from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8034, base_ring=CyclotomicField(34))
M = H._module
chi = DirichletCharacter(H, M([0,17,10]))
pari: [g,chi] = znchar(Mod(3613,8034))
Basic properties
Modulus: | \(8034\) | |
Conductor: | \(1339\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(34\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{1339}(935,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8034.co
\(\chi_{8034}(493,\cdot)\) \(\chi_{8034}(961,\cdot)\) \(\chi_{8034}(1039,\cdot)\) \(\chi_{8034}(1741,\cdot)\) \(\chi_{8034}(3223,\cdot)\) \(\chi_{8034}(3613,\cdot)\) \(\chi_{8034}(3769,\cdot)\) \(\chi_{8034}(4081,\cdot)\) \(\chi_{8034}(4237,\cdot)\) \(\chi_{8034}(5173,\cdot)\) \(\chi_{8034}(5329,\cdot)\) \(\chi_{8034}(5641,\cdot)\) \(\chi_{8034}(6811,\cdot)\) \(\chi_{8034}(6967,\cdot)\) \(\chi_{8034}(7591,\cdot)\) \(\chi_{8034}(7825,\cdot)\)
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: | \(\Q(\zeta_{17})\) |
Fixed field: | Number field defined by a degree 34 polynomial |
Values on generators
\((5357,1237,5773)\) → \((1,-1,e\left(\frac{5}{17}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
\( \chi_{ 8034 }(3613, a) \) | \(1\) | \(1\) | \(e\left(\frac{27}{34}\right)\) | \(e\left(\frac{23}{34}\right)\) | \(e\left(\frac{15}{34}\right)\) | \(e\left(\frac{10}{17}\right)\) | \(e\left(\frac{1}{34}\right)\) | \(e\left(\frac{1}{17}\right)\) | \(e\left(\frac{10}{17}\right)\) | \(e\left(\frac{5}{17}\right)\) | \(e\left(\frac{9}{34}\right)\) | \(e\left(\frac{8}{17}\right)\) |
sage: chi.jacobi_sum(n)