from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8085, base_ring=CyclotomicField(42))
M = H._module
chi = DirichletCharacter(H, M([0,0,31,21]))
pari: [g,chi] = znchar(Mod(241,8085))
Basic properties
Modulus: | \(8085\) | |
Conductor: | \(539\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(42\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{539}(241,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8085.em
\(\chi_{8085}(241,\cdot)\) \(\chi_{8085}(1396,\cdot)\) \(\chi_{8085}(2056,\cdot)\) \(\chi_{8085}(2551,\cdot)\) \(\chi_{8085}(3211,\cdot)\) \(\chi_{8085}(4366,\cdot)\) \(\chi_{8085}(4861,\cdot)\) \(\chi_{8085}(5521,\cdot)\) \(\chi_{8085}(6016,\cdot)\) \(\chi_{8085}(6676,\cdot)\) \(\chi_{8085}(7171,\cdot)\) \(\chi_{8085}(7831,\cdot)\)
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: | Number field defined by a degree 42 polynomial |
Values on generators
\((2696,4852,1816,3676)\) → \((1,1,e\left(\frac{31}{42}\right),-1)\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(13\) | \(16\) | \(17\) | \(19\) | \(23\) | \(26\) | \(29\) |
\( \chi_{ 8085 }(241, a) \) | \(1\) | \(1\) | \(e\left(\frac{29}{42}\right)\) | \(e\left(\frac{8}{21}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{16}{21}\right)\) | \(e\left(\frac{20}{21}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{1}{21}\right)\) | \(e\left(\frac{23}{42}\right)\) | \(e\left(\frac{11}{14}\right)\) |
sage: chi.jacobi_sum(n)