from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7600, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([0,15,3,10]))
pari: [g,chi] = znchar(Mod(729,7600))
Basic properties
Modulus: | \(7600\) | |
Conductor: | \(3800\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(30\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{3800}(2629,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 7600.fs
\(\chi_{7600}(729,\cdot)\) \(\chi_{7600}(809,\cdot)\) \(\chi_{7600}(2329,\cdot)\) \(\chi_{7600}(3769,\cdot)\) \(\chi_{7600}(5289,\cdot)\) \(\chi_{7600}(5369,\cdot)\) \(\chi_{7600}(6809,\cdot)\) \(\chi_{7600}(6889,\cdot)\)
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: | \(\Q(\zeta_{15})\) |
Fixed field: | Number field defined by a degree 30 polynomial |
Values on generators
\((4751,5701,5777,401)\) → \((1,-1,e\left(\frac{1}{10}\right),e\left(\frac{1}{3}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(21\) | \(23\) | \(27\) | \(29\) |
\( \chi_{ 7600 }(729, a) \) | \(1\) | \(1\) | \(e\left(\frac{8}{15}\right)\) | \(-1\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{1}{10}\right)\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{23}{30}\right)\) | \(e\left(\frac{3}{5}\right)\) | \(e\left(\frac{11}{30}\right)\) |
sage: chi.jacobi_sum(n)