from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(453, base_ring=CyclotomicField(50))
M = H._module
chi = DirichletCharacter(H, M([25,6]))
pari: [g,chi] = znchar(Mod(275,453))
Basic properties
| Modulus: | \(453\) | |
| Conductor: | \(453\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(50\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 453.t
\(\chi_{453}(20,\cdot)\) \(\chi_{453}(29,\cdot)\) \(\chi_{453}(44,\cdot)\) \(\chi_{453}(50,\cdot)\) \(\chi_{453}(68,\cdot)\) \(\chi_{453}(86,\cdot)\) \(\chi_{453}(98,\cdot)\) \(\chi_{453}(110,\cdot)\) \(\chi_{453}(125,\cdot)\) \(\chi_{453}(242,\cdot)\) \(\chi_{453}(245,\cdot)\) \(\chi_{453}(275,\cdot)\) \(\chi_{453}(278,\cdot)\) \(\chi_{453}(299,\cdot)\) \(\chi_{453}(311,\cdot)\) \(\chi_{453}(374,\cdot)\) \(\chi_{453}(380,\cdot)\) \(\chi_{453}(383,\cdot)\) \(\chi_{453}(386,\cdot)\) \(\chi_{453}(425,\cdot)\)
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: | \(\Q(\zeta_{25})\) |
| Fixed field: | Number field defined by a degree 50 polynomial |
Values on generators
\((152,157)\) → \((-1,e\left(\frac{3}{25}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
| \( \chi_{ 453 }(275, a) \) | \(-1\) | \(1\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{47}{50}\right)\) | \(e\left(\frac{1}{25}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{21}{25}\right)\) | \(e\left(\frac{39}{50}\right)\) | \(e\left(\frac{23}{25}\right)\) | \(e\left(\frac{47}{50}\right)\) | \(e\left(\frac{3}{5}\right)\) |
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)