from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8001, base_ring=CyclotomicField(14))
M = H._module
chi = DirichletCharacter(H, M([7,0,13]))
pari: [g,chi] = znchar(Mod(1079,8001))
Basic properties
Modulus: | \(8001\) | |
Conductor: | \(381\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(14\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{381}(317,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8001.ev
\(\chi_{8001}(1079,\cdot)\) \(\chi_{8001}(1268,\cdot)\) \(\chi_{8001}(1520,\cdot)\) \(\chi_{8001}(2024,\cdot)\) \(\chi_{8001}(3032,\cdot)\) \(\chi_{8001}(5048,\cdot)\)
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: | \(\Q(\zeta_{7})\) |
Fixed field: | 14.14.4889868222780801055774484791029.1 |
Values on generators
\((3557,1144,7750)\) → \((-1,1,e\left(\frac{13}{14}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
\( \chi_{ 8001 }(1079, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{14}\right)\) | \(e\left(\frac{5}{7}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{1}{14}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{9}{14}\right)\) | \(e\left(\frac{2}{7}\right)\) | \(e\left(\frac{3}{7}\right)\) | \(e\left(\frac{11}{14}\right)\) | \(1\) |
sage: chi.jacobi_sum(n)