from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4015, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([9,18,17]))
pari: [g,chi] = znchar(Mod(692,4015))
Basic properties
Modulus: | \(4015\) | |
Conductor: | \(4015\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(36\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4015.dr
\(\chi_{4015}(692,\cdot)\) \(\chi_{4015}(857,\cdot)\) \(\chi_{4015}(1143,\cdot)\) \(\chi_{4015}(1253,\cdot)\) \(\chi_{4015}(2397,\cdot)\) \(\chi_{4015}(2507,\cdot)\) \(\chi_{4015}(2793,\cdot)\) \(\chi_{4015}(2958,\cdot)\) \(\chi_{4015}(3673,\cdot)\) \(\chi_{4015}(3717,\cdot)\) \(\chi_{4015}(3948,\cdot)\) \(\chi_{4015}(3992,\cdot)\)
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: | \(\Q(\zeta_{36})\) |
Fixed field: | Number field defined by a degree 36 polynomial |
Values on generators
\((1607,2191,881)\) → \((i,-1,e\left(\frac{17}{36}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(12\) | \(13\) | \(14\) |
\( \chi_{ 4015 }(692, a) \) | \(1\) | \(1\) | \(e\left(\frac{19}{36}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{23}{36}\right)\) | \(e\left(\frac{1}{9}\right)\) | \(e\left(\frac{31}{36}\right)\) |
sage: chi.jacobi_sum(n)