from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8034, base_ring=CyclotomicField(34))
M = H._module
chi = DirichletCharacter(H, M([17,17,25]))
pari: [g,chi] = znchar(Mod(389,8034))
Basic properties
| Modulus: | \(8034\) | |
| Conductor: | \(4017\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(34\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{4017}(389,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8034.cp
\(\chi_{8034}(233,\cdot)\) \(\chi_{8034}(389,\cdot)\) \(\chi_{8034}(1325,\cdot)\) \(\chi_{8034}(1481,\cdot)\) \(\chi_{8034}(1793,\cdot)\) \(\chi_{8034}(1949,\cdot)\) \(\chi_{8034}(2339,\cdot)\) \(\chi_{8034}(3821,\cdot)\) \(\chi_{8034}(4523,\cdot)\) \(\chi_{8034}(4601,\cdot)\) \(\chi_{8034}(5069,\cdot)\) \(\chi_{8034}(5771,\cdot)\) \(\chi_{8034}(6005,\cdot)\) \(\chi_{8034}(6629,\cdot)\) \(\chi_{8034}(6785,\cdot)\) \(\chi_{8034}(7955,\cdot)\)
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
\((5357,1237,5773)\) → \((-1,-1,e\left(\frac{25}{34}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 8034 }(389, a) \) | \(1\) | \(1\) | \(e\left(\frac{25}{34}\right)\) | \(e\left(\frac{15}{34}\right)\) | \(e\left(\frac{29}{34}\right)\) | \(e\left(\frac{33}{34}\right)\) | \(e\left(\frac{11}{34}\right)\) | \(e\left(\frac{5}{34}\right)\) | \(e\left(\frac{8}{17}\right)\) | \(e\left(\frac{25}{34}\right)\) | \(e\left(\frac{7}{17}\right)\) | \(e\left(\frac{3}{17}\right)\) |
sage: chi.jacobi_sum(n)