from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1157, base_ring=CyclotomicField(66))
M = H._module
chi = DirichletCharacter(H, M([44,18]))
pari: [g,chi] = znchar(Mod(217,1157))
Basic properties
Modulus: | \(1157\) | |
Conductor: | \(1157\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(33\) | 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.bg
\(\chi_{1157}(16,\cdot)\) \(\chi_{1157}(217,\cdot)\) \(\chi_{1157}(256,\cdot)\) \(\chi_{1157}(269,\cdot)\) \(\chi_{1157}(334,\cdot)\) \(\chi_{1157}(360,\cdot)\) \(\chi_{1157}(477,\cdot)\) \(\chi_{1157}(484,\cdot)\) \(\chi_{1157}(490,\cdot)\) \(\chi_{1157}(523,\cdot)\) \(\chi_{1157}(536,\cdot)\) \(\chi_{1157}(542,\cdot)\) \(\chi_{1157}(601,\cdot)\) \(\chi_{1157}(627,\cdot)\) \(\chi_{1157}(744,\cdot)\) \(\chi_{1157}(757,\cdot)\) \(\chi_{1157}(776,\cdot)\) \(\chi_{1157}(809,\cdot)\) \(\chi_{1157}(906,\cdot)\) \(\chi_{1157}(1043,\cdot)\)
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 33 polynomial |
Values on generators
\((535,92)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{3}{11}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
\( \chi_{ 1157 }(217, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{33}\right)\) | \(e\left(\frac{31}{33}\right)\) | \(e\left(\frac{2}{33}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{32}{33}\right)\) | \(e\left(\frac{14}{33}\right)\) | \(e\left(\frac{1}{11}\right)\) | \(e\left(\frac{29}{33}\right)\) | \(e\left(\frac{4}{33}\right)\) | \(e\left(\frac{19}{33}\right)\) |
sage: chi.jacobi_sum(n)