from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4032, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([0,9,32,0]))
pari: [g,chi] = znchar(Mod(2941,4032))
Basic properties
| Modulus: | \(4032\) | |
| Conductor: | \(576\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(48\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{576}(61,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4032.hg
\(\chi_{4032}(85,\cdot)\) \(\chi_{4032}(421,\cdot)\) \(\chi_{4032}(589,\cdot)\) \(\chi_{4032}(925,\cdot)\) \(\chi_{4032}(1093,\cdot)\) \(\chi_{4032}(1429,\cdot)\) \(\chi_{4032}(1597,\cdot)\) \(\chi_{4032}(1933,\cdot)\) \(\chi_{4032}(2101,\cdot)\) \(\chi_{4032}(2437,\cdot)\) \(\chi_{4032}(2605,\cdot)\) \(\chi_{4032}(2941,\cdot)\) \(\chi_{4032}(3109,\cdot)\) \(\chi_{4032}(3445,\cdot)\) \(\chi_{4032}(3613,\cdot)\) \(\chi_{4032}(3949,\cdot)\)
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
\((127,3781,1793,577)\) → \((1,e\left(\frac{3}{16}\right),e\left(\frac{2}{3}\right),1)\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(11\) | \(13\) | \(17\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(37\) |
| \( \chi_{ 4032 }(2941, a) \) | \(1\) | \(1\) | \(e\left(\frac{25}{48}\right)\) | \(e\left(\frac{29}{48}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(i\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{23}{24}\right)\) | \(e\left(\frac{1}{24}\right)\) | \(e\left(\frac{35}{48}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{11}{16}\right)\) |
sage: chi.jacobi_sum(n)