from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(740, base_ring=CyclotomicField(36))
M = H._module
chi = DirichletCharacter(H, M([18,9,23]))
pari: [g,chi] = znchar(Mod(227,740))
Basic properties
Modulus: | \(740\) | |
Conductor: | \(740\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(36\) | 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 740.cb
\(\chi_{740}(87,\cdot)\) \(\chi_{740}(143,\cdot)\) \(\chi_{740}(163,\cdot)\) \(\chi_{740}(167,\cdot)\) \(\chi_{740}(203,\cdot)\) \(\chi_{740}(207,\cdot)\) \(\chi_{740}(227,\cdot)\) \(\chi_{740}(283,\cdot)\) \(\chi_{740}(387,\cdot)\) \(\chi_{740}(483,\cdot)\) \(\chi_{740}(627,\cdot)\) \(\chi_{740}(723,\cdot)\)
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: | \(\Q(\zeta_{36})\) |
Fixed field: | 36.0.3947574212825584447047556613608375798697755145862339017216000000000000000000000000000.2 |
Values on generators
\((371,297,261)\) → \((-1,i,e\left(\frac{23}{36}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(3\) | \(7\) | \(9\) | \(11\) | \(13\) | \(17\) | \(19\) | \(21\) | \(23\) | \(27\) |
\( \chi_{ 740 }(227, a) \) | \(-1\) | \(1\) | \(e\left(\frac{31}{36}\right)\) | \(e\left(\frac{7}{36}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{2}{3}\right)\) | \(e\left(\frac{7}{9}\right)\) | \(e\left(\frac{13}{18}\right)\) | \(e\left(\frac{13}{36}\right)\) | \(e\left(\frac{1}{18}\right)\) | \(e\left(\frac{5}{6}\right)\) | \(e\left(\frac{7}{12}\right)\) |
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)