from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7488, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([0,21,32,24]))
pari: [g,chi] = znchar(Mod(493,7488))
Basic properties
Modulus: | \(7488\) | |
Conductor: | \(7488\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(48\) | 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 7488.lv
\(\chi_{7488}(493,\cdot)\) \(\chi_{7488}(805,\cdot)\) \(\chi_{7488}(1429,\cdot)\) \(\chi_{7488}(1741,\cdot)\) \(\chi_{7488}(2365,\cdot)\) \(\chi_{7488}(2677,\cdot)\) \(\chi_{7488}(3301,\cdot)\) \(\chi_{7488}(3613,\cdot)\) \(\chi_{7488}(4237,\cdot)\) \(\chi_{7488}(4549,\cdot)\) \(\chi_{7488}(5173,\cdot)\) \(\chi_{7488}(5485,\cdot)\) \(\chi_{7488}(6109,\cdot)\) \(\chi_{7488}(6421,\cdot)\) \(\chi_{7488}(7045,\cdot)\) \(\chi_{7488}(7357,\cdot)\)
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: | \(\Q(\zeta_{48})\) |
Fixed field: | Number field defined by a degree 48 polynomial |
Values on generators
\((703,6085,5825,5761)\) → \((1,e\left(\frac{7}{16}\right),e\left(\frac{2}{3}\right),-1)\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
\( \chi_{ 7488 }(493, a) \) | \(1\) | \(1\) | \(e\left(\frac{13}{48}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{17}{48}\right)\) | \(i\) | \(e\left(\frac{9}{16}\right)\) | \(e\left(\frac{11}{24}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{23}{48}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{13}{16}\right)\) |
sage: chi.jacobi_sum(n)