from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1050, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,33,10]))
pari: [g,chi] = znchar(Mod(73,1050))
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 \(\chi_{175}(73,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1050.bv
\(\chi_{1050}(73,\cdot)\) \(\chi_{1050}(103,\cdot)\) \(\chi_{1050}(187,\cdot)\) \(\chi_{1050}(283,\cdot)\) \(\chi_{1050}(313,\cdot)\) \(\chi_{1050}(367,\cdot)\) \(\chi_{1050}(397,\cdot)\) \(\chi_{1050}(523,\cdot)\) \(\chi_{1050}(577,\cdot)\) \(\chi_{1050}(703,\cdot)\) \(\chi_{1050}(733,\cdot)\) \(\chi_{1050}(787,\cdot)\) \(\chi_{1050}(817,\cdot)\) \(\chi_{1050}(913,\cdot)\) \(\chi_{1050}(997,\cdot)\) \(\chi_{1050}(1027,\cdot)\)
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
\((701,127,451)\) → \((1,e\left(\frac{11}{20}\right),e\left(\frac{1}{6}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
| \( \chi_{ 1050 }(73, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{19}{20}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(i\) |
sage: chi.jacobi_sum(n)