from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1400, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([30,30,51,10]))
pari: [g,chi] = znchar(Mod(1347,1400))
Basic properties
Modulus: | \(1400\) | |
Conductor: | \(1400\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1400.dq
\(\chi_{1400}(3,\cdot)\) \(\chi_{1400}(187,\cdot)\) \(\chi_{1400}(227,\cdot)\) \(\chi_{1400}(283,\cdot)\) \(\chi_{1400}(467,\cdot)\) \(\chi_{1400}(523,\cdot)\) \(\chi_{1400}(563,\cdot)\) \(\chi_{1400}(747,\cdot)\) \(\chi_{1400}(787,\cdot)\) \(\chi_{1400}(803,\cdot)\) \(\chi_{1400}(1027,\cdot)\) \(\chi_{1400}(1067,\cdot)\) \(\chi_{1400}(1083,\cdot)\) \(\chi_{1400}(1123,\cdot)\) \(\chi_{1400}(1347,\cdot)\) \(\chi_{1400}(1363,\cdot)\)
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: | \(\Q(\zeta_{60})\) |
Fixed field: | Number field defined by a degree 60 polynomial |
Values on generators
\((351,701,1177,801)\) → \((-1,-1,e\left(\frac{17}{20}\right),e\left(\frac{1}{6}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(27\) | \(29\) | \(31\) |
\( \chi_{ 1400 }(1347, a) \) | \(-1\) | \(1\) | \(e\left(\frac{7}{60}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{4}{15}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{7}{15}\right)\) |
sage: chi.jacobi_sum(n)