from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2601, base_ring=CyclotomicField(34))
M = H._module
chi = DirichletCharacter(H, M([0,11]))
pari: [g,chi] = znchar(Mod(1954,2601))
Basic properties
| Modulus: | \(2601\) | |
| Conductor: | \(289\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(34\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{289}(220,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2601.t
\(\chi_{2601}(118,\cdot)\) \(\chi_{2601}(271,\cdot)\) \(\chi_{2601}(424,\cdot)\) \(\chi_{2601}(730,\cdot)\) \(\chi_{2601}(883,\cdot)\) \(\chi_{2601}(1036,\cdot)\) \(\chi_{2601}(1189,\cdot)\) \(\chi_{2601}(1342,\cdot)\) \(\chi_{2601}(1495,\cdot)\) \(\chi_{2601}(1648,\cdot)\) \(\chi_{2601}(1801,\cdot)\) \(\chi_{2601}(1954,\cdot)\) \(\chi_{2601}(2107,\cdot)\) \(\chi_{2601}(2260,\cdot)\) \(\chi_{2601}(2413,\cdot)\) \(\chi_{2601}(2566,\cdot)\)
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: | \(\Q(\zeta_{17})\) |
| Fixed field: | Number field defined by a degree 34 polynomial |
Values on generators
\((290,2026)\) → \((1,e\left(\frac{11}{34}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 2601 }(1954, a) \) | \(1\) | \(1\) | \(e\left(\frac{8}{17}\right)\) | \(e\left(\frac{16}{17}\right)\) | \(e\left(\frac{3}{34}\right)\) | \(e\left(\frac{5}{34}\right)\) | \(e\left(\frac{7}{17}\right)\) | \(e\left(\frac{19}{34}\right)\) | \(e\left(\frac{15}{34}\right)\) | \(e\left(\frac{7}{17}\right)\) | \(e\left(\frac{21}{34}\right)\) | \(e\left(\frac{15}{17}\right)\) |
sage: chi.jacobi_sum(n)