from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(153, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([8,27]))
pari: [g,chi] = znchar(Mod(65,153))
Basic properties
| Modulus: | \(153\) | |
| Conductor: | \(153\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(48\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 153.s
\(\chi_{153}(5,\cdot)\) \(\chi_{153}(11,\cdot)\) \(\chi_{153}(14,\cdot)\) \(\chi_{153}(20,\cdot)\) \(\chi_{153}(23,\cdot)\) \(\chi_{153}(29,\cdot)\) \(\chi_{153}(41,\cdot)\) \(\chi_{153}(56,\cdot)\) \(\chi_{153}(65,\cdot)\) \(\chi_{153}(74,\cdot)\) \(\chi_{153}(92,\cdot)\) \(\chi_{153}(95,\cdot)\) \(\chi_{153}(113,\cdot)\) \(\chi_{153}(122,\cdot)\) \(\chi_{153}(131,\cdot)\) \(\chi_{153}(146,\cdot)\)
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
\((137,37)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{9}{16}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 153 }(65, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{31}{48}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{5}{48}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{1}{6}\right)\) |
sage: chi.jacobi_sum(n)
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
Jacobi sum
sage: chi.jacobi_sum(n)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)