from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1152, base_ring=CyclotomicField(32))
M = H._module
chi = DirichletCharacter(H, M([0,25,0]))
pari: [g,chi] = znchar(Mod(37,1152))
Basic properties
Modulus: | \(1152\) | |
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 \(\chi_{128}(37,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1152.bl
\(\chi_{1152}(37,\cdot)\) \(\chi_{1152}(109,\cdot)\) \(\chi_{1152}(181,\cdot)\) \(\chi_{1152}(253,\cdot)\) \(\chi_{1152}(325,\cdot)\) \(\chi_{1152}(397,\cdot)\) \(\chi_{1152}(469,\cdot)\) \(\chi_{1152}(541,\cdot)\) \(\chi_{1152}(613,\cdot)\) \(\chi_{1152}(685,\cdot)\) \(\chi_{1152}(757,\cdot)\) \(\chi_{1152}(829,\cdot)\) \(\chi_{1152}(901,\cdot)\) \(\chi_{1152}(973,\cdot)\) \(\chi_{1152}(1045,\cdot)\) \(\chi_{1152}(1117,\cdot)\)
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
\((127,901,641)\) → \((1,e\left(\frac{25}{32}\right),1)\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
\( \chi_{ 1152 }(37, a) \) | \(1\) | \(1\) | \(e\left(\frac{25}{32}\right)\) | \(e\left(\frac{13}{16}\right)\) | \(e\left(\frac{13}{32}\right)\) | \(e\left(\frac{23}{32}\right)\) | \(e\left(\frac{7}{8}\right)\) | \(e\left(\frac{31}{32}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{3}{32}\right)\) | \(i\) |
sage: chi.jacobi_sum(n)