from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1050, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,51,20]))
pari: [g,chi] = znchar(Mod(247,1050))
Basic properties
Modulus: | \(1050\) | |
Conductor: | \(175\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{175}(72,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1050.bu
\(\chi_{1050}(37,\cdot)\) \(\chi_{1050}(67,\cdot)\) \(\chi_{1050}(163,\cdot)\) \(\chi_{1050}(247,\cdot)\) \(\chi_{1050}(277,\cdot)\) \(\chi_{1050}(373,\cdot)\) \(\chi_{1050}(403,\cdot)\) \(\chi_{1050}(487,\cdot)\) \(\chi_{1050}(583,\cdot)\) \(\chi_{1050}(613,\cdot)\) \(\chi_{1050}(667,\cdot)\) \(\chi_{1050}(697,\cdot)\) \(\chi_{1050}(823,\cdot)\) \(\chi_{1050}(877,\cdot)\) \(\chi_{1050}(1003,\cdot)\) \(\chi_{1050}(1033,\cdot)\)
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: | \(\Q(\zeta_{60})\) |
Fixed field: | Number field defined by a degree 60 polynomial |
Values on generators
\((701,127,451)\) → \((1,e\left(\frac{17}{20}\right),e\left(\frac{1}{3}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
\( \chi_{ 1050 }(247, a) \) | \(-1\) | \(1\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(-i\) |
sage: chi.jacobi_sum(n)