from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(450, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([20,21]))
pari: [g,chi] = znchar(Mod(103,450))
Basic properties
| Modulus: | \(450\) | |
| Conductor: | \(225\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{225}(103,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 450.x
\(\chi_{450}(13,\cdot)\) \(\chi_{450}(67,\cdot)\) \(\chi_{450}(97,\cdot)\) \(\chi_{450}(103,\cdot)\) \(\chi_{450}(133,\cdot)\) \(\chi_{450}(187,\cdot)\) \(\chi_{450}(223,\cdot)\) \(\chi_{450}(247,\cdot)\) \(\chi_{450}(277,\cdot)\) \(\chi_{450}(283,\cdot)\) \(\chi_{450}(313,\cdot)\) \(\chi_{450}(337,\cdot)\) \(\chi_{450}(367,\cdot)\) \(\chi_{450}(373,\cdot)\) \(\chi_{450}(403,\cdot)\) \(\chi_{450}(427,\cdot)\)
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: | \(\Q(\zeta_{60})\) |
| Fixed field: | Number field defined by a degree 60 polynomial |
Values on generators
\((101,127)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{7}{20}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
| \( \chi_{ 450 }(103, a) \) | \(-1\) | \(1\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{31}{60}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{1}{15}\right)\) |
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)