from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1408, base_ring=CyclotomicField(32))
M = H._module
chi = DirichletCharacter(H, M([16,1,16]))
pari: [g,chi] = znchar(Mod(1275,1408))
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: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1408.bh
\(\chi_{1408}(43,\cdot)\) \(\chi_{1408}(131,\cdot)\) \(\chi_{1408}(219,\cdot)\) \(\chi_{1408}(307,\cdot)\) \(\chi_{1408}(395,\cdot)\) \(\chi_{1408}(483,\cdot)\) \(\chi_{1408}(571,\cdot)\) \(\chi_{1408}(659,\cdot)\) \(\chi_{1408}(747,\cdot)\) \(\chi_{1408}(835,\cdot)\) \(\chi_{1408}(923,\cdot)\) \(\chi_{1408}(1011,\cdot)\) \(\chi_{1408}(1099,\cdot)\) \(\chi_{1408}(1187,\cdot)\) \(\chi_{1408}(1275,\cdot)\) \(\chi_{1408}(1363,\cdot)\)
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: | 32.32.144215564533589000876246801170130951941346253827716612240069988132090544128.1 |
Values on generators
\((639,133,1025)\) → \((-1,e\left(\frac{1}{32}\right),-1)\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) | \(23\) |
\( \chi_{ 1408 }(1275, a) \) | \(1\) | \(1\) | \(e\left(\frac{19}{32}\right)\) | \(e\left(\frac{1}{32}\right)\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{31}{32}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{23}{32}\right)\) | \(e\left(\frac{29}{32}\right)\) | \(e\left(\frac{15}{16}\right)\) |
sage: chi.jacobi_sum(n)