from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1408, base_ring=CyclotomicField(32))
M = H._module
chi = DirichletCharacter(H, M([0,7,16]))
pari: [g,chi] = znchar(Mod(813,1408))
Basic properties
| Modulus: | \(1408\) | |
| Conductor: | \(1408\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(32\) | 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 1408.bj
\(\chi_{1408}(21,\cdot)\) \(\chi_{1408}(109,\cdot)\) \(\chi_{1408}(197,\cdot)\) \(\chi_{1408}(285,\cdot)\) \(\chi_{1408}(373,\cdot)\) \(\chi_{1408}(461,\cdot)\) \(\chi_{1408}(549,\cdot)\) \(\chi_{1408}(637,\cdot)\) \(\chi_{1408}(725,\cdot)\) \(\chi_{1408}(813,\cdot)\) \(\chi_{1408}(901,\cdot)\) \(\chi_{1408}(989,\cdot)\) \(\chi_{1408}(1077,\cdot)\) \(\chi_{1408}(1165,\cdot)\) \(\chi_{1408}(1253,\cdot)\) \(\chi_{1408}(1341,\cdot)\)
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: | \(\Q(\zeta_{32})\) |
| Fixed field: | 32.0.144215564533589000876246801170130951941346253827716612240069988132090544128.1 |
Values on generators
\((639,133,1025)\) → \((1,e\left(\frac{7}{32}\right),-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) | \(23\) |
| \( \chi_{ 1408 }(813, a) \) | \(-1\) | \(1\) | \(e\left(\frac{21}{32}\right)\) | \(e\left(\frac{7}{32}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{25}{32}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{17}{32}\right)\) | \(e\left(\frac{11}{32}\right)\) | \(e\left(\frac{1}{16}\right)\) |
sage: chi.jacobi_sum(n)