from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(690, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([22,33,38]))
pari: [g,chi] = znchar(Mod(53,690))
Basic properties
Modulus: | \(690\) | |
Conductor: | \(345\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{345}(53,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 690.u
\(\chi_{690}(17,\cdot)\) \(\chi_{690}(53,\cdot)\) \(\chi_{690}(83,\cdot)\) \(\chi_{690}(107,\cdot)\) \(\chi_{690}(113,\cdot)\) \(\chi_{690}(143,\cdot)\) \(\chi_{690}(203,\cdot)\) \(\chi_{690}(227,\cdot)\) \(\chi_{690}(263,\cdot)\) \(\chi_{690}(287,\cdot)\) \(\chi_{690}(293,\cdot)\) \(\chi_{690}(383,\cdot)\) \(\chi_{690}(467,\cdot)\) \(\chi_{690}(497,\cdot)\) \(\chi_{690}(503,\cdot)\) \(\chi_{690}(527,\cdot)\) \(\chi_{690}(557,\cdot)\) \(\chi_{690}(563,\cdot)\) \(\chi_{690}(617,\cdot)\) \(\chi_{690}(677,\cdot)\)
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: | Number field defined by a degree 44 polynomial |
Values on generators
\((461,277,511)\) → \((-1,-i,e\left(\frac{19}{22}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
\( \chi_{ 690 }(53, a) \) | \(-1\) | \(1\) | \(e\left(\frac{7}{44}\right)\) | \(e\left(\frac{3}{11}\right)\) | \(e\left(\frac{15}{44}\right)\) | \(e\left(\frac{13}{44}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{2}{11}\right)\) | \(e\left(\frac{39}{44}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{25}{44}\right)\) |
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)