from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6027, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,37,0]))
pari: [g,chi] = znchar(Mod(3650,6027))
Basic properties
Modulus: | \(6027\) | |
Conductor: | \(147\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{147}(122,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6027.cu
\(\chi_{6027}(206,\cdot)\) \(\chi_{6027}(698,\cdot)\) \(\chi_{6027}(1067,\cdot)\) \(\chi_{6027}(1559,\cdot)\) \(\chi_{6027}(1928,\cdot)\) \(\chi_{6027}(2789,\cdot)\) \(\chi_{6027}(3281,\cdot)\) \(\chi_{6027}(3650,\cdot)\) \(\chi_{6027}(4142,\cdot)\) \(\chi_{6027}(4511,\cdot)\) \(\chi_{6027}(5003,\cdot)\) \(\chi_{6027}(5864,\cdot)\)
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: | \(\Q(\zeta_{147})^+\) |
Values on generators
\((4019,493,2794)\) → \((-1,e\left(\frac{37}{42}\right),1)\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
\( \chi_{ 6027 }(3650, a) \) | \(1\) | \(1\) | \(e\left(\frac{17}{42}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{5}{6}\right)\) |
sage: chi.jacobi_sum(n)