from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(345, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([22,33,38]))
pari: [g,chi] = znchar(Mod(53,345))
Basic properties
| Modulus: | \(345\) | |
| Conductor: | \(345\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(44\) | 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 345.u
\(\chi_{345}(17,\cdot)\) \(\chi_{345}(38,\cdot)\) \(\chi_{345}(53,\cdot)\) \(\chi_{345}(83,\cdot)\) \(\chi_{345}(107,\cdot)\) \(\chi_{345}(113,\cdot)\) \(\chi_{345}(122,\cdot)\) \(\chi_{345}(143,\cdot)\) \(\chi_{345}(152,\cdot)\) \(\chi_{345}(158,\cdot)\) \(\chi_{345}(182,\cdot)\) \(\chi_{345}(203,\cdot)\) \(\chi_{345}(212,\cdot)\) \(\chi_{345}(218,\cdot)\) \(\chi_{345}(227,\cdot)\) \(\chi_{345}(263,\cdot)\) \(\chi_{345}(272,\cdot)\) \(\chi_{345}(287,\cdot)\) \(\chi_{345}(293,\cdot)\) \(\chi_{345}(332,\cdot)\)
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: | \(\Q(\zeta_{44})\) |
| Fixed field: | Number field defined by a degree 44 polynomial |
Values on generators
\((116,277,166)\) → \((-1,-i,e\left(\frac{19}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 345 }(53, a) \) | \(-1\) | \(1\) | \(e\left(\frac{43}{44}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{7}{44}\right)\) | \(e\left(\frac{41}{44}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{15}{44}\right)\) | \(e\left(\frac{3}{22}\right)\) | \(e\left(\frac{10}{11}\right)\) | \(e\left(\frac{13}{44}\right)\) | \(e\left(\frac{5}{11}\right)\) |
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)