from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2601, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([40,3]))
pari: [g,chi] = znchar(Mod(1958,2601))
Basic properties
| Modulus: | \(2601\) | |
| Conductor: | \(153\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(48\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{153}(122,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2601.w
\(\chi_{2601}(65,\cdot)\) \(\chi_{2601}(131,\cdot)\) \(\chi_{2601}(158,\cdot)\) \(\chi_{2601}(329,\cdot)\) \(\chi_{2601}(653,\cdot)\) \(\chi_{2601}(932,\cdot)\) \(\chi_{2601}(1091,\cdot)\) \(\chi_{2601}(1370,\cdot)\) \(\chi_{2601}(1694,\cdot)\) \(\chi_{2601}(1865,\cdot)\) \(\chi_{2601}(1892,\cdot)\) \(\chi_{2601}(1958,\cdot)\) \(\chi_{2601}(2063,\cdot)\) \(\chi_{2601}(2237,\cdot)\) \(\chi_{2601}(2387,\cdot)\) \(\chi_{2601}(2561,\cdot)\)
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: | \(\Q(\zeta_{48})\) |
| Fixed field: | Number field defined by a degree 48 polynomial |
Values on generators
\((290,2026)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{1}{16}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 2601 }(1958, a) \) | \(1\) | \(1\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{23}{48}\right)\) | \(e\left(\frac{1}{48}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{13}{48}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{35}{48}\right)\) | \(e\left(\frac{5}{6}\right)\) |
sage: chi.jacobi_sum(n)