from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1728, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([0,15,8]))
pari: [g,chi] = znchar(Mod(629,1728))
Basic properties
Modulus: | \(1728\) | |
Conductor: | \(576\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(48\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{576}(245,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1728.bz
\(\chi_{1728}(125,\cdot)\) \(\chi_{1728}(197,\cdot)\) \(\chi_{1728}(341,\cdot)\) \(\chi_{1728}(413,\cdot)\) \(\chi_{1728}(557,\cdot)\) \(\chi_{1728}(629,\cdot)\) \(\chi_{1728}(773,\cdot)\) \(\chi_{1728}(845,\cdot)\) \(\chi_{1728}(989,\cdot)\) \(\chi_{1728}(1061,\cdot)\) \(\chi_{1728}(1205,\cdot)\) \(\chi_{1728}(1277,\cdot)\) \(\chi_{1728}(1421,\cdot)\) \(\chi_{1728}(1493,\cdot)\) \(\chi_{1728}(1637,\cdot)\) \(\chi_{1728}(1709,\cdot)\)
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: | \(\Q(\zeta_{48})\) |
Fixed field: | Number field defined by a degree 48 polynomial |
Values on generators
\((703,325,1217)\) → \((1,e\left(\frac{5}{16}\right),e\left(\frac{1}{6}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) |
\( \chi_{ 1728 }(629, a) \) | \(-1\) | \(1\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{35}{48}\right)\) | \(e\left(\frac{1}{48}\right)\) | \(i\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{7}{24}\right)\) | \(e\left(\frac{29}{48}\right)\) | \(e\left(\frac{5}{6}\right)\) |
sage: chi.jacobi_sum(n)