from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2601, base_ring=CyclotomicField(24))
M = H._module
chi = DirichletCharacter(H, M([8,15]))
pari: [g,chi] = znchar(Mod(688,2601))
Basic properties
| Modulus: | \(2601\) | |
| Conductor: | \(153\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(24\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{153}(76,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 2601.s
\(\chi_{2601}(688,\cdot)\) \(\chi_{2601}(733,\cdot)\) \(\chi_{2601}(1555,\cdot)\) \(\chi_{2601}(1579,\cdot)\) \(\chi_{2601}(1600,\cdot)\) \(\chi_{2601}(1624,\cdot)\) \(\chi_{2601}(2446,\cdot)\) \(\chi_{2601}(2491,\cdot)\)
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: | \(\Q(\zeta_{24})\) |
| Fixed field: | 24.24.128028748427622359924863503266793533356497.1 |
Values on generators
\((290,2026)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{5}{8}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 2601 }(688, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(i\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{17}{24}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{1}{3}\right)\) |
sage: chi.jacobi_sum(n)