from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8085, base_ring=CyclotomicField(28))
M = H._module
chi = DirichletCharacter(H, M([0,21,4,14]))
pari: [g,chi] = znchar(Mod(43,8085))
Basic properties
| Modulus: | \(8085\) | |
| Conductor: | \(2695\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(28\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{2695}(43,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8085.dr
\(\chi_{8085}(43,\cdot)\) \(\chi_{8085}(967,\cdot)\) \(\chi_{8085}(1198,\cdot)\) \(\chi_{8085}(2122,\cdot)\) \(\chi_{8085}(3277,\cdot)\) \(\chi_{8085}(3508,\cdot)\) \(\chi_{8085}(4432,\cdot)\) \(\chi_{8085}(4663,\cdot)\) \(\chi_{8085}(5818,\cdot)\) \(\chi_{8085}(6742,\cdot)\) \(\chi_{8085}(6973,\cdot)\) \(\chi_{8085}(7897,\cdot)\)
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
\((2696,4852,1816,3676)\) → \((1,-i,e\left(\frac{1}{7}\right),-1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(8\) | \(13\) | \(16\) | \(17\) | \(19\) | \(23\) | \(26\) | \(29\) |
| \( \chi_{ 8085 }(43, a) \) | \(1\) | \(1\) | \(e\left(\frac{27}{28}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{25}{28}\right)\) | \(e\left(\frac{13}{28}\right)\) | \(e\left(\frac{6}{7}\right)\) | \(e\left(\frac{23}{28}\right)\) | \(1\) | \(e\left(\frac{19}{28}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{4}{7}\right)\) |
sage: chi.jacobi_sum(n)