from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1001, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,54,35]))
pari: [g,chi] = znchar(Mod(50,1001))
Basic properties
| Modulus: | \(1001\) | |
| Conductor: | \(143\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{143}(50,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1001.ei
\(\chi_{1001}(50,\cdot)\) \(\chi_{1001}(85,\cdot)\) \(\chi_{1001}(106,\cdot)\) \(\chi_{1001}(162,\cdot)\) \(\chi_{1001}(288,\cdot)\) \(\chi_{1001}(358,\cdot)\) \(\chi_{1001}(414,\cdot)\) \(\chi_{1001}(435,\cdot)\) \(\chi_{1001}(470,\cdot)\) \(\chi_{1001}(596,\cdot)\) \(\chi_{1001}(722,\cdot)\) \(\chi_{1001}(743,\cdot)\) \(\chi_{1001}(778,\cdot)\) \(\chi_{1001}(799,\cdot)\) \(\chi_{1001}(904,\cdot)\) \(\chi_{1001}(981,\cdot)\)
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
\((430,365,925)\) → \((1,e\left(\frac{9}{10}\right),e\left(\frac{7}{12}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(8\) | \(9\) | \(10\) | \(12\) | \(15\) |
| \( \chi_{ 1001 }(50, a) \) | \(1\) | \(1\) | \(e\left(\frac{29}{60}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{9}{20}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(-1\) | \(e\left(\frac{23}{60}\right)\) |
sage: chi.jacobi_sum(n)