from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1792, base_ring=CyclotomicField(32))
M = H._module
chi = DirichletCharacter(H, M([0,29,0]))
pari: [g,chi] = znchar(Mod(57,1792))
Basic properties
Modulus: | \(1792\) | |
Conductor: | \(128\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(32\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{128}(85,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1792.bn
\(\chi_{1792}(57,\cdot)\) \(\chi_{1792}(169,\cdot)\) \(\chi_{1792}(281,\cdot)\) \(\chi_{1792}(393,\cdot)\) \(\chi_{1792}(505,\cdot)\) \(\chi_{1792}(617,\cdot)\) \(\chi_{1792}(729,\cdot)\) \(\chi_{1792}(841,\cdot)\) \(\chi_{1792}(953,\cdot)\) \(\chi_{1792}(1065,\cdot)\) \(\chi_{1792}(1177,\cdot)\) \(\chi_{1792}(1289,\cdot)\) \(\chi_{1792}(1401,\cdot)\) \(\chi_{1792}(1513,\cdot)\) \(\chi_{1792}(1625,\cdot)\) \(\chi_{1792}(1737,\cdot)\)
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: | \(\Q(\zeta_{128})^+\) |
Values on generators
\((1023,1541,1025)\) → \((1,e\left(\frac{29}{32}\right),1)\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) |
\( \chi_{ 1792 }(57, a) \) | \(1\) | \(1\) | \(e\left(\frac{23}{32}\right)\) | \(e\left(\frac{29}{32}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{1}{32}\right)\) | \(e\left(\frac{19}{32}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{27}{32}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{13}{16}\right)\) |
sage: chi.jacobi_sum(n)