from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(605, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([33,42]))
pari: [g,chi] = znchar(Mod(208,605))
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: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 605.r
\(\chi_{605}(32,\cdot)\) \(\chi_{605}(43,\cdot)\) \(\chi_{605}(87,\cdot)\) \(\chi_{605}(98,\cdot)\) \(\chi_{605}(142,\cdot)\) \(\chi_{605}(153,\cdot)\) \(\chi_{605}(197,\cdot)\) \(\chi_{605}(208,\cdot)\) \(\chi_{605}(252,\cdot)\) \(\chi_{605}(263,\cdot)\) \(\chi_{605}(307,\cdot)\) \(\chi_{605}(318,\cdot)\) \(\chi_{605}(373,\cdot)\) \(\chi_{605}(417,\cdot)\) \(\chi_{605}(428,\cdot)\) \(\chi_{605}(472,\cdot)\) \(\chi_{605}(527,\cdot)\) \(\chi_{605}(538,\cdot)\) \(\chi_{605}(582,\cdot)\) \(\chi_{605}(593,\cdot)\)
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: | 44.44.2885428559557085084648615903962269104974580506944665166312236845353556846511909399754484184086322784423828125.1 |
Values on generators
\((122,486)\) → \((-i,e\left(\frac{21}{22}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(7\) | \(8\) | \(9\) | \(12\) | \(13\) | \(14\) |
\( \chi_{ 605 }(208, a) \) | \(1\) | \(1\) | \(e\left(\frac{31}{44}\right)\) | \(i\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{19}{44}\right)\) | \(e\left(\frac{5}{44}\right)\) | \(-1\) | \(e\left(\frac{29}{44}\right)\) | \(e\left(\frac{29}{44}\right)\) | \(e\left(\frac{3}{22}\right)\) |
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)