from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(605, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([33,28]))
pari: [g,chi] = znchar(Mod(23,605))
Basic properties
Modulus: | \(605\) | |
Conductor: | \(605\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | yes | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 605.q
\(\chi_{605}(12,\cdot)\) \(\chi_{605}(23,\cdot)\) \(\chi_{605}(67,\cdot)\) \(\chi_{605}(78,\cdot)\) \(\chi_{605}(133,\cdot)\) \(\chi_{605}(177,\cdot)\) \(\chi_{605}(188,\cdot)\) \(\chi_{605}(232,\cdot)\) \(\chi_{605}(287,\cdot)\) \(\chi_{605}(298,\cdot)\) \(\chi_{605}(342,\cdot)\) \(\chi_{605}(353,\cdot)\) \(\chi_{605}(397,\cdot)\) \(\chi_{605}(408,\cdot)\) \(\chi_{605}(452,\cdot)\) \(\chi_{605}(463,\cdot)\) \(\chi_{605}(507,\cdot)\) \(\chi_{605}(518,\cdot)\) \(\chi_{605}(562,\cdot)\) \(\chi_{605}(573,\cdot)\)
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: | \(\Q(\zeta_{44})\) |
Fixed field: | 44.0.23846517021132934583872858710431976074170086834253431126547411945070717739767846278962679207324981689453125.1 |
Values on generators
\((122,486)\) → \((-i,e\left(\frac{7}{11}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(12\) | \(13\) | \(14\) |
\( \chi_{ 605 }(23, a) \) | \(-1\) | \(1\) | \(e\left(\frac{17}{44}\right)\) | \(i\) | \(e\left(\frac{17}{22}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{7}{44}\right)\) | \(-1\) | \(e\left(\frac{1}{44}\right)\) | \(e\left(\frac{23}{44}\right)\) | \(e\left(\frac{13}{22}\right)\) |
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)