from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6042, base_ring=CyclotomicField(26))
M = H._module
chi = DirichletCharacter(H, M([0,13,5]))
pari: [g,chi] = znchar(Mod(2773,6042))
Basic properties
| Modulus: | \(6042\) | |
| Conductor: | \(1007\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
| Order: | \(26\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
| Real: | no | |
| Primitive: | no, induced from \(\chi_{1007}(759,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
| Minimal: | yes | |
| Parity: | odd | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 6042.bl
\(\chi_{6042}(37,\cdot)\) \(\chi_{6042}(835,\cdot)\) \(\chi_{6042}(1177,\cdot)\) \(\chi_{6042}(1633,\cdot)\) \(\chi_{6042}(1861,\cdot)\) \(\chi_{6042}(2317,\cdot)\) \(\chi_{6042}(2659,\cdot)\) \(\chi_{6042}(2773,\cdot)\) \(\chi_{6042}(2887,\cdot)\) \(\chi_{6042}(3343,\cdot)\) \(\chi_{6042}(4141,\cdot)\) \(\chi_{6042}(5965,\cdot)\)
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
\((2015,4771,2281)\) → \((1,-1,e\left(\frac{5}{26}\right))\)
First values
| \(a\) | \(-1\) | \(1\) | \(5\) | \(7\) | \(11\) | \(13\) | \(17\) | \(23\) | \(25\) | \(29\) | \(31\) | \(35\) |
| \( \chi_{ 6042 }(2773, a) \) | \(-1\) | \(1\) | \(e\left(\frac{1}{26}\right)\) | \(e\left(\frac{9}{13}\right)\) | \(e\left(\frac{2}{13}\right)\) | \(e\left(\frac{3}{26}\right)\) | \(e\left(\frac{12}{13}\right)\) | \(-1\) | \(e\left(\frac{1}{13}\right)\) | \(e\left(\frac{9}{26}\right)\) | \(e\left(\frac{11}{13}\right)\) | \(e\left(\frac{19}{26}\right)\) |
sage: chi.jacobi_sum(n)