from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4212, base_ring=CyclotomicField(54))
M = H._module
chi = DirichletCharacter(H, M([0,50,27]))
pari: [g,chi] = znchar(Mod(805,4212))
Basic properties
| Modulus: | \(4212\) | |
| Conductor: | \(1053\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(54\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1053}(805,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4212.dx
\(\chi_{4212}(25,\cdot)\) \(\chi_{4212}(337,\cdot)\) \(\chi_{4212}(493,\cdot)\) \(\chi_{4212}(805,\cdot)\) \(\chi_{4212}(961,\cdot)\) \(\chi_{4212}(1273,\cdot)\) \(\chi_{4212}(1429,\cdot)\) \(\chi_{4212}(1741,\cdot)\) \(\chi_{4212}(1897,\cdot)\) \(\chi_{4212}(2209,\cdot)\) \(\chi_{4212}(2365,\cdot)\) \(\chi_{4212}(2677,\cdot)\) \(\chi_{4212}(2833,\cdot)\) \(\chi_{4212}(3145,\cdot)\) \(\chi_{4212}(3301,\cdot)\) \(\chi_{4212}(3613,\cdot)\) \(\chi_{4212}(3769,\cdot)\) \(\chi_{4212}(4081,\cdot)\)
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
\((2107,3485,3889)\) → \((1,e\left(\frac{25}{27}\right),-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 4212 }(805, a) \) | \(1\) | \(1\) | \(e\left(\frac{43}{54}\right)\) | \(e\left(\frac{17}{54}\right)\) | \(e\left(\frac{29}{54}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{5}{27}\right)\) | \(e\left(\frac{16}{27}\right)\) | \(e\left(\frac{7}{27}\right)\) | \(e\left(\frac{1}{54}\right)\) | \(e\left(\frac{1}{9}\right)\) |
sage: chi.jacobi_sum(n)