from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1157, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([22,39]))
pari: [g,chi] = znchar(Mod(1004,1157))
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.bp
\(\chi_{1157}(22,\cdot)\) \(\chi_{1157}(81,\cdot)\) \(\chi_{1157}(87,\cdot)\) \(\chi_{1157}(100,\cdot)\) \(\chi_{1157}(133,\cdot)\) \(\chi_{1157}(139,\cdot)\) \(\chi_{1157}(146,\cdot)\) \(\chi_{1157}(263,\cdot)\) \(\chi_{1157}(289,\cdot)\) \(\chi_{1157}(354,\cdot)\) \(\chi_{1157}(367,\cdot)\) \(\chi_{1157}(406,\cdot)\) \(\chi_{1157}(607,\cdot)\) \(\chi_{1157}(737,\cdot)\) \(\chi_{1157}(874,\cdot)\) \(\chi_{1157}(971,\cdot)\) \(\chi_{1157}(1004,\cdot)\) \(\chi_{1157}(1023,\cdot)\) \(\chi_{1157}(1036,\cdot)\) \(\chi_{1157}(1153,\cdot)\)
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
\((535,92)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{13}{22}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
\( \chi_{ 1157 }(1004, a) \) | \(1\) | \(1\) | \(e\left(\frac{26}{33}\right)\) | \(e\left(\frac{61}{66}\right)\) | \(e\left(\frac{19}{33}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{47}{66}\right)\) | \(e\left(\frac{35}{66}\right)\) | \(e\left(\frac{4}{11}\right)\) | \(e\left(\frac{28}{33}\right)\) | \(e\left(\frac{5}{33}\right)\) | \(e\left(\frac{32}{33}\right)\) |
sage: chi.jacobi_sum(n)