from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8085, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([0,45,40,6]))
pari: [g,chi] = znchar(Mod(508,8085))
Basic properties
Modulus: | \(8085\) | |
Conductor: | \(385\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(60\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{385}(123,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8085.fg
\(\chi_{8085}(508,\cdot)\) \(\chi_{8085}(667,\cdot)\) \(\chi_{8085}(1108,\cdot)\) \(\chi_{8085}(1537,\cdot)\) \(\chi_{8085}(1843,\cdot)\) \(\chi_{8085}(2272,\cdot)\) \(\chi_{8085}(2713,\cdot)\) \(\chi_{8085}(3313,\cdot)\) \(\chi_{8085}(3742,\cdot)\) \(\chi_{8085}(4342,\cdot)\) \(\chi_{8085}(5077,\cdot)\) \(\chi_{8085}(5518,\cdot)\) \(\chi_{8085}(5947,\cdot)\) \(\chi_{8085}(6388,\cdot)\) \(\chi_{8085}(6547,\cdot)\) \(\chi_{8085}(7123,\cdot)\)
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: | \(\Q(\zeta_{60})\) |
Fixed field: | Number field defined by a degree 60 polynomial |
Values on generators
\((2696,4852,1816,3676)\) → \((1,-i,e\left(\frac{2}{3}\right),e\left(\frac{1}{10}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(13\) | \(16\) | \(17\) | \(19\) | \(23\) | \(26\) | \(29\) |
\( \chi_{ 8085 }(508, a) \) | \(1\) | \(1\) | \(e\left(\frac{11}{60}\right)\) | \(e\left(\frac{11}{30}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{7}{20}\right)\) | \(e\left(\frac{11}{15}\right)\) | \(e\left(\frac{19}{60}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{7}{12}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{1}{5}\right)\) |
sage: chi.jacobi_sum(n)