from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4032, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([24,3,24,32]))
pari: [g,chi] = znchar(Mod(3707,4032))
Basic properties
Modulus: | \(4032\) | |
Conductor: | \(1344\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(48\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{1344}(1019,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | no | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4032.gz
\(\chi_{4032}(107,\cdot)\) \(\chi_{4032}(179,\cdot)\) \(\chi_{4032}(611,\cdot)\) \(\chi_{4032}(683,\cdot)\) \(\chi_{4032}(1115,\cdot)\) \(\chi_{4032}(1187,\cdot)\) \(\chi_{4032}(1619,\cdot)\) \(\chi_{4032}(1691,\cdot)\) \(\chi_{4032}(2123,\cdot)\) \(\chi_{4032}(2195,\cdot)\) \(\chi_{4032}(2627,\cdot)\) \(\chi_{4032}(2699,\cdot)\) \(\chi_{4032}(3131,\cdot)\) \(\chi_{4032}(3203,\cdot)\) \(\chi_{4032}(3635,\cdot)\) \(\chi_{4032}(3707,\cdot)\)
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
\((127,3781,1793,577)\) → \((-1,e\left(\frac{1}{16}\right),-1,e\left(\frac{2}{3}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
\( \chi_{ 4032 }(3707, a) \) | \(1\) | \(1\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{15}{16}\right)\) | \(e\left(\frac{11}{12}\right)\) | \(e\left(\frac{13}{48}\right)\) | \(e\left(\frac{5}{24}\right)\) | \(e\left(\frac{19}{24}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{43}{48}\right)\) |
sage: chi.jacobi_sum(n)