from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7524, base_ring=CyclotomicField(30))
M = H._module
chi = DirichletCharacter(H, M([15,5,6,10]))
pari: [g,chi] = znchar(Mod(6923,7524))
Basic properties
Modulus: | \(7524\) | |
Conductor: | \(7524\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(30\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 7524.gd
\(\chi_{7524}(2291,\cdot)\) \(\chi_{7524}(2819,\cdot)\) \(\chi_{7524}(2975,\cdot)\) \(\chi_{7524}(3503,\cdot)\) \(\chi_{7524}(4343,\cdot)\) \(\chi_{7524}(4871,\cdot)\) \(\chi_{7524}(6395,\cdot)\) \(\chi_{7524}(6923,\cdot)\)
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
\((3763,6689,4105,2377)\) → \((-1,e\left(\frac{1}{6}\right),e\left(\frac{1}{5}\right),e\left(\frac{1}{3}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(13\) | \(17\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) | \(37\) |
\( \chi_{ 7524 }(6923, a) \) | \(1\) | \(1\) | \(e\left(\frac{29}{30}\right)\) | \(e\left(\frac{17}{30}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{19}{30}\right)\) | \(1\) | \(e\left(\frac{14}{15}\right)\) | \(e\left(\frac{7}{30}\right)\) | \(e\left(\frac{1}{30}\right)\) | \(e\left(\frac{8}{15}\right)\) | \(e\left(\frac{2}{5}\right)\) |
sage: chi.jacobi_sum(n)