) \(\chi_{4033}(1773,\cdot)\) \(\chi_{4033}(1846,\cdot)\) \(\chi_{4033}(1957,\cdot)\) \(\chi_{4033}(2458,\cdot)\) \(\chi_{4033}(2745,\cdot)\) \(\chi_{4033}(3004,\cdot)\) \(\chi_{4033}(3043,\cdot)\) \(\chi_{4033}(3189,\cdot)\) \(\chi_{4033}(3235,\cdot)\) \(\chi_{4033}(3790,\cdot)\) \(\chi_{4033}(4030,\cdot)\)
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
\((1963,2295)\) → \((e\left(\frac{8}{9}\right),e\left(\frac{41}{54}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
\( \chi_{ 4033 }(2745, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{16}{27}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{4}{27}\right)\) | \(e\left(\frac{41}{54}\right)\) | \(e\left(\frac{22}{27}\right)\) | \(-1\) | \(e\left(\frac{5}{27}\right)\) | \(e\left(\frac{17}{54}\right)\) | \(e\left(\frac{37}{54}\right)\) |
sage: chi.jacobi_sum(n)