from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(690, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([0,11,6]))
pari: [g,chi] = znchar(Mod(217,690))
Basic properties
Modulus: | \(690\) | |
Conductor: | \(115\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{115}(102,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 690.w
\(\chi_{690}(7,\cdot)\) \(\chi_{690}(37,\cdot)\) \(\chi_{690}(43,\cdot)\) \(\chi_{690}(67,\cdot)\) \(\chi_{690}(97,\cdot)\) \(\chi_{690}(103,\cdot)\) \(\chi_{690}(157,\cdot)\) \(\chi_{690}(217,\cdot)\) \(\chi_{690}(247,\cdot)\) \(\chi_{690}(283,\cdot)\) \(\chi_{690}(313,\cdot)\) \(\chi_{690}(337,\cdot)\) \(\chi_{690}(343,\cdot)\) \(\chi_{690}(373,\cdot)\) \(\chi_{690}(433,\cdot)\) \(\chi_{690}(457,\cdot)\) \(\chi_{690}(493,\cdot)\) \(\chi_{690}(517,\cdot)\) \(\chi_{690}(523,\cdot)\) \(\chi_{690}(613,\cdot)\)
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: | \(\Q(\zeta_{115})^+\) |
Values on generators
\((461,277,511)\) → \((1,i,e\left(\frac{3}{22}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(7\) | \(11\) | \(13\) | \(17\) | \(19\) | \(29\) | \(31\) | \(37\) | \(41\) | \(43\) |
\( \chi_{ 690 }(217, a) \) | \(1\) | \(1\) | \(e\left(\frac{37}{44}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{29}{44}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{9}{11}\right)\) | \(e\left(\frac{5}{44}\right)\) | \(e\left(\frac{7}{11}\right)\) | \(e\left(\frac{19}{44}\right)\) |
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)