from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(356, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([22,5]))
pari: [g,chi] = znchar(Mod(131,356))
Basic properties
| Modulus: | \(356\) | |
| Conductor: | \(356\) | 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 356.n
\(\chi_{356}(47,\cdot)\) \(\chi_{356}(71,\cdot)\) \(\chi_{356}(79,\cdot)\) \(\chi_{356}(99,\cdot)\) \(\chi_{356}(107,\cdot)\) \(\chi_{356}(131,\cdot)\) \(\chi_{356}(183,\cdot)\) \(\chi_{356}(187,\cdot)\) \(\chi_{356}(195,\cdot)\) \(\chi_{356}(199,\cdot)\) \(\chi_{356}(227,\cdot)\) \(\chi_{356}(231,\cdot)\) \(\chi_{356}(247,\cdot)\) \(\chi_{356}(287,\cdot)\) \(\chi_{356}(303,\cdot)\) \(\chi_{356}(307,\cdot)\) \(\chi_{356}(335,\cdot)\) \(\chi_{356}(339,\cdot)\) \(\chi_{356}(347,\cdot)\) \(\chi_{356}(351,\cdot)\)
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: | 44.0.11724385642028745774656376755923007752612263949364698259094951647151017819768316744951538808520704.1 |
Values on generators
\((179,181)\) → \((-1,e\left(\frac{5}{44}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 356 }(131, a) \) | \(-1\) | \(1\) | \(e\left(\frac{27}{44}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{31}{44}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{27}{44}\right)\) | \(e\left(\frac{25}{44}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{21}{44}\right)\) | \(e\left(\frac{7}{22}\right)\) |
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)