from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8085, base_ring=CyclotomicField(28))
M = H._module
chi = DirichletCharacter(H, M([0,7,10,0]))
pari: [g,chi] = znchar(Mod(727,8085))
Basic properties
Modulus: | \(8085\) | |
Conductor: | \(245\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(28\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{245}(237,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8085.ds
\(\chi_{8085}(727,\cdot)\) \(\chi_{8085}(958,\cdot)\) \(\chi_{8085}(1882,\cdot)\) \(\chi_{8085}(2113,\cdot)\) \(\chi_{8085}(3268,\cdot)\) \(\chi_{8085}(4192,\cdot)\) \(\chi_{8085}(4423,\cdot)\) \(\chi_{8085}(5347,\cdot)\) \(\chi_{8085}(5578,\cdot)\) \(\chi_{8085}(6502,\cdot)\) \(\chi_{8085}(6733,\cdot)\) \(\chi_{8085}(7657,\cdot)\)
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: | \(\Q(\zeta_{28})\) |
Fixed field: | Number field defined by a degree 28 polynomial |
Values on generators
\((2696,4852,1816,3676)\) → \((1,i,e\left(\frac{5}{14}\right),1)\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(13\) | \(16\) | \(17\) | \(19\) | \(23\) | \(26\) | \(29\) |
\( \chi_{ 8085 }(727, a) \) | \(1\) | \(1\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{17}{28}\right)\) | \(e\left(\frac{15}{28}\right)\) | \(e\left(\frac{1}{7}\right)\) | \(e\left(\frac{5}{28}\right)\) | \(1\) | \(e\left(\frac{9}{28}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{13}{14}\right)\) |
sage: chi.jacobi_sum(n)