from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(612, base_ring=CyclotomicField(48))
M = H._module
chi = DirichletCharacter(H, M([0,32,9]))
pari: [g,chi] = znchar(Mod(61,612))
Basic properties
| Modulus: | \(612\) | |
| Conductor: | \(153\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(48\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{153}(61,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 612.bn
\(\chi_{612}(61,\cdot)\) \(\chi_{612}(97,\cdot)\) \(\chi_{612}(133,\cdot)\) \(\chi_{612}(193,\cdot)\) \(\chi_{612}(241,\cdot)\) \(\chi_{612}(265,\cdot)\) \(\chi_{612}(277,\cdot)\) \(\chi_{612}(301,\cdot)\) \(\chi_{612}(313,\cdot)\) \(\chi_{612}(337,\cdot)\) \(\chi_{612}(385,\cdot)\) \(\chi_{612}(445,\cdot)\) \(\chi_{612}(481,\cdot)\) \(\chi_{612}(517,\cdot)\) \(\chi_{612}(589,\cdot)\) \(\chi_{612}(601,\cdot)\)
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
\((307,137,37)\) → \((1,e\left(\frac{2}{3}\right),e\left(\frac{3}{16}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(19\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 612 }(61, a) \) | \(-1\) | \(1\) | \(e\left(\frac{13}{48}\right)\) | \(e\left(\frac{35}{48}\right)\) | \(e\left(\frac{47}{48}\right)\) | \(e\left(\frac{1}{12}\right)\) | \(e\left(\frac{5}{8}\right)\) | \(e\left(\frac{7}{48}\right)\) | \(e\left(\frac{13}{24}\right)\) | \(e\left(\frac{5}{48}\right)\) | \(e\left(\frac{1}{48}\right)\) | \(1\) |
sage: chi.jacobi_sum(n)
Gauss sum
sage: chi.gauss_sum(a)
pari: znchargauss(g,chi,a)
Jacobi sum
sage: chi.jacobi_sum(n)
Kloosterman sum
sage: chi.kloosterman_sum(a,b)