from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1037, base_ring=CyclotomicField(60))
M = H._module
chi = DirichletCharacter(H, M([45,34]))
pari: [g,chi] = znchar(Mod(106,1037))
Basic properties
| Modulus: | \(1037\) | |
| Conductor: | \(1037\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(60\) | 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 1037.ch
\(\chi_{1037}(4,\cdot)\) \(\chi_{1037}(106,\cdot)\) \(\chi_{1037}(293,\cdot)\) \(\chi_{1037}(310,\cdot)\) \(\chi_{1037}(344,\cdot)\) \(\chi_{1037}(370,\cdot)\) \(\chi_{1037}(412,\cdot)\) \(\chi_{1037}(446,\cdot)\) \(\chi_{1037}(463,\cdot)\) \(\chi_{1037}(472,\cdot)\) \(\chi_{1037}(659,\cdot)\) \(\chi_{1037}(676,\cdot)\) \(\chi_{1037}(710,\cdot)\) \(\chi_{1037}(778,\cdot)\) \(\chi_{1037}(812,\cdot)\) \(\chi_{1037}(829,\cdot)\)
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: | \(\Q(\zeta_{60})\) |
| Fixed field: | Number field defined by a degree 60 polynomial |
Values on generators
\((428,307)\) → \((-i,e\left(\frac{17}{30}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 1037 }(106, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{15}\right)\) | \(e\left(\frac{3}{20}\right)\) | \(e\left(\frac{2}{15}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{13}{60}\right)\) | \(e\left(\frac{1}{60}\right)\) | \(e\left(\frac{1}{5}\right)\) | \(e\left(\frac{3}{10}\right)\) | \(e\left(\frac{17}{60}\right)\) | \(-i\) |
sage: chi.jacobi_sum(n)