from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6027, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([0,23,21]))
pari: [g,chi] = znchar(Mod(40,6027))
Basic properties
| Modulus: | \(6027\) | |
| Conductor: | \(2009\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{2009}(40,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6027.cw
\(\chi_{6027}(40,\cdot)\) \(\chi_{6027}(409,\cdot)\) \(\chi_{6027}(1270,\cdot)\) \(\chi_{6027}(1762,\cdot)\) \(\chi_{6027}(2131,\cdot)\) \(\chi_{6027}(2623,\cdot)\) \(\chi_{6027}(2992,\cdot)\) \(\chi_{6027}(3484,\cdot)\) \(\chi_{6027}(4345,\cdot)\) \(\chi_{6027}(4714,\cdot)\) \(\chi_{6027}(5206,\cdot)\) \(\chi_{6027}(5575,\cdot)\)
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: | \(\Q(\zeta_{21})\) |
| Fixed field: | Number field defined by a degree 42 polynomial |
Values on generators
\((4019,493,2794)\) → \((1,e\left(\frac{23}{42}\right),-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 6027 }(40, a) \) | \(-1\) | \(1\) | \(e\left(\frac{5}{21}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{4}{7}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{4}{21}\right)\) | \(e\left(\frac{2}{3}\right)\) |
sage: chi.jacobi_sum(n)