from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3381, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([21,29,0]))
pari: [g,chi] = znchar(Mod(2945,3381))
Basic properties
Modulus: | \(3381\) | |
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 \(\chi_{147}(5,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3381.bk
\(\chi_{3381}(47,\cdot)\) \(\chi_{3381}(185,\cdot)\) \(\chi_{3381}(530,\cdot)\) \(\chi_{3381}(1013,\cdot)\) \(\chi_{3381}(1151,\cdot)\) \(\chi_{3381}(1496,\cdot)\) \(\chi_{3381}(1634,\cdot)\) \(\chi_{3381}(2117,\cdot)\) \(\chi_{3381}(2462,\cdot)\) \(\chi_{3381}(2600,\cdot)\) \(\chi_{3381}(2945,\cdot)\) \(\chi_{3381}(3083,\cdot)\)
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
\((2255,346,442)\) → \((-1,e\left(\frac{29}{42}\right),1)\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
\( \chi_{ 3381 }(2945, a) \) | \(1\) | \(1\) | \(e\left(\frac{19}{42}\right)\) | \(e\left(\frac{19}{21}\right)\) | \(e\left(\frac{11}{21}\right)\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{41}{42}\right)\) | \(e\left(\frac{5}{42}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(e\left(\frac{17}{21}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{1}{6}\right)\) |
sage: chi.jacobi_sum(n)