from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4028, base_ring=CyclotomicField(26))
M = H._module
chi = DirichletCharacter(H, M([0,0,10]))
pari: [g,chi] = znchar(Mod(77,4028))
Basic properties
| Modulus: | \(4028\) | |
| Conductor: | \(53\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(13\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{53}(24,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 4028.z
\(\chi_{4028}(77,\cdot)\) \(\chi_{4028}(153,\cdot)\) \(\chi_{4028}(381,\cdot)\) \(\chi_{4028}(685,\cdot)\) \(\chi_{4028}(837,\cdot)\) \(\chi_{4028}(1141,\cdot)\) \(\chi_{4028}(1369,\cdot)\) \(\chi_{4028}(1901,\cdot)\) \(\chi_{4028}(1977,\cdot)\) \(\chi_{4028}(3193,\cdot)\) \(\chi_{4028}(3269,\cdot)\) \(\chi_{4028}(3725,\cdot)\)
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 13 polynomial |
Values on generators
\((2015,2757,2281)\) → \((1,1,e\left(\frac{5}{13}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(3\) | \(5\) | \(7\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(21\) | \(23\) |
| \( \chi_{ 4028 }(77, a) \) | \(1\) | \(1\) | \(e\left(\frac{7}{13}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{5}{13}\right)\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{4}{13}\right)\) | \(e\left(\frac{3}{13}\right)\) | \(e\left(\frac{8}{13}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(1\) |
sage: chi.jacobi_sum(n)