from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4032, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([0,39,24,40]))
pari: [g,chi] = znchar(Mod(341,4032))
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 \(\chi_{1344}(341,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4032.gy
\(\chi_{4032}(269,\cdot)\) \(\chi_{4032}(341,\cdot)\) \(\chi_{4032}(773,\cdot)\) \(\chi_{4032}(845,\cdot)\) \(\chi_{4032}(1277,\cdot)\) \(\chi_{4032}(1349,\cdot)\) \(\chi_{4032}(1781,\cdot)\) \(\chi_{4032}(1853,\cdot)\) \(\chi_{4032}(2285,\cdot)\) \(\chi_{4032}(2357,\cdot)\) \(\chi_{4032}(2789,\cdot)\) \(\chi_{4032}(2861,\cdot)\) \(\chi_{4032}(3293,\cdot)\) \(\chi_{4032}(3365,\cdot)\) \(\chi_{4032}(3797,\cdot)\) \(\chi_{4032}(3869,\cdot)\)
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
\((127,3781,1793,577)\) → \((1,e\left(\frac{13}{16}\right),-1,e\left(\frac{5}{6}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 4032 }(341, a) \) | \(1\) | \(1\) | \(e\left(\frac{23}{48}\right)\) | \(e\left(\frac{43}{48}\right)\) | \(e\left(\frac{11}{16}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{41}{48}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{1}{3}\right)\) | \(e\left(\frac{47}{48}\right)\) |
sage: chi.jacobi_sum(n)