from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2960, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([18,9,0,20]))
pari: [g,chi] = znchar(Mod(1291,2960))
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 \(\chi_{592}(107,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2960.gt
\(\chi_{2960}(451,\cdot)\) \(\chi_{2960}(571,\cdot)\) \(\chi_{2960}(811,\cdot)\) \(\chi_{2960}(971,\cdot)\) \(\chi_{2960}(1011,\cdot)\) \(\chi_{2960}(1291,\cdot)\) \(\chi_{2960}(1931,\cdot)\) \(\chi_{2960}(2051,\cdot)\) \(\chi_{2960}(2291,\cdot)\) \(\chi_{2960}(2451,\cdot)\) \(\chi_{2960}(2491,\cdot)\) \(\chi_{2960}(2771,\cdot)\)
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
\((2591,741,1777,2481)\) → \((-1,i,1,e\left(\frac{5}{9}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
| \( \chi_{ 2960 }(1291, a) \) | \(-1\) | \(1\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{7}{18}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{25}{36}\right)\) | \(e\left(\frac{17}{36}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{12}\right)\) |
sage: chi.jacobi_sum(n)