from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(712, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([0,0,5]))
pari: [g,chi] = znchar(Mod(665,712))
Basic properties
Modulus: | \(712\) | |
Conductor: | \(89\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{89}(42,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 712.ba
\(\chi_{712}(9,\cdot)\) \(\chi_{712}(17,\cdot)\) \(\chi_{712}(49,\cdot)\) \(\chi_{712}(129,\cdot)\) \(\chi_{712}(161,\cdot)\) \(\chi_{712}(169,\cdot)\) \(\chi_{712}(225,\cdot)\) \(\chi_{712}(249,\cdot)\) \(\chi_{712}(257,\cdot)\) \(\chi_{712}(361,\cdot)\) \(\chi_{712}(377,\cdot)\) \(\chi_{712}(409,\cdot)\) \(\chi_{712}(425,\cdot)\) \(\chi_{712}(465,\cdot)\) \(\chi_{712}(481,\cdot)\) \(\chi_{712}(513,\cdot)\) \(\chi_{712}(529,\cdot)\) \(\chi_{712}(633,\cdot)\) \(\chi_{712}(641,\cdot)\) \(\chi_{712}(665,\cdot)\)
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
\((535,357,537)\) → \((1,1,e\left(\frac{5}{44}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(21\) |
\( \chi_{ 712 }(665, a) \) | \(1\) | \(1\) | \(e\left(\frac{5}{44}\right)\) | \(e\left(\frac{21}{22}\right)\) | \(e\left(\frac{9}{44}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{6}{11}\right)\) | \(e\left(\frac{27}{44}\right)\) | \(e\left(\frac{3}{44}\right)\) | \(e\left(\frac{15}{22}\right)\) | \(e\left(\frac{43}{44}\right)\) | \(e\left(\frac{7}{22}\right)\) |
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)