from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1800, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,30,40,27]))
pari: [g,chi] = znchar(Mod(637,1800))
Basic properties
Modulus: | \(1800\) | |
Conductor: | \(1800\) | 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 1800.dl
\(\chi_{1800}(13,\cdot)\) \(\chi_{1800}(133,\cdot)\) \(\chi_{1800}(277,\cdot)\) \(\chi_{1800}(373,\cdot)\) \(\chi_{1800}(517,\cdot)\) \(\chi_{1800}(637,\cdot)\) \(\chi_{1800}(733,\cdot)\) \(\chi_{1800}(853,\cdot)\) \(\chi_{1800}(877,\cdot)\) \(\chi_{1800}(997,\cdot)\) \(\chi_{1800}(1213,\cdot)\) \(\chi_{1800}(1237,\cdot)\) \(\chi_{1800}(1453,\cdot)\) \(\chi_{1800}(1573,\cdot)\) \(\chi_{1800}(1597,\cdot)\) \(\chi_{1800}(1717,\cdot)\)
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
\((1351,901,1001,577)\) → \((1,-1,e\left(\frac{2}{3}\right),e\left(\frac{9}{20}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) |
\( \chi_{ 1800 }(637, a) \) | \(-1\) | \(1\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{2}{15}\right)\) |
sage: chi.jacobi_sum(n)