from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1113, base_ring=CyclotomicField(26))
M = H._module
chi = DirichletCharacter(H, M([0,0,11]))
pari: [g,chi] = znchar(Mod(43,1113))
Basic properties
| Modulus: | \(1113\) | |
| Conductor: | \(53\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(26\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{53}(43,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 1113.be
\(\chi_{1113}(43,\cdot)\) \(\chi_{1113}(64,\cdot)\) \(\chi_{1113}(274,\cdot)\) \(\chi_{1113}(358,\cdot)\) \(\chi_{1113}(400,\cdot)\) \(\chi_{1113}(484,\cdot)\) \(\chi_{1113}(547,\cdot)\) \(\chi_{1113}(568,\cdot)\) \(\chi_{1113}(589,\cdot)\) \(\chi_{1113}(673,\cdot)\) \(\chi_{1113}(799,\cdot)\) \(\chi_{1113}(820,\cdot)\)
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: | \(\Q(\zeta_{13})\) |
| Fixed field: | Number field defined by a degree 26 polynomial |
Values on generators
\((743,955,1009)\) → \((1,1,e\left(\frac{11}{26}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(4\) | \(5\) | \(8\) | \(10\) | \(11\) | \(13\) | \(16\) | \(17\) | \(19\) |
| \( \chi_{ 1113 }(43, a) \) | \(1\) | \(1\) | \(e\left(\frac{11}{26}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{23}{26}\right)\) | \(e\left(\frac{7}{26}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{17}{26}\right)\) |
sage: chi.jacobi_sum(n)