from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6006, base_ring=CyclotomicField(10))
M = H._module
chi = DirichletCharacter(H, M([5,5,6,5]))
pari: [g,chi] = znchar(Mod(1637,6006))
Basic properties
Modulus: | \(6006\) | |
Conductor: | \(3003\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(10\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{3003}(1637,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6006.ds
\(\chi_{6006}(1637,\cdot)\) \(\chi_{6006}(2183,\cdot)\) \(\chi_{6006}(3821,\cdot)\) \(\chi_{6006}(5459,\cdot)\)
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: | \(\Q(\zeta_{5})\) |
Fixed field: | 10.10.325053402565947488433.1 |
Values on generators
\((2003,3433,4369,925)\) → \((-1,-1,e\left(\frac{3}{5}\right),-1)\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
\( \chi_{ 6006 }(1637, a) \) | \(1\) | \(1\) | \(e\left(\frac{9}{10}\right)\) | \(e\left(\frac{2}{5}\right)\) | \(e\left(\frac{4}{5}\right)\) | \(-1\) | \(e\left(\frac{4}{5}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{7}{10}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(1\) |
sage: chi.jacobi_sum(n)