from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6272, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,0,13]))
pari: [g,chi] = znchar(Mod(255,6272))
Basic properties
Modulus: | \(6272\) | |
Conductor: | \(196\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{196}(59,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6272.cf
\(\chi_{6272}(255,\cdot)\) \(\chi_{6272}(383,\cdot)\) \(\chi_{6272}(1151,\cdot)\) \(\chi_{6272}(1279,\cdot)\) \(\chi_{6272}(2047,\cdot)\) \(\chi_{6272}(2943,\cdot)\) \(\chi_{6272}(3071,\cdot)\) \(\chi_{6272}(3839,\cdot)\) \(\chi_{6272}(3967,\cdot)\) \(\chi_{6272}(4863,\cdot)\) \(\chi_{6272}(5631,\cdot)\) \(\chi_{6272}(5759,\cdot)\)
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: | \(\Q(\zeta_{196})^+\) |
Values on generators
\((4607,3333,4609)\) → \((-1,1,e\left(\frac{13}{42}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) |
\( \chi_{ 6272 }(255, a) \) | \(1\) | \(1\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{37}{42}\right)\) | \(e\left(\frac{3}{14}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{11}{42}\right)\) | \(e\left(\frac{20}{21}\right)\) |
sage: chi.jacobi_sum(n)