from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1157, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([11,48]))
pari: [g,chi] = znchar(Mod(550,1157))
Basic properties
Modulus: | \(1157\) | |
Conductor: | \(1157\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(66\) | 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 1157.bo
\(\chi_{1157}(4,\cdot)\) \(\chi_{1157}(121,\cdot)\) \(\chi_{1157}(134,\cdot)\) \(\chi_{1157}(153,\cdot)\) \(\chi_{1157}(186,\cdot)\) \(\chi_{1157}(283,\cdot)\) \(\chi_{1157}(420,\cdot)\) \(\chi_{1157}(550,\cdot)\) \(\chi_{1157}(751,\cdot)\) \(\chi_{1157}(790,\cdot)\) \(\chi_{1157}(803,\cdot)\) \(\chi_{1157}(868,\cdot)\) \(\chi_{1157}(894,\cdot)\) \(\chi_{1157}(1011,\cdot)\) \(\chi_{1157}(1018,\cdot)\) \(\chi_{1157}(1024,\cdot)\) \(\chi_{1157}(1057,\cdot)\) \(\chi_{1157}(1070,\cdot)\) \(\chi_{1157}(1076,\cdot)\) \(\chi_{1157}(1135,\cdot)\)
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: | \(\Q(\zeta_{33})\) |
Fixed field: | Number field defined by a degree 66 polynomial |
Values on generators
\((535,92)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{8}{11}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
\( \chi_{ 1157 }(550, a) \) | \(1\) | \(1\) | \(e\left(\frac{53}{66}\right)\) | \(e\left(\frac{13}{33}\right)\) | \(e\left(\frac{20}{33}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{13}{66}\right)\) | \(e\left(\frac{49}{66}\right)\) | \(e\left(\frac{9}{22}\right)\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{7}{33}\right)\) | \(e\left(\frac{17}{66}\right)\) |
sage: chi.jacobi_sum(n)