from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1125, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([40,57]))
pari: [g,chi] = znchar(Mod(43,1125))
Basic properties
| Modulus: | \(1125\) | |
| Conductor: | \(225\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{225}(88,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1125.bb
\(\chi_{1125}(7,\cdot)\) \(\chi_{1125}(43,\cdot)\) \(\chi_{1125}(157,\cdot)\) \(\chi_{1125}(232,\cdot)\) \(\chi_{1125}(268,\cdot)\) \(\chi_{1125}(382,\cdot)\) \(\chi_{1125}(418,\cdot)\) \(\chi_{1125}(457,\cdot)\) \(\chi_{1125}(493,\cdot)\) \(\chi_{1125}(607,\cdot)\) \(\chi_{1125}(643,\cdot)\) \(\chi_{1125}(718,\cdot)\) \(\chi_{1125}(832,\cdot)\) \(\chi_{1125}(868,\cdot)\) \(\chi_{1125}(907,\cdot)\) \(\chi_{1125}(1093,\cdot)\)
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
\((1001,127)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{19}{20}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(7\) | \(8\) | \(11\) | \(13\) | \(14\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 1125 }(43, a) \) | \(-1\) | \(1\) | \(e\left(\frac{37}{60}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{17}{20}\right)\) | \(e\left(\frac{13}{15}\right)\) | \(e\left(\frac{23}{60}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{7}{15}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{1}{10}\right)\) |
sage: chi.jacobi_sum(n)