from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8800, base_ring=CyclotomicField(40))
M = H._module
chi = DirichletCharacter(H, M([0,5,20,16]))
pari: [g,chi] = znchar(Mod(7749,8800))
Basic properties
| Modulus: | \(8800\) | |
| Conductor: | \(1760\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(40\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1760}(709,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | no | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 8800.mq
\(\chi_{8800}(949,\cdot)\) \(\chi_{8800}(1149,\cdot)\) \(\chi_{8800}(1549,\cdot)\) \(\chi_{8800}(2149,\cdot)\) \(\chi_{8800}(3149,\cdot)\) \(\chi_{8800}(3349,\cdot)\) \(\chi_{8800}(3749,\cdot)\) \(\chi_{8800}(4349,\cdot)\) \(\chi_{8800}(5349,\cdot)\) \(\chi_{8800}(5549,\cdot)\) \(\chi_{8800}(5949,\cdot)\) \(\chi_{8800}(6549,\cdot)\) \(\chi_{8800}(7549,\cdot)\) \(\chi_{8800}(7749,\cdot)\) \(\chi_{8800}(8149,\cdot)\) \(\chi_{8800}(8749,\cdot)\)
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: | \(\Q(\zeta_{40})\) |
| Fixed field: | Number field defined by a degree 40 polynomial |
Values on generators
\((2751,3301,4577,5601)\) → \((1,e\left(\frac{1}{8}\right),-1,e\left(\frac{2}{5}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) | \(29\) |
| \( \chi_{ 8800 }(7749, a) \) | \(1\) | \(1\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{11}{20}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{31}{40}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{3}{40}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(i\) | \(e\left(\frac{9}{40}\right)\) | \(e\left(\frac{7}{40}\right)\) |
sage: chi.jacobi_sum(n)