from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6003, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([28,0,27]))
pari: [g,chi] = znchar(Mod(4624,6003))
Basic properties
Modulus: | \(6003\) | |
Conductor: | \(261\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{261}(187,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6003.bx
\(\chi_{6003}(760,\cdot)\) \(\chi_{6003}(1588,\cdot)\) \(\chi_{6003}(2209,\cdot)\) \(\chi_{6003}(2416,\cdot)\) \(\chi_{6003}(2623,\cdot)\) \(\chi_{6003}(2761,\cdot)\) \(\chi_{6003}(3589,\cdot)\) \(\chi_{6003}(3658,\cdot)\) \(\chi_{6003}(4210,\cdot)\) \(\chi_{6003}(4417,\cdot)\) \(\chi_{6003}(4624,\cdot)\) \(\chi_{6003}(5659,\cdot)\)
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: | 42.42.565343212441678035532894502003808167878401992443661947648452445739810658542578516149.1 |
Values on generators
\((668,3133,4555)\) → \((e\left(\frac{2}{3}\right),1,e\left(\frac{9}{14}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
\( \chi_{ 6003 }(4624, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{42}\right)\) | \(e\left(\frac{13}{21}\right)\) | \(e\left(\frac{10}{21}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{31}{42}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{5}{21}\right)\) |
sage: chi.jacobi_sum(n)