from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1037, base_ring=CyclotomicField(16))
M = H._module
chi = DirichletCharacter(H, M([9,4]))
pari: [g,chi] = znchar(Mod(133,1037))
Basic properties
| Modulus: | \(1037\) | |
| Conductor: | \(1037\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(16\) | 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.be
\(\chi_{1037}(133,\cdot)\) \(\chi_{1037}(377,\cdot)\) \(\chi_{1037}(538,\cdot)\) \(\chi_{1037}(721,\cdot)\) \(\chi_{1037}(743,\cdot)\) \(\chi_{1037}(804,\cdot)\) \(\chi_{1037}(843,\cdot)\) \(\chi_{1037}(1026,\cdot)\)
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: | \(\Q(\zeta_{16})\) |
| Fixed field: | 16.16.7597869690780846180466826362661244163553.1 |
Values on generators
\((428,307)\) → \((e\left(\frac{9}{16}\right),i)\)
First values
| \(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(5\) | \(6\) | \(7\) | \(8\) | \(9\) | \(10\) | \(11\) |
| \( \chi_{ 1037 }(133, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{1}{16}\right)\) | \(i\) | \(e\left(\frac{5}{16}\right)\) | \(e\left(\frac{3}{16}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{3}{8}\right)\) | \(e\left(\frac{1}{8}\right)\) | \(e\left(\frac{7}{16}\right)\) | \(e\left(\frac{11}{16}\right)\) |
sage: chi.jacobi_sum(n)