from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7803, base_ring=CyclotomicField(34))
M = H._module
chi = DirichletCharacter(H, M([0,30]))
pari: [g,chi] = znchar(Mod(6886,7803))
Basic properties
Modulus: | \(7803\) | |
Conductor: | \(289\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(17\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{289}(239,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 7803.r
\(\chi_{7803}(460,\cdot)\) \(\chi_{7803}(919,\cdot)\) \(\chi_{7803}(1378,\cdot)\) \(\chi_{7803}(1837,\cdot)\) \(\chi_{7803}(2296,\cdot)\) \(\chi_{7803}(2755,\cdot)\) \(\chi_{7803}(3214,\cdot)\) \(\chi_{7803}(3673,\cdot)\) \(\chi_{7803}(4132,\cdot)\) \(\chi_{7803}(4591,\cdot)\) \(\chi_{7803}(5050,\cdot)\) \(\chi_{7803}(5509,\cdot)\) \(\chi_{7803}(5968,\cdot)\) \(\chi_{7803}(6427,\cdot)\) \(\chi_{7803}(6886,\cdot)\) \(\chi_{7803}(7345,\cdot)\)
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 17 polynomial |
Values on generators
\((2891,2026)\) → \((1,e\left(\frac{15}{17}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(7\) | \(8\) | \(10\) | \(11\) | \(13\) | \(14\) | \(16\) |
\( \chi_{ 7803 }(6886, a) \) | \(1\) | \(1\) | \(e\left(\frac{11}{17}\right)\) | \(e\left(\frac{5}{17}\right)\) | \(e\left(\frac{1}{17}\right)\) | \(e\left(\frac{13}{17}\right)\) | \(e\left(\frac{16}{17}\right)\) | \(e\left(\frac{12}{17}\right)\) | \(e\left(\frac{5}{17}\right)\) | \(e\left(\frac{16}{17}\right)\) | \(e\left(\frac{7}{17}\right)\) | \(e\left(\frac{10}{17}\right)\) |
sage: chi.jacobi_sum(n)