from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(736, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([0,33,30]))
pari: [g,chi] = znchar(Mod(617,736))
Basic properties
| Modulus: | \(736\) | |
| Conductor: | \(368\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{368}(157,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 736.z
\(\chi_{736}(57,\cdot)\) \(\chi_{736}(89,\cdot)\) \(\chi_{736}(153,\cdot)\) \(\chi_{736}(201,\cdot)\) \(\chi_{736}(217,\cdot)\) \(\chi_{736}(249,\cdot)\) \(\chi_{736}(281,\cdot)\) \(\chi_{736}(297,\cdot)\) \(\chi_{736}(313,\cdot)\) \(\chi_{736}(329,\cdot)\) \(\chi_{736}(425,\cdot)\) \(\chi_{736}(457,\cdot)\) \(\chi_{736}(521,\cdot)\) \(\chi_{736}(569,\cdot)\) \(\chi_{736}(585,\cdot)\) \(\chi_{736}(617,\cdot)\) \(\chi_{736}(649,\cdot)\) \(\chi_{736}(665,\cdot)\) \(\chi_{736}(681,\cdot)\) \(\chi_{736}(697,\cdot)\)
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: | 44.0.4141890260646712580912980965306954513336276372715662057543551492310346739946349214617837764608.1 |
Values on generators
\((415,645,97)\) → \((1,-i,e\left(\frac{15}{22}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
| \( \chi_{ 736 }(617, a) \) | \(-1\) | \(1\) | \(e\left(\frac{7}{44}\right)\) | \(e\left(\frac{19}{44}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{7}{22}\right)\) | \(e\left(\frac{39}{44}\right)\) | \(e\left(\frac{35}{44}\right)\) | \(e\left(\frac{13}{22}\right)\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{21}{44}\right)\) | \(e\left(\frac{27}{44}\right)\) |
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)