from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2960, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([0,27,0,11]))
pari: [g,chi] = znchar(Mod(2381,2960))
Basic properties
Modulus: | \(2960\) | |
Conductor: | \(592\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{592}(13,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2960.he
\(\chi_{2960}(261,\cdot)\) \(\chi_{2960}(301,\cdot)\) \(\chi_{2960}(461,\cdot)\) \(\chi_{2960}(501,\cdot)\) \(\chi_{2960}(661,\cdot)\) \(\chi_{2960}(701,\cdot)\) \(\chi_{2960}(1021,\cdot)\) \(\chi_{2960}(1461,\cdot)\) \(\chi_{2960}(1541,\cdot)\) \(\chi_{2960}(2381,\cdot)\) \(\chi_{2960}(2461,\cdot)\) \(\chi_{2960}(2901,\cdot)\)
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: | 36.0.4886860176107258124616704873602845327686728999915307588219200292503475176863258640384.1 |
Values on generators
\((2591,741,1777,2481)\) → \((1,-i,1,e\left(\frac{11}{36}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
\( \chi_{ 2960 }(2381, a) \) | \(-1\) | \(1\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{5}{18}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{5}{36}\right)\) | \(e\left(\frac{17}{18}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{7}{12}\right)\) |
sage: chi.jacobi_sum(n)