from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2960, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([18,9,27,17]))
pari: [g,chi] = znchar(Mod(203,2960))
Basic properties
Modulus: | \(2960\) | |
Conductor: | \(2960\) | 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: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2960.gr
\(\chi_{2960}(203,\cdot)\) \(\chi_{2960}(227,\cdot)\) \(\chi_{2960}(283,\cdot)\) \(\chi_{2960}(387,\cdot)\) \(\chi_{2960}(627,\cdot)\) \(\chi_{2960}(947,\cdot)\) \(\chi_{2960}(1643,\cdot)\) \(\chi_{2960}(1963,\cdot)\) \(\chi_{2960}(2203,\cdot)\) \(\chi_{2960}(2307,\cdot)\) \(\chi_{2960}(2363,\cdot)\) \(\chi_{2960}(2387,\cdot)\)
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
\((2591,741,1777,2481)\) → \((-1,i,-i,e\left(\frac{17}{36}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
\( \chi_{ 2960 }(203, a) \) | \(-1\) | \(1\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{5}{9}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{3}\right)\) |
sage: chi.jacobi_sum(n)