from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3040, base_ring=CyclotomicField(12))
M = H._module
chi = DirichletCharacter(H, M([0,0,9,2]))
pari: [g,chi] = znchar(Mod(673,3040))
Basic properties
Modulus: | \(3040\) | |
Conductor: | \(95\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(12\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{95}(8,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3040.dj
\(\chi_{3040}(673,\cdot)\) \(\chi_{3040}(1057,\cdot)\) \(\chi_{3040}(2273,\cdot)\) \(\chi_{3040}(2497,\cdot)\)
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: | \(\Q(\zeta_{12})\) |
Fixed field: | 12.12.11974738784767578125.1 |
Values on generators
\((191,2661,1217,1921)\) → \((1,1,-i,e\left(\frac{1}{6}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(21\) | \(23\) | \(27\) | \(29\) |
\( \chi_{ 3040 }(673, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{12}\right)\) | \(-i\) | \(e\left(\frac{5}{6}\right)\) | \(1\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{5}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(i\) | \(e\left(\frac{1}{3}\right)\) |
sage: chi.jacobi_sum(n)