from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3040, base_ring=CyclotomicField(18))
M = H._module
chi = DirichletCharacter(H, M([9,9,0,5]))
pari: [g,chi] = znchar(Mod(431,3040))
Basic properties
Modulus: | \(3040\) | |
Conductor: | \(152\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(18\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{152}(51,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 3040.ed
\(\chi_{3040}(431,\cdot)\) \(\chi_{3040}(591,\cdot)\) \(\chi_{3040}(751,\cdot)\) \(\chi_{3040}(1231,\cdot)\) \(\chi_{3040}(2351,\cdot)\) \(\chi_{3040}(2511,\cdot)\)
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: | \(\Q(\zeta_{9})\) |
Fixed field: | 18.18.735565072612935262326166126592.1 |
Values on generators
\((191,2661,1217,1921)\) → \((-1,-1,1,e\left(\frac{5}{18}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(21\) | \(23\) | \(27\) | \(29\) |
\( \chi_{ 3040 }(431, a) \) | \(1\) | \(1\) | \(e\left(\frac{11}{18}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{2}{9}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{8}{9}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{2}{9}\right)\) |
sage: chi.jacobi_sum(n)